Representations of Reductive p-adic Groups: International Conference, IISER, Pune, India, 2017 [1st ed.] 978-981-13-6627-7;978-981-13-6628-4

Table of contents :
Front Matter ....Pages i-xiii
Local Langlands and Springer Correspondences (Anne-Marie Aubert)....Pages 1-37
Arithmetic of Cuspidal Representations (Colin J. Bushnell)....Pages 39-126
Root Data with Group Actions (Jeffrey D. Adler, Joshua M. Lansky)....Pages 127-143
The Character of a Simple Supercuspidal Representation of SL(2, F) (Moshe Adrian)....Pages 145-159
Classification of Strongly Positive Representations of Even General Unitary Groups (Yeansu Kim, Ivan Matić)....Pages 161-174
On the Unicity of Types for Toral Supercuspidal Representations (Peter Latham, Monica Nevins)....Pages 175-190
Local Gamma Factors, Converse Theorems and Related Problems (Chufeng Nien)....Pages 191-205
On Completions of Hecke Algebras (Maarten Solleveld)....Pages 207-262
On Relatively Tempered Representations for p-adic Symmetric Spaces (Shuichiro Takeda)....Pages 263-289

Citation preview

Progress in Mathematics 328

Anne-Marie Aubert Manish Mishra Alan Roche Steven Spallone Editors

Representations of Reductive p -adic Groups International Conference, IISER, Pune, India, 2017

Progress in Mathematics Volume 328

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Imperial College London, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Anne-Marie Aubert Manish Mishra Alan Roche Steven Spallone •

•

•

Editors

Representations of Reductive p-adic Groups International Conference, IISER, Pune, India, 2017

Editors Anne-Marie Aubert Institut de Mathématiques de Jussieu Paris, France Alan Roche Department of Mathematics University of Oklahoma Norman, OK, USA

Manish Mishra Department of Mathematics Indian Institute of Science Education and Research Pune Pune, Maharashtra, India Steven Spallone Department of Mathematics Indian Institute of Science Education and Research Pune Pune, Maharashtra, India

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-981-13-6627-7 ISBN 978-981-13-6628-4 (eBook) https://doi.org/10.1007/978-981-13-6628-4 Library of Congress Control Number: 2019932681 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This volume is based around the workshop and conference Representations of p-adic groups held at IISER Pune from July 10 to 19, 2017. The first two chapters give expanded accounts of two of the workshop minicourses. Each is concerned with aspects of the local Langlands conjecture or correspondence (LLC). This predicts a close relation between irreducible representations of reductive p-adic groups and suitable relatives of Galois representations, now known as Langlands parameters. Its influence pervades our subject and lies at the heart of applications to automorphic forms. The chapter “Local Langlands and Springer Correspondences,” by Anne-Marie Aubert, builds toward a precise formulation of the conjecture in a general setting, sketching in the considerable background that this requires. At the same time, it provides a valuable window on the author’s work with Ahmed Moussaoui and Maarten Solleveld on links between the LLC and (generalized) Springer correspondences. It also explores a connection with the world of non-commutative geometry via the Aubert-Baum-Plymen-Solleveld conjecture. Among interrelated results and conjectures, the chapter includes, in particular, a conjectural description of the manner in which irreducible cuspidal (or supercuspidal in Harish-Chandra’s terminology) representations fit into and generate the LLC. The chapter “Arithmetic of Cuspidal Representations,” by Colin Bushnell, looks at the Langlands correspondence for the family of groups GLn ðFÞ for F a non-Archimedean local field. In this case, via the Zelevinsky classification and properties of local constants, the correspondence can be viewed as a canonical bijection between equivalence classes of irreducible cuspidal representations of GLn ðFÞ and equivalence classes of irreducible n-dimensional representations of the Weil group of F. Its existence has been known for over 20 years. Many mysteries, however, remain. For example, can one describe the correspondence explicitly, or at least in some detail, in terms of the classification via types of the irreducible cuspidal representations of GLn ðFÞ due to Bushnell and Kutzko? Indeed, the linchpin of that classification—the notion of simple character—is unmistakably arithmetic in nature. It is natural then to attempt to understand the correspondence along such lines. Bushnell and Henniart, in a long series of papers, have made v

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Preface

considerable progress in this direction. Bushnell’s write-up provides a characteristically cogent account of their work. With one exception, the remaining chapters present original papers. The exception is a pleasant survey by Chufeng Nien on local converse theorems. We give below a brief description of the original papers, ignoring restrictions on characteristic or residual characteristic. Please see the introductions to the papers for a more careful and detailed overview or—best of all—read the papers themselves! – The contribution by Jeff Adler and Josh Lansky discusses group actions on root data. It fits into the authors’ continuing study of liftings of representations of reductive groups over finite and local non-Archimedean fields. – Moshe Adrian’s paper gives a complete character formula, valid in all residual characteristics, for a simple supercuspidal representation of SL2 ðFÞ. – Yeansu Kim and Ivan Matić classify strongly positive discrete series representations of even unitary and general unitary groups by exploiting and extending a formalism due to Tadić. – Peter Latham and Monica Nevins look at questions concerning uniqueness of types for a broad class of supercuspidal representations of a semisimple and simply connected p-adic group. – Maarten Solleveld looks at completions of Hecke algebras that arise naturally in studying the tempered representations of a reductive p-adic group. As an application, he computes the topological K-theory of the reduced C  -algebra of many classical p-adic groups. – Shuichiro Takeda’s paper is concerned with p-adic symmetric spaces. It establishes an analogue in this setting of the classical result that irreducible tempered representations embed as direct summands of modules that are induced from discrete series representations of Levi subgroups. We are enormously grateful to the authors for their labors. We would like to acknowledge also the work of the various referees who read carefully through the submissions and offered many detailed and constructive comments. Funding for the workshop and conference was provided by the National Board for Higher Mathematics (NBHM) in India and by IISER Pune. We are thankful for their support. Finally, a special word of thanks to the participants for many stimulating conversations and discussions, mathematical and otherwise, over a memorable week and a half in Pune. Paris, France Pune, India Norman, USA Pune, India

Anne-Marie Aubert Manish Mishra Alan Roche Steven Spallone

Participants and Speakers

 Jeff Adler, American University, USA  Moshe Adrian, CUNY, USA  U. K. Anandavardhanan, IIT Mumbai, India Michael Arnold, University of East Anglia, UK  Anne-Marie Aubert, IMJ-PRG, France Mugur (Peter) Badea, Radboud Universiteit Nijmegen, The Netherlands Kumar Balasubramanian, IISER Bhopal, India Subham Bhakta, Chennai Mathematical Institute, India Gautam Borisagar, IITRAM, India  Colin Bushnell, King’s College London, UK Kolhatkar Sampada Chandrashekhar, IIS Bangalore, India Jesua Epequin Chavez, IMJ-PRG, France Jonathan Cohen, University of Maryland, USA  Tyrone Crisp, Radboud Universiteit Nijmegen, The Netherlands Peiyi Cui, Université Pierre et Marie Curie and Université de Rennes 1, France Chandan S. Dalawat, HRI Allahabad, India Bastian Drevon, Université de Versailles Saint-Quentin-en-Yvelines, France Melissa Emory, University of Missouri, USA Mikolaj Fraczyck, Université Paris-Sud, France Venketasubramanian C. G., IISER Tirupati, India Jessica Fintzen, University of Michigan, USA Radhika Ganapathy, TIFR, Mumbai, India Zahi Hazan, Tel Aviv University, Israel Lu Hengfei, TIFR, Mumbai, India Trias Justin, IMJ-PRG, France  Yeansu Kim, Chonnam National University, South Korea Neha Kwatra, IISER Mohali, India Luis Lomeli, Instituto de Matemáticas PUCV, Chile Sampath Lonka, University of Hyderabad, India  Jia-jun Ma, Shanghai Jiao Tong University, China Arnaud Mayeux, IMJ-PRG, France vii

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Participants and Speakers

Amiya Mondal, TIFR, Mumbai, India  Ahmed Moussaoui, Université de Versailles Saint-Quentin-en-Yvelines, France Santosh Nadimpalli, TIFR, Mumbai, India  Monica Nevins, University of Ottawa, Canada  Chufeng Nien, National Cheng Kung University, Taiwan  Eric Opdam, University of Amsterdam, The Netherlands  Dipendra Prasad, TIFR, Mumbai, India  Alan Roche, University of Oklahoma, USA Beth Romano, University of Cambridge, UK Lauren C. Ruth, University of California, Riverside, USA Claudia Schoemann, Mathematical Institute, University of Goettingen, Germany  Vincent Sécherre, Université de Versailles Saint-Quentin-en-Yvelines, France  Maarten Solleveld, Radboud Universiteit Nijmegen, The Netherlands Anna Szumowicz, Université Pierre et Marie Curie, France, and Durham University, UK  Shuichiro Takeda, University of Missouri, USA Lanard Thomas, Université Pierre et Marie Curie, France Sandeep Varma, TIFR, Mumbai, India Tian An Wong, IISER Pune, India Hongjie Yu, Université Paris 7, France  = speaker

Contents

Local Langlands and Springer Correspondences . . . . . . . . . . . . . . . . . Anne-Marie Aubert

1

Arithmetic of Cuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . . Colin J. Bushnell

39

Root Data with Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jeffrey D. Adler and Joshua M. Lansky

127

The Character of a Simple Supercuspidal Representation of SL(2, F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moshe Adrian

145

Classification of Strongly Positive Representations of Even General Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yeansu Kim and Ivan Matić

161

On the Unicity of Types for Toral Supercuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Latham and Monica Nevins

175

Local Gamma Factors, Converse Theorems and Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chufeng Nien

191

On Completions of Hecke Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maarten Solleveld On Relatively Tempered Representations for p-adic Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuichiro Takeda

207

263

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About the Editors

Anne-Marie Aubert is Research Director in Mathematics at the Centre National de la Recherche Scientifique (CNRS) at the Institut Mathématiques de Jussieu Paris Rive-Gauche, Paris, France. She is a member of the Comité National de la Recherche Scientifique, and has served on the editorial board of the Bulletin and the Memoirs of the French Mathematical Society. Manish Mishra is Assistant Professor of Mathematics at the Indian Institute of Science Education and Research (IISER), Pune, India. He previously held postdoctoral positions at the Heidelberg University and the Hebrew University of Jerusalem. He completed his BTech at the Indian Institute of Technology Kanpur and his Ph.D. at Purdue University, USA. Alan Roche is Associate Professor of Mathematics at the University of Oklahoma, USA. He previously held visiting positions at Purdue University and Oklahoma State University, USA. Steven Spallone is Associate Professor of Mathematics at the Indian Institute of Science Education and Research (IISER), Pune, India. He graduated from the University of Pennsylvania and completed his Ph.D. at the University of Chicago, USA, in 1998 and 2004, respectively. His research interests include number theory and representation theory.

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Local Langlands and Springer Correspondences Anne-Marie Aubert

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Langlands Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Weil Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The L-Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Definitions for Langlands Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 L-Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Enhanced Langlands Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Group Sφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Definition of Enhanced Langlands Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Desiderata for the Local Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . 3.4 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Generalized Springer Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Cuspidal Enhanced Unipotent Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Partition of the Enhancement of the Unipotent Variety of G ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Twisted Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 A Partition of the Enhancement of the Unipotent Variety of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cuspidality for Enhanced L-Parameters: Definition and Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A Partition of the Set of Enhanced Langlands Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Inertial Classes for Enhanced Langlands Parameters . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Cuspidal Support Map for Enhanced Langlands Parameters . . . . . . . . . . . . . 6.3 A Generalized Springer Correspondence for Enhanced L-Parameters . . . . . . . . . 7 A Galois Version of the ABPS Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Twisted Extended Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Bernstein Decomposition of the Category of Smooth Representations . . . . . 7.3 The ABPS Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A Version for Enhanced L-Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 A Conjectural Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 4 6 8 10 11 11 13 14 17 19 19 22 23 23 24 25 25 27 28 29 29 30 32 33 34 35

It is a pleasure to acknowledge the excellent comments and questions of the workshop participants. A.-M. Aubert (B) Institut de Mathématiques de Jussieu—Paris Rive Gauche, IMJ-PRG, C.N.R.S, Sorbonne Université, Université Paris Diderot, 75005 Paris, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_1

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Abstract These notes give an overview of results obtained jointly with Ahmed Moussaoui and Maarten Solleveld on the local Langlands correspondence, focusing on the links of the latter with both the generalized Springer correspondence and the geometric conjecture, the so-called ABPS Conjecture, introduced in collaboration with Paul Baum, Roger Plymen and Maarten Solleveld. 2010 Mathematics Subject Classification 20C08 · 14F43 · 20G20

1 Introduction The local Langlands correspondence predicts a relation between two rather different kinds of objects: on the one side irreducible representations of reductive groups over a local field F, and on the other side certain analogs of Galois representations, called Langlands parameters. Let G be the group of F-rational points of a connected reductive algebraic group G over a non-Archimedean local field F. Slightly more precisely, the local Langlands conjecture for G asserts the existence a finite-to-one surjection, subject to certain properties, recG : Irr(G) → (G), from the set Irr(G) of (isomorphism classes of) irreducible smooth representations of G to the set (G) of equivalence classes (for a certain equivalence relation) of Langlands parameters for G. The (conjectural) fibers of recG are called L-packets for G. The irreducible representations π inside a given L-packet are said to be L-indistinguishable. To distinguish them, the idea (see Lusztig [44], Vogan [61], Arthur [2]) is to “enhance” φ by an additional datum ρ, which is an irreducible representation of a certain finite group Sφ [defined by Eq. (2)]. The pair (φ, ρ) will be called an enhanced Langlands parameter for G. Let e (G  ) be the collection of equivalence classes of such pairs (φ, ρ) with ρ ∈ Irr(Sφ ). Then the LLC for G should be an injective map Irr(G) → e (G),

(1)

which satisfies several natural properties. The map will almost never be surjective, but for every φ which is relevant for G the image should contain at least one pair (φ, ρ). A remarkable aspect of the Langlands conjecture [61] is that it is better to consider not just one reductive group G at a time, but all inner forms (more precisely all inner twists) G ϑ of G simultaneously. Inner twists share the same Langlands dual group. The hope is that one can turn (1) into a bijection by defining a suitable equivalence relation on the set of inner twists and taking the corresponding union

Local Langlands and Springer Correspondences

3

of the sets Irr(G ϑ ) on the left-hand side. Such a statement was proven for unipotent representations (also known as representations with unipotent reduction) of simple p-adic groups in [48]. The main goal of these notes is to provide an overview of the main results obtained jointly with Moussaoui and Solleveld in [15] on the local Langlands correspondence, focusing on the links of the latter with both the generalized Springer correspondence and the geometric conjecture, the so-called ABPS Conjecture, introduced in collaboration with Paul Baum, Roger Plymen and Maarten Solleveld and studied in several articles, notably [12]. We also give an account of the known results regarding the preservation (and non-preservation) of the depth by the LLC. For maximal generality, we adhere to the setup for L-parameters used by Arthur in [2]. Let W F be the Weil group of F and let L G = G ∨  W F be the L-group of G. We set W F := W F × SL2 (C). Let G ∨ad be the adjoint group of G ∨ and G ∨sc be the simply connected cover of the derived group of G ∨ . Let φ : W F → L G be an L-parameter. We denote by ZG ∨ad (φ) the centralizer in ∨ 1 ∨ ∨ G ad of φ(W F ) and we define ZG ∨ (φ) to be its inverse image under G sc → G ad . Let sc 1 Sφ := π0 (ZG ∨ (φ)), sc

(2)

1 denote the component group of ZG ∨ (φ). It coincides with the group considered by sc both Arthur in [2] and Kaletha in [39, § 4.6]. We set 1 (3) Gφ := ZG ∨ (φ| W F ). sc

A remarkable fact is that the finite group Sφ is isomorphic to the component group AGφ (u φ ) := π0 (ZGφ (u φ )),

(4)

  of the centralizer in Gφ of unipotent element u φ := φ(1, 01 11 ). It provides a way to plug the generalized Springer correspondences for the groups Gφ , where φ runs over (G), in the study of the local Langlands correspondence for G. In particular, it allows to transfer the Lusztig notion of cuspidality for representations of the groups AGφ (u φ ) into a notion of cuspidality for enhanced L-parameters for G: An enhanced L-parameter (φ, ρ) for G is called cuspidal if u φ and ρ, considered as data for the complex reductive group Gφ , form a cuspidal pair. By definition, this means that the restriction of ρ from AGφ (u φ ) to AGφ◦ (u φ ) is a direct sum of cuspidal representations in Lusztig’s sense [45]. Intuitively, it says that ρ or ρ| AG◦ (u φ ) cannot φ be obtained (via an appropriate notion of parabolic induction) from any pair (u  , ρ  ) that can arise from a proper Levi subgroup of Gφ◦ . It is essential to use L-parameters enhanced with a representation of a suitable component group, for cuspidality cannot be detected from the L-parameter alone. In particular, we conjecture in [15, § 6] (generalizing to G arbitrary the conjecture stated by Moussaoui when G is F-split: [52, Conjecture 1.2]) that the cuspidal en-

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hanced Langlands parameters correspond by the LLC to the irreducible supercuspidal representations of G. The validity of this conjecture is proved for representations with unipotent reduction of the group G of the F-rational points of any connected reductive algebraic group which splits over an unramified extension of F in [26, Theorem 2] (when G is simple of adjoint type, it is a special case of [47, 48]), for the Deligne-Lusztig depthzero supercuspidal representations (as a consequence of [25]), and also for general linear groups and split classical p-adic groups (any representation) (see [52]), for inner forms of linear groups and of special linear groups, and for quasi-split unitary p-adic groups (any representation) (see [15, § 6]). We thank Maarten Solleveld and the referee for helpful and constructive comments that greatly contributed to improving the final version of the paper.

2 Langlands Parameters 2.1 The Weil Group Let F be a local non-Archimedean field with finite residual field k F = Fq . Let Fsep be a fixed separable closure of F, and let  F denote the Galois group of Fsep /F. The field F admits a unique unramified extension Fm /F of degree m and contained in Fsep , for each integer m ≥ 1. The composite of all the fields Fm is the unique maximal unramified extension of F contained in Fsep and will be denoted by Fur . The extension Fur will allow us to decompose the study of  F in two steps of a different nature by considering separately the group Gal(Fur /F) and the group I F := Gal(Fsep /Fur ), called the inertia group of F: We have an exact sequence of topological groups 1 → I F →  F → Gal(Fur /F) → 0.

(5)

The extension Fm /F is Galois and the group Gal(Fm /F) is cyclic. An F-automorphism of Fm is determined by its action on the residual field k Fm  Fq m of Fm , and there is a unique element σm of Gal(Fm /F) which acts on k Fm by λ → λq . We set Fr m := σm−1 . The map Fr m → 1 + mZ gives a canonical isomorphism from Gal(Fm /F) onto Z/mZ. Taking the inverse limit over m, we get a canonical isomorphism of topological groups between Gal(Fur /F) and  Z := lim Z/mZ, ← − m≥1

and a unique element Fr F ∈ Gal(Fur /F) which acts on Fm as Fr m , for all m. The element Fr F is called the geometric Frobenius substitution on Fur (its inverse σ F is the arithmetic Frobenius substitution). An element of  F is called a geometric Frobenius element (over F) if its image in Gal(Fur /F) is Fr F . The Chinese Remainder Theorem

Local Langlands and Springer Correspondences

5

 gives a canonical isomorphism of topological groups  Z   Z , where  range over all prime numbers, and Z is the (additive) group of -adic integers. We recall some properties of the ramification groups (with respect to the upper numbering) of  F , as defined in [57, Remark IV.3.1]: 0 •  −1 F :=  F and  F := I F , the inertia group. • For every r ∈ R≥0 , lF is the compact subgroup of I F that consists of all γ ∈  F which, for every finite Galois extension E of F contained in Fsep , act trivially on the ring o E /pi(r,E) (where i(r, E) ∈ Z≥0 can be found with [57, § IV.3]). E • r ∈ R≥0 is called a jump of the filtration if

rF+ :=



 tF

t>r

does not equal rF . The set of jumps of the filtration is countably infinite and need not consist of integers. In order to formulate the Langlands correspondence, we need to introduce the Weil group W F , a subgroup of  F . Let a W F denote the inverse image in  F of the cyclic subgroup of Gal(Fur /F) generated by Fr F . Then a W F is the dense subgroup of  F generated by the geometric Frobenius elements. It is normal in  F and fits into an exact sequence (of abstract groups) 1 → I F → a W F → Z → 0.

(6)

The Weil group W F of F (relative to Fsep ) is the topological group, with underlying abstract group a W F , so that I F is an open subgroup of W F , and the topology on I F , as subspace of W F , coincides with its natural topology as Gal(Fsep /Fur ) ⊂  F . Thus W F is locally profinite, and the inclusion ι F : W F ⊂  F is continuous. The definition of W F does depend on the choice of Fsep /F, but only up to inner automorphism of  F . We will need some basic aspects of the representation theory of W F . The group  F being profinite, its smooth representations are semisimple. On the contrary, W F has smooth representations which are not semisimple. The irreducible representations of W F are quite closely related to those of  F (in particular, they have finite dimension): (1) if τ is an irreducible representation of  F , then τ ◦ ι F is an irreducible smooth representation of W F , (2) for any irreducible smooth representation σ of W F , there is an unramified character χ of W F (i.e., ker(χ ) contains I F ) such that χ ⊗ σ  τ ◦ ι F , for some irreducible smooth representations τ of  F .

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2.2 The L-Group Let G be the group of F-points of a connected reductive algebraic group G defined over F. The group G is split over Fsep and is classified by its root data. Let T be a maximal torus of G, defined and split over Fsep , and let X (T) := Hom F (T, Gm ) be the group of Fsep -rational characters of T. We set X ∨ (T) := Hom F (Gm , T). Let R(G, T) and R ∨ (G, T) denote the sets of roots and coroots of G with respect to T, respectively. The corresponding root datum for G over Fsep is R(G) := (X (T), R(G, T), X ∨ (T), R(G, T)∨ ). Definition 2.1 The Langlands dual group of G is the the reductive connected algebraic group G∨ , defined over C, whose roots (resp. coroots) are the coroots (resp. roots) of G, so that a root datum for G∨ is R(G∨ ) := (X ∨ (T), R ∨ (G, T), X (T), R(G, T)). We denote by G ∨ the C-points of the group G∨ . Examples 2.2 We have • • • •

GLn (F)∨ = GLn (C), SLn (F)∨ = PGLn (C), and PGLn (F)∨ = SLn (C); Sp2n (F)∨ = SO2n+1 (C), SO2n+1 (F)∨ = Sp2n (C); SO2n (F)∨ = SO2n (C); if G a group of exceptional type, then G and G∨ are of the same type (e.g. G2 (F)∨ = G2 (C)).

Of special importance is the group G = GLn . Indeed, any complex reductive group G may be embedded into GLn (C) for some n. We choose a Borel subgroup B ⊃ T of G defined over Fsep , and let  and ∨ denote the corresponding basis of R(G, T) and R ∨ (G, T). A based root datum for G is R0 (G) := (X (T), , X ∨ (T), ∨ ). We choose a pinning (G, B, T, {xα }α ∈ ), which induces a splitting of the exact sequence 1 −→ Int(G) −→ Aut(G) −→ Aut(R0 (G)) −→ 1. The F structure of G is given by a morphism of θ :  F −→ Aut(G) which descends to θ :  F −→ Aut(R0 (G)). To obtain the L-group as a group that actually sees G = G(F) and not just G, we have to work in the Galois action of  F on the group G∨ . Since Aut(R0 (G))  Aut(R0 (G)∨ )  Aut(R0 (G∨ )), we have θ :  F −→ Aut(R0 (G)) −→ Aut(G∨ ).

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Definition 2.3 The (Weil form of the) L-group of G is the group L

G := G ∨ θ W F .

Remark 2.4 In the case when G is F-split, we have a direct product, that is: L G = G∨ × WF . Definition 2.5 Let G and H be two connected reductive algebraic groups over F. A homomorphism η : L H → L G is called an L-homomorphism if • η is continuous; • the diagram

L

H

η

L

G commutes;

WF • the restriction of η to H ∨ is morphism of algebraic groups from H ∨ to G ∨ . General connected reductive F-groups need not be quasi-split, but they are always forms of split F-groups. Definition 2.6 Two F-groups G = G(F) and H = H(F) are called forms of each other if G is isomorphic to H as algebraic groups, or equivalently if G(Fsep ) ∼ = H(Fsep ) as Fsep -groups. An isomorphism ϑ : H → G determines a 1-cocycle γϑ :

 F → Aut(G) σ → ϑσ ϑ −1 σ −1 .

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From γϑ , one can recover H (up to isomorphism) as H∼ = {g ∈ G(Fsep ) : (γϑ (σ ) ◦ σ )g = g ∀σ ∈ Gal(Fsep /F)}. Given another form ϑ  : H → G, the groups H and H  are F-isomorphic if and only if the 1-cocycles γϑ and γϑ  are cohomologous, that is, if there exists a f ∈ Aut(G) such that (8) γϑ (σ ) = f −1 γϑ  (σ ) σ f σ −1 ∀σ ∈ Gal(Fsep /F). In this way, the isomorphism classes of forms of G are in bijection with the Galois cohomology group H 1 (F, Aut(G)). Definition 2.7 An F-group H is an inner form of G if the cocycle γϑ takes values in the group of inner automorphisms Inn(G). On the other hand, if the values of γϑ are not contained in Inn(G), then H is called an outer form of G. Proposition 2.8 [59, § 16.4] Every connected reductive F-group G is an inner form of a unique quasi-split F-group G ∗ .

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Example 2.9 Let D be a division algebra with center F, of dimension d 2 over F. Then G = GLm (D) is an inner form of GLn (F) = G ∗ where n = dm. There is a reduced norm map Nrd : GLm (D) → F × and the derived group SLm (D) := ker(Nrd : GLm (D) → F × ) is an inner form of SLn (F). Every inner form of GLn (F) or SLn (F) is isomorphic to one of this kind. When n = 2, the only possibilities for d are 1 or 2, and so the inner forms are, up to isomorphism, GL2 (F) and D × , and SL2 (F) and SL1 (D). Definition 2.10 An inner twist of G consists of a pair (H, ϑ) as above, where H = ∼ H(F) and ϑ : H −→ G are such that im(γϑ ) ⊂ Gad . Two inner twists of G are equivalent if (8) holds for some f ∈ Inn(G). Let IT(G) denote the set of equivalence classes of inner twists of G. Remark 2.11 It is quite possible that two inequivalent inner twists (H, ϑ) and (H  , ϑ  ) share the same group H ∼ = H  . This happens precisely when γϑ and γϑ are in the same orbit of Aut(G)/Inn(G) on H 1 (F, Gad ). Kottwitz has found an important alternative description of H 1 (F, G). Recall that the complex dual group G ∨ = G∨ (C) is endowed with an action of Gal(Fsep /F). There exists a natural isomorphism    ∼ κG : H 1 (F, G) −→ Irr π0 Z(G ∨ )W F , see [41, Proposition 6.4]. This is particularly useful in the following way. An inner twist of G is the same thing as an inner twist of the unique quasi-split inner form ∗ ∨ (F) be the adjoint group of G ∗ . Let Gsc be the simply G ∗ = G∗ (F). Let G ∗ad = Gad ∨ ∨ ∨ connected cover of the derived group G der of G . We have G sc = (G ad )∨ , and   ∼ ∗ ) −→ Irr Z(G ∨sc )W F . κG ∗ad : H 1 (F, Gad

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The equivalence classes of inner twists of G are parametrized by the Galois ∗ ∗ ). For every ϑ ∈ H 1 (F, Gad ), we will denote by G ϑ cohomology group H 1 (F, Gad ∗ an inner twist of G which is parametrized by ϑ, and we will take G ∗ to be G 1 . All the inner twists of a given group G share the same L-group, because the action of W F on G ∨ is only uniquely defined up to inner automorphisms. This also works the other way round: From the Langlands dual group L G, one can recover the inner form class of G. Hence, it is natural to consider all the inner twists of a given group simultaneously.

2.3 Definitions for Langlands Parameters We write W F := W F × SL2 (C) (as version of the Weil-Deligne group of F).

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9

Definition 2.12 A Langlands parameter (or L-parameter, for short) for L G is a smooth morphism φ : W F → L G such that W F

φ

L

G commutes,

WF φ(w) is semisimple for each w ∈ W F (i.e., r (φ(w)) is semisimple for every finite dimensional representation r of L G), and φ|SL2 (C) is a morphism of complex algebraic groups. The group G ∨ acts on the set ( L G) of such φ’s by conjugation. Let ( L G) denote the set of G ∨ -orbits in ( L G) Definition 2.13 An L-parameter φ : W F → L G is said to be • • • •

unramified if it has trivial restriction to I F ; tame if it has trivial restriction to the wild ramification group PF of F; essentially tame if φ(PF ) lies in a maximal torus of G ∨ ; discrete (or elliptic) if there is no proper W F -stable Levi subgroup L ∨ ⊂ G ∨ such that φ(W F ) ⊂ L L. • bounded if φ  (W F ) ⊂ G ∨ is bounded, where φ(w) = (φ  (w), w). (This is equivalent to φ  (Frob) being a compact element of G ∨ .

2.3.1

Relevance for Langlands Parameters

We will call a Levi factor of a parabolic subgroup of G simply a Levi subgroup of G. The bijection R(G, T) ←→ R ∨ (G, T) = R(G ∨ , T ∨ ) gives a basis ∨ and provides a canonical bijection between the sets of conjugacy classes of parabolic subgroups of G and of G ∨ . Definition 2.14 [21, § 3] A parabolic subgroup P ∨ of G ∨ is F-relevant if the corresponding class of parabolic subgroups of G contains an element P which is defined over F. Similarly, a Levi subgroup L ∨ ⊂ G ∨ is F − r elevant if it is a Levi factor of a parabolic subgroup P ∨ ⊂ G ∨ which is F-relevant. Definition 2.15 A parabolic subgroup P ∨ of G ∨ is W F − quasi − stable if the projection NL G (P ∨ ) → W F is surjective. Remark 2.16 The W F -quasi-stable parabolic subgroups of G ∨ are precisely the neutral components of what Borel [21, § 3] calls parabolic subgroups of L G. Definition 2.17 Let φ ∈ ( L G)) and let P ∨ be a W F -quasi-stable parabolic subgroup of G ∨ with a Levi factor L ∨ such that • the image of φ is contained in NL P (L ∨ ), where L P := P ∨  W F ;

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• P ∨ is a minimal for this property. The L-parameter φ is G-relevant (or is an L-parameter for G) if P ∨ is F-relevant. Notation 2.18 We denote by (G) the subset of ( L G) of the G∨ -conjugacy classes of L-parameters that are G-relevant. A Langlands classification of L-parameters for G is obtained in [58]. We have ( L G) = (G ∗ ), where G ∗ is the quasi-split inner form of G as defined in Proposition 2.8, and it is expected that in general φ (G) is nonempty if and only if φ is G-relevant. Example 2.19 Let D be a division algebra with center F such that dim F (D) = 4. Then G = D × is the unique non-split inner form of G ∗ = GL2 (F). The only Levi subgroup of D × defined over F is D × itself, and it corresponds to the Levi subgroup GL2 (C) on the complex side. Let φ ∈ (GL2 (F)) = (D × ) be the embedding W F → GL2 (C) × W F . No proper parabolic subgroup of GL2 (C) contains φ(SL2 (C)) = SL2 (C), so φ is relevant for both D × and GL2 (F).

2.4 L-Packets 2.4.1

For a Given Group G

The local Langlands conjecture (LLC) asserts that the set Irr(G) of isomorphism classes of irreducible smooth representations of G can be parametrized by the set (G). This parametrization is usually not a bijection. In fact, it is conjectured that each conjugacy class φ ∈ (G) is associated with a finite set φ (G) of isomorphism classes of irreducible smooth representations of G, and that they give a disjoint decomposition of Irr(G); φ (G). (10) Irr(G) = φ∈(G)

Such finite sets are called L-packets for G. This parametrization is based on the belief that there should be certain arithmetic invariants (e.g., L-factors) defined on both the representation side and the parameter side that are preserved by the correspondence. From this point of view, one can think that the L-packet φ attached to some φ ∈ (G) consists of all irreducible smooth representations of G whose arithmetic invariants match those of φ.

Local Langlands and Springer Correspondences

2.4.2

11

For All the Inner Twists of G

The local Langlands correspondence predicts the existence of a partition of the set Irr(IT(G) of equivalence classes of the irreducible smooth representations of all the groups G i in IT(G) into finite subsets:

Irr(IT(G)) =

φ ( L G),

(11)

φ∈( L G)

where each L-packet φ ( L G) is the union of L-packets for the groups G i . Example 2.20 We keep the notation of Example 2.19. We have φ (GL2 (F)) = {StGL2 (F) }

and

φ (D × ) = {St D× = triv D× }.

Thus here (11) is φ ( L G) = φ (GL2 (F)) ∪ φ (D × ) = {St GL2 (F) , St D× }. Let φ  ∈ (GL2 (C)) such that φ  (SL2 (C)) = 1 and φ  (W F ) ⊂ diag(GL2 (C)) × W F . Then L ∨ = diag(GL2 (C)) is the minimal Levi subgroup such that L L contains the image of φ2 . Thus the standard Borel subgroup P ∨ of GL2 (C) satisfies the conditions in Definition 2.17. But its conjugacy class does not correspond to any parabolic subgroup of D × , so φ  is not relevant for D × .

be a pair of quasi-split connected reductive groups, defined over F, Let G ⊂ G with the same derived group. It is expected that the L-packets of G = G(F) are

:= G(F) restrictions of L-packets of G in the sense that for each L-packet (G)



for G there is a packet (G) for G whose restriction to G is equal to . Typically, one uses this last expectation to construct L-packets for G from the knowledge of

e.g., in the cases of SLn ⊂ GLn and Sp2n ⊂ GSp2n . In some cases, L-packets of G, however, we wish to move in the other direction and use the knowledge of L-packets

see [62]. of G to obtain structural information about the L-packets of G,

3 Enhanced Langlands Parameters 3.1 The Group Sφ To parametrize the irreducible representations in a given L-packet, we need more information than just the Langlands parameter itself. Let ZG∨ (φ) denote the centralizer of φ(W F ) in G∨ . This is a reductive algebraic group over C, in general disconnected. We denote by ZG∨ the center of G∨ and we write

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  Rφ := π0 ZG∨ (φ)/Z(G∨ )W F .

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Remark 3.1 It is expected that φ (G) is in bijection with Irr(Rφ ) if G is quasi-split. However, for G not quasi-split, this is not always the case. ∨ ∨ Let Gad be the quotient G∨ /ZG∨ (i.e., the group Gad is the adjoint group of G∨ ). ∨ ∨ W F Since ZG∨ (φ) ∩ Z(G ) = Z(G ) ,

ZG∨ (φ)/Z(G∨ )W F ∼ = ZG∨ (φ)Z(G∨ )/Z(G∨ ).

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∨ . Let The right-hand side can be considered as a subgroup of the adjoint group Gad 1 ∨ ∨ ZGsc∨ (φ) be its inverse image under the quotient map Gsc → Gad . We can also characterize it as

 ∨ g ∈ Gsc : gφg −1 = φ ag for some ag ∈ B 1 (W F , Z(G∨ )) (14)    −1 1 ∨ = g ∈ ZGsc∨ φ(SL2 (C)) : gφ|W F g = φ|W F ag for some ag ∈ B (W F , Z(G ))   1 = ZG ∨ (φ| W F ) ∩ ZG∨ φ(SL 2 (C)) , sc sc 1 ZG ∨ (φ) = sc

where B 1 (W F , Z(G∨ )) denotes the set of 1-coboundaries for group cohomology, that is, the maps W F → Z(G∨ ) of the form w → zwz −1 w −1 with z ∈ Z(G∨ ). The 1 ◦ neutral component of ZG ∨ (φ) is ZG∨ (φ) , so it is a complex reductive group. sc sc 1 Remark 3.2 We have ZG whenever ∨ (φ) = ZG∨ (φ) sc sc in particular if G is an inner twist of a split group .

∨ WF ∨ Z(Gsc ) = Z(Gsc ),

Following Arthur [2], given φ, we define the group Sφ as the component group 1 of ZG ∨ (φ):  1  (15) Sφ := π0 ZG ∨ (φ) . sc ∨ ∨ → Gad induces a homomorphism Sφ → Rφ . We set Via (13), the map Gsc ∨ ∨ )/Z(Gsc ) ∩ ZGsc∨ (φ)◦ . Zφ := Z(Gsc

(16)

Then (see [12, Lemma 1.7]) Sφ is a central extension of Rφ by Zφ : 1 → Zφ → Sφ → Rφ → 1.

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∨ ∨ ∨ is a central extension of Gad = G∨ /Z(G∨ ), the conjugation action of Gsc Since Gsc ∨ on itself and on Sφ descends to an action of Gad . Via the canonical quotient map, also G∨ acts on Sφ by conjugation. We attach to a given L-parameter φ ∈ ( L G) the following (possibly disconnected) complex reductive group: 1 Gφ := ZG ∨ (φ(W F )). sc

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Local Langlands and Springer Correspondences

13

The following proposition provides a way to link the local Langlands correspondence for G to the generalized Springer correspondence for the groups Gφ ’s. Proposition 3.3 [15, (92)] The group Sφ is isomorphic to the group AGφ (u φ ) := π0 (ZGφ (u φ )), where u φ := φ(1,

11 0 1 ).

3.2 Definition of Enhanced Langlands Parameters Definition 3.4 An enhanced Langlands parameter (or enhanced L-parameter) for L G is a pair (φ, ρ), where φ ∈ ( L G) and ρ is an irreducible representation of the group Sφ defined in (15). Remark 3.5 The enhanced Langlands parameter (φ, ρ) is already determined by φ|W F , u φ and ρ. Sometimes, we will write (φ|W F , u φ , ρ) instead of (φ, ρ). We let G∨ act on the set of enhanced L-parameters for L G by g · (φ, ρ) = (gφg −1 , g · ρ)

where

(g · ρ)(a) = ρ(g −1 ag).

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Notation 3.6 Let e ( L G) denote the set of G∨ -conjugacy classes of enhanced Langlands parameters for L G. We set

ZφW F := Z(G ∨sc )W F

According to [2, § 4]

  Z(G ∨sc )W F ∩ Z G ∨sc (φ)◦ .

Z(G ∨sc ) ∩ ZG ∨sc (φ)◦ ⊂ Z(G ∨sc )W F .

(20)

(21)

Hence, ZφW F can be regarded as a subgroup of Zφ and Zφ /ZφW F ∼ = Z(G ∨sc )/Z(G ∨sc )W F .

(22)

By Schur’s lemma, every enhanced Langlands parameter (φ, ρ) restricts to a char∨ WF ) . With the acter ρ|Z W F of ZφW F . This can be inflated to a character ζρ of Z(Gsc φ

∗ Kottwitz isomorphism (9), we get an element ϑ := κG−1∗ (ζρ ) ∈ H 1 (F, Gad ). In this ad way, (φ, ρ) determines a unique inner twist G ϑ of G. This can be regarded as an alternative way to specify for which inner twists of G an enhanced Langlands parameter is relevant. Fortunately, it turns out that it agrees with the earlier definition of relevance of Langlands parameters. Indeed for φ ∈ ( L G), the following are equivalent (see [12, Prop. 1.8])

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(1) φ is relevant for G ϑ ; (2) Z (G ∨sc )W F ∩ ZG ∨sc (φ)◦ ⊂ ker ζ ;  (3) there exists a ρ ∈ Irr(Sφ ) such that ζ is the lift of ρ Z W F to Z(G ∨sc )W F , φ

where ζ ∈ via (9).

Irr(Z(G ∨sc )W F ) and

G ϑ is the inner twist of G associated with ϑ = κG−1∗ (ζ ) ad

Notation 3.7 We denote by e (G) the set of G∨ -equivalence classes of enhanced L-parameters for L G that are relevant for G.  Let (φ, ρ) ∈ e ( L G). If ρ is relevant for G, then the character ζρ ∈ Irr Z(G ∨sc )W F ) can be extended in precisely [Z(G ∨sc ) : Z(G ∨sc )W F ] ways to a character of Z(G ∨sc ). We choose such an extension and we denote it by ζG . Of course we pick ζG = triv when  G = G ∗ . Every φ ∈ ( L G) can be enhanced with a ρ ∈ Irr(Sφ ) such that ρ  Zφ

inflates to ζG . We define ∨ ) → Sφ )}, e,ζG (G) := {(φ, ρ) ∈ e (G) : ζG id Vρ = ρ ◦ (Z(Gsc

(23)

where Vρ is the vector space underlying the representation ρ. Since Sφ /Zφ ∼ = Rφ , we obtain e,triv (G) = {(φ, ρ) : φ ∈ (G), ρ ∈ Irr(Rφ )},

if G is quasi-split.

3.3 Desiderata for the Local Langlands Correspondence We are ready to formulate the version of the conjectural local Langlands correspondence stated in [12]. It is inspired by many sources, in particular [21, § 10], [61, § 4], [2, § 3] and [33, § 5.2]. Conjecture 3.8 Let (G, ϑ) be an inner twist of a quasi-split F-group G ∗ . There exists a surjection e ( L G) −→ Irr(G) : (φ, ρ) → πφ,ρ , which becomes bijective when restricted to e,ζG (G). We write its inverse as Irr(G) −→ e (G) : π → (φπ , ρπ ). Then the map Irr(G) → (G) : π → φπ is canonical. These maps satisfy the properties (1)–(7) listed below. Remark 3.9 The above bijection becomes more elegant if one considers the union over inner twists. It then says that there exists a surjection

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15

{(φ, ρ) : φ ∈ (G ∗ ), ρ ∈ Irr(Sφ )} → {(G, ϑ, π ) : (G, ϑ) inner twist of G ∗ , π ∈ Irr(G)} whose fibers have exactly [Z(G ∨sc ) : Z(G ∨sc )W F ] elements. Properties: (1) The central character of π equals the character of Z(G) constructed from φπ in [21, § 10.1]. (2) Let z ∈ Hc1 (W F , Z(G∨ )) be a class in continuous group cohomology, and let χz : G → C× be the character associated with it in [21, § 10.2]. Thus zφπ ∈ ˜ (G) and Szφπ = Sφπ . Then the LLC should satisfy (zφπ , ρπ ) = (φχz π , ρχz π ). (3) π is essentially square-integrable if and only if φπ is discrete. (4) π is tempered if and only if φπ is bounded. (5) Let P be a parabolic subgroup of G with Levi factor L. Suppose that g ∈ NG (L) and gˇ ∈ NG ∨ (L ∨ ) are such that Ad(g) : L → L and Ad(ˇg) : L ∨ → L ∨ form a corresponding pair of homomorphisms, in the sense of [21, § 2]. Then ˇ π , gˇ · ρπ ) (φg·π , ρg·π ) = (Ad(g)φ

for all

π ∈ Irr(L).

(6) Suppose that (φ L , ρ L ) ∈ e (L) is bounded. Then  {πφ,ρ : φ = φ L composed with L L → L G, ρ S L contains ρ L } φ

(24)

equals the set of irreducible constituents of the parabolically induced representation IGP (πφ L ,ρ L ). (7) If φ L is discrete but not necessarily bounded, then (24) is the set of Langlands constituents of IGP (πφ L ,ρ L ), as in [6, p. 30].

3.3.1

Inner Forms of GLn (F)

The local Langlands correspondence for supercuspidal representations of GLn (F) was established first for F of positive characteristic in [42], and later for F of characteristic zero in [34, 35, 56], independently. Together with the Jacquet–Langlands correspondence, this provides the LLC for essentially square-integrable representations of inner forms G = GLm (D) of GLn (F). It is extended to all irreducible G-representations via the Zelevinsky classification [64], see [9, 36]. For these groups, every L-packet is a singleton and the LLC is a canonical bijective map recGLm (D) : Irr(GLm (D)) → (GLm (D)).

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A.-M. Aubert

Inner Forms of SLn (F)

The local Langlands correspondence for SLm (D) was established [36] (for F of characteristic zero) and [9] (for F of positive characteristic). It is derived from the LLC for GLm (D), in the sense that every L-packet for SLm (D) consists of the irreducible constituents of GLm (D) (φ (GLm (D))), ResSL m (D)

with φ ∈ (GLm (D)). Of course these L-packets have more than one element in general (as in Example 2.20). 3.3.3

The Local Langlands Conjecture for Tempered Representation

Let Irr t (G) denote the subset of Irr(G) consisting all irreducible smooth representations π whose matrix coefficients are L 2+ on G modulo the center Z(G), and let bd (G) be the subset of (G) consisting G∨ -conjugacy classes of L-parameters φ such that the φ(W F × SU2 ) is bounded modulo the center. The local Langlands conjecture for tempered representation asserts the existence of a surjective map (26) rectG : Irr t (G) → bd (G), whose fibers are the tempered L-packets. For each φ ∈ bd (G), the corresponding tempered L-packet tφ (G) is expected to be parametrized by Irr(Sφ )ζG , and Irr t (G) to be a disjoint union of the tφ (G). In the case when the characteristic of F is zero, this conjecture has been proved by Arthur in [3] for quasi-split orthogonal groups and symplectic groups, by Mok in [51] for quasi-split unitary groups, and by Kaletha, Minguez, Shin and White for non in [40]. Note that the unitary group U E/F (n) admits a non-quasi-split inner form exactly when n is even. The LLC was also proved for the groups GSp4 (F) [27, 30], its inner form [29], the group Sp4 (F) [28] and its inner form [23], and for the groups GSpin4 , GSpin6 , and their inner forms [4]. In [31], Ganapathy and Varma used Arthur’s results to lift the LLC for split symplectic and special orthogonal groups on a non-Archimedean field of odd positive characteristic, but using a “Gan-Takeda type” characterization instead of the theory of endoscopy. They proved (see [31, Theorem 13.6.1]) that given a tempered representation π of G, there exists a unique bounded Langlands parameter φπ , defined by suitable compatibility conditions on Langlands-Shahidi L-functions and γ -factors, and on Plancherel measures, together with the requirement that φπ is discrete if π belongs to the discrete series. For a pleasant and precise description of the state of the art with respect to the local Langlands correspondence, the reader should consult [37, 38, 43], the latter including global aspects and the link with the Ramanujan conjecture. A survey on other aspects of the Langlands correspondence may be found in [5] and the references therein.

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17

3.4 Depth Another invariant that makes sense on both sides of the LLC is the depth. The depth d(π ) of an irreducible smooth representation π of a reductive p-adic group G was defined by Moy and Prasad [54] in terms of filtrations G x,r (with x a point in the Bruhat-Tits building of G and r ∈ R≥0 ) of its parahoric subgroups G x,0 . The depth of a Langlands parameter φ is defined to be the smallest number d(φ) ≥ 0 such that φ is trivial on r for all r > d(φ). Yu [63, § 7.10] proved that the depth is preserved by the LLC for unramified tori. Recently, Mishra and Pattanayak proved that it is not preserved for wildly ramified tori [50].

3.4.1

Inner Forms of General Linear Groups

Let GLm (D) an inner form of GLn (F). Let k D = o D /p D be the residual field of D. Let A be a hereditary o F -order in Mm (D). The Jacobson radical of A will be denoted by P. Let r = e D (A) and e = e F (A) denote the integers defined by p D A = Pr and p F A = Pe , respectively. We have e F (A) = d e D (A). The normalizer in G of A× will be denoted by  K(A) := g ∈ G : g −1 A× g = A× . Define a sequence of compact open subgroups of G = GLm (D) by U 0 (A) := A× ,

and

U j (A) := 1 + P j , j ≥ 1.

Then A× is a parahoric subgroup of G and U 1 (A) is its pro-unipotent radical. We define the normalized level of an irreducible representation π of G to be (π ) := min { j/e F (A)} ,

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where ( j, A) ranges over all pairs consisting an integer j ≥ 0 and a hereditary o F order A in Mm (D) such that π contains the trivial character of U j+1 (A). Then (see, for instance, [8, Proposition 2.5]) the normalized level of π ∈ Irr(G) equals its Moy– Prasad depth: (π ) = d(π ). Let ψ be a nontrivial character of F and let c(ψ) be the largest integer c such that p−c F ⊂ ker ψ. The  factor of φ (and ψ) was defined in [60]. It takes the form (s, φ, ψ) = (0, φ, ψ)q −(a(φ)+nc(ψ))s with (0, φ, ψ) ∈ C× .

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Here a(φ) ∈ Z≥0 is the Artin conductor of φ (called f (φ) in [57, § VI.2]). For any elliptic φ ∈ (G L n (F)), we have (see [8, Lemma 2.3])

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 0 d(φ) = a(φ) n −1

if I F ⊂ ker(φ),

(29)

otherwise.

Let π be an irreducible representation of GLm (D). Let (s, π, ψ) denote the Godement–Jacquet local constant [32]. It takes the form (s, π, ψ) = (0, π, ψ) q − f (π,ψ)s ,

where (0, π, ψ) ∈ C× .

(30)

A representation of D × is called unramified if it is trivial on o×D . An unramified representation of D × is a character and has depth zero. Let π be a supercuspidal irreducible representation of G. We have (see [8, Proposition 2.6]:  n (c(ψ) + 1) − 1 f (π, ψ) = n (d(π ) + 1 + c(ψ))

if m = 1 and π is unramified, otherwise.

(31)

We set f (π ) := f (π, ψ) − nc(ψ).

(32)

Theorem 3.10 [8, Theorem 2.7] The depth d(π ) and the conductor f (π ) of each essentially square-integrable irreducible representation π of GLm (D) are linked by the following relation:  d(π ) =

0 f (π ) − n n

if π is an unramified twist of St GLm (D) otherwise.

In particular

 d(π ) = max

 f (π ) − n ,0 . n

,

(33)

(34)

Theorem 3.10 is a key ingredient in the proof of the following result: Theorem 3.11 [8, Theorem 2.9] The LLC for G = GLm (D) preserves the depth, that is: d(π ) = d(φπ ),

3.4.2

where φπ = recG (π ).

Inner Forms of Special General Linear Groups

The situation is different for SLm (D). All the irreducible representations in a given L-packet φ have the same depth, so the depth is an invariant of the L-packet, say d(φ ). We have d(φ ) = d(ϕ) where ϕ is a lift of φ which has minimal depth among the lifts of φ, and the following holds:

Local Langlands and Springer Correspondences

d(φ) ≤ d(φ )

19

(35)

for any Langlands parameter φ for SLm (D) [8, Proposition 3.4 and Corollary 3.4]. Moreover (35) is an equality if φ is essentially tame (in the terminology of Definition 2.13). Remark 3.12 The notion of essentially tameness is consistent with the usual notion for Langlands parameters for GLn (F). Indeed, any lift ϕ : W F → GLn (C) of φ, is called essentially tame if its restriction to PF is a direct sum of characters. We denote by t (ϕ) the torsion number of ϕ, that is, the number of unramified characters χ of W F such ϕχ ∼ = ϕ. Then φ and ϕ are essentially tame if and only if the residual characteristic p of F does not divide n/t (ϕ) [22, Appendix]. However, let F be a local non-Archimedean field of characteristic 2, that is, F is of the form F = Fq ((t)), the field of Laurent series with coefficients in Fq , with q = 2 f . This case is particularly interesting because there are countably many quadratic extensions of Fq ((t)). Then equality holds in (35) only if φ is essentially tame (i.e., t (ϕ) = 2), as proved in [14]. In the case when G is a classical group, the characteristic of F is zero (i.e., F is a finite extension of Q p ) and p is odd, Ganapathy and Varma proved in [31, Lemma 8.2.3] that the following inequality holds d(φπ ) ≤ d(π ) + 1,

(36)

where φ the Langlands parameter attached to π by Arthur. If moreover p is sufficiently large with respect to G, then it is shown in [31] that d(π ) ≤ d(φπ ).

(37)

Very recently, it was proved in [55], for a quasi-split classical group over F, with F of characteristic equal to zero and p sufficiently large, that the depth of representations in each L-packet equals the depth of the corresponding L-parameter, and that, for quasi-split unitary groups, the depth is constant in each L-packet.

4 Generalized Springer Correspondence 4.1 Cuspidal Enhanced Unipotent Classes Let G be a complex (possibly disconnected) reductive group. Let G ◦ be its identity component. For u ∈ G unipotent, we denote by AG (u) the component group of the centralizer of u in G. We denote by U(G) the unipotent variety of G.

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Definition 4.1 The enhancement of U(G) is the set Ue (G) of G-conjugacy classes of pairs (u, ρ), with u ∈ G unipotent and ρ ∈ Irr(AG (u). We call a pair (u, ρ) an enhanced unipotent class. Let DGb (U(G)) denote the constructible G-equivariant-derived category on U(G) as in [19] and let PervG (U(G)) be its subcategory of G-equivariant perverse sheaves. By a P-resolution of an algebraic variety X , we mean a variety Y endowed with a free P-action and a smooth P-equivariant morphism Y → X . Definition 4.2 The integration functor is the functor b γPG : DP (U(G)) → DGb (U(G)),

defined by

(γPG A)(Y ) := (qY )! A(Y )[2 dim G/P],

b (U(G)) and Y a G-resolution of U(G), where qY : P\Y → G\Y for A any object of DP is the quotient functor and A(Y ) is defined by regarding Y as a P-resolution of U(G).

Definition 4.3 Let P ◦ = L◦ U be a parabolic subgroup of G ◦ , with L◦ a Levi component and U the unipotent radical. Let m : U(P ◦ ) → U(G ◦ )

and

p : U(P ◦ ) → U(L◦ )

denote inclusion and projection, respectively. The parabolic induction functor is the functor ◦

◦

G G ∗ ◦ ◦ iL ◦ ⊂P ◦ := γP ◦ ◦ m ! ◦ p : PervL◦ (U(L )) → PervG ◦ (U(G )). ◦

G Remarks 4.4 (1) The functor iL ◦ ⊂P ◦ commutes with Verdier duality. It is left adjoint G◦ G◦ ∗ ! to rL◦ ⊂P ◦ := p! ◦ m and right adjoint to  rL ◦ ⊂P ◦ := p∗ ◦ m . These functors are exchanged by Verdier duality. G◦ (2) If F L is a simple object in PervL◦ (U(L◦ )), then iL ◦ ⊂P ◦ (FL◦ ) is semisimple.

Definition 4.5 A simple object F in PervG ◦ (U(G ◦ )) is cuspidal if for any simple G◦ object FL◦ in PervL◦ (U(L◦ )), F does not occur in iL ◦ ⊂P ◦ (FL◦ ) (equivalently, if ◦ ◦ G  G rL◦ ⊂P ◦ (F) = 0, resp. rL◦ ⊂P ◦ (F) = 0) for any proper parabolic subgroup P ◦ of G ◦ with Levi factor L◦ . Remarks 4.6 (1) Cuspidality is preserved by Verdier duality (see, for instance, [1, Remark 2.3]). (2) The above definition of cuspidality is inspired by [46] (see also [1]). It is equivalent to the original definition given by Lusztig in [45] (as shown in [49, 23.2. (b)]). For u a given unipotent element in G ◦ , let AG ◦ (u) denote the component group of the centralizer ZG ◦ (u) of u in G ◦ . Let Ou = (u)G ◦ be the G ◦ -conjugacy class of u.

Local Langlands and Springer Correspondences

21

With a slight abuse of notation, we will sometimes write AG ◦ (Ou ) := AG ◦ (u). We will denote by (38) ρ → Eρ the bijection between Irr(AG ◦ (u)) and the irreducible G ◦ -equivariant local systems E on Ou . We denote by (39) E → ρE the inverse bijection. Definition 4.7 (1) A character ρ ∈ Irr(AG ◦ (u)) is cuspidal if the perverse sheaf IC(O, Eρ ) is cuspidal. (2) An enhanced unipotent class (O, ρ) in G ◦ is cuspidal if ρ is cuspidal. Proposition 4.8 [45, Proposition 2.8] If (Ou , ρ) is cuspidal, then u is a distinguished unipotent element in G ◦ (i.e., u does not meet U(L◦ ) for any proper Levi subgroup L of G ◦ ). Remark 4.9 In general, not every distinguished unipotent element supports a cuspidal representation. Example 4.10 For G = SLn (C), the unipotent classes in G are in bijection with the partitions λ = (λ1 , λ2 , . . . , λr ) of n: The corresponding G-conjugacy class Oλ consists of unipotent matrices with Jordan blocks of sizes λ1 , λ2 , . . .,λr . We identify the center Z(G) with the group μn of complex n-th roots of unity. For u ∈ Oλ , the natural homomorphism Z(G) → AG (u) is surjective with kernel μn/ gcd(λ) , where gcd(λ) := gcd(λ1 , λ2 , . . . , λr ). Hence, the irreducible G-equivariant local systems on Oλ all have rank one, and they are distinguished by their central characters, which range over those χ ∈ Irr(μn ) such gcd(λ) is a multiple of the order of χ . We denote these local systems by Eλ,χ . The unique distinguished unipotent class in G is the regular unipotent class O(n) , consisting unipotent matrices with a single Jordan block. The cuspidal irreducible G-equivariant local systems are supported on O(n) (by Proposition 4.8) and are of the form E(n),χ , with χ ∈ Irr(μn ) of order n (see [45, (10.3.2)]). We will now extend the above notion of cuspidality from G ◦ to G: Definition 4.11 An enhanced unipotent class (O, ρ) with O = (u)G and ρ ∈ Irr(AG (u)) is cuspidal if the restriction of ρ to AG ◦ (u) is a direct sum of cuspidal irreducible representations of AG ◦ (u). Notation 4.12 We set Irr cusp (AG (u)) := {ε ∈ Irr(AG (u)) such that ε is cuspidal} . Definition 4.13 A quasi-Levi subgroup of G is a subgroup M of the form M = ZG (Z(L)◦ ), with L a Levi subgroup of G ◦ . The group M is said to be cuspidal if there exists a cuspidal enhanced unipotent pair in M.

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Notation 4.14 Let B(Ue (G)) be the set of G-conjugacy classes of pairs (M, (O, ε)), where M is a cuspidal quasi-Levi subgroup of G, and (O, ε) is a cuspidal enhanced unipotent pair in M. We will write the elements of B(Ue (G)) as t := [M, v, ε]G .

4.2 A Partition of the Enhancement of the Unipotent Variety of G ◦ The purpose of this section is to describe a theory à la Harish-Chandra for Ue (G ◦ ), the first step being to define a cuspidal support map for enhanced unipotent classes of G. The enhancement Ue (G ◦ ) of U(G ◦ ) parametrizes the isomorphism classes of simple objects of PervG ◦ (U(G ◦ )). Indeed, the simple objects in PervG ◦ (U(G ◦ )) are the IC(O, E), where O is a unipotent class in G ◦ and E is an irreducible G ◦ -equivariant Q -local system on O. Let (O, ρ) ∈ Ue (G ◦ ) be an arbitrary enhanced unipotent class, and set Fρ := IC(O, Eρ ). Then Fρ occurs as a summand of iL◦ ⊂P ◦ (IC(O0 , E0 )), for some quadruple (P ◦ , L◦ , O0 , E0 ), where P ◦ is a parabolic subgroup of G ◦ with Levi subgroup L◦ and (O0 , E0 ) is a cuspidal enhanced unipotent class in L◦ (see [45, § 6.2] and [1, Cor. 2.7]) and, moreover, (P ◦ , L◦ , O0 , E0 ) is unique up to G ◦ -conjugation (see [45, Prop. 6.3]). We set ε0 := ρE0 using the bijection (39), and we denote by t◦ := (L◦ , (O0 , ε0 ))G ◦ , the G ◦ -conjugacy class of (L◦ , (O0 , ε0 )) and we call it the cuspidal support of the enhanced unipotent class (O, ρ). The center Z(G ◦ ) of G ◦ maps naturally to AG ◦ (O) and to AL◦ (O0 ). By construction [45, Theorem 6.5.a] ρ and ε have the same Z(G ◦ ) -character.

(40)

Definition 4.15 The cuspidal support map for Ue (G ◦ ) is the map G ◦ : Ue (G ◦ ) → B(Ue (G ◦ )), where B(Ue (G ◦ )) is as in Notation 4.14, which sends the G ◦ -conjugacy class of (O, ρ) to its cuspidal support t◦ = (L◦ , (O0 , ε0 ))G ◦ . By (40), the map G ◦ preserves the Z(G ◦ )-characters of the involved representations. Notation 4.16 Let t◦ ∈ B(Ue (G ◦ )). ◦

(1) We denote by Ue (G ◦ )t the fiber of t under the map G ◦ . (2) Let WL◦ := NG ◦ (L◦ )/L◦ , and let Wt◦ := NG ◦ (t◦ )/L◦ .

Local Langlands and Springer Correspondences

23

Theorem 4.17 [45] (1) The group WL◦ is a Weyl group and it coincides with Wt◦ for every t◦ ∈ B(Ue (G ◦ )). (2) We have ◦ Ue (G ◦ )t , Ue (G ◦ ) = t◦ ∈B(Ue (G ◦ )) ◦

and Ue (G ◦ )t is in bijection with Irr(WL◦ ). The goal of the next two subsections is to generalize the previous notation and terminology from G ◦ to G.

4.3 Twisted Group Algebras Throughout this section,  is a finite group and K is an algebraically closed field whose characteristic does not divide the order of . Suppose that κ :  ×  → K × is a 2-cocycle, that is, κ(γ1 , γ2 γ3 )κ(γ2 , γ3 ) = κ(γ1 , γ2 )κ(γ1 γ2 , γ3 ) ∀γ1 , γ2 , γ3 ∈ .

(41)

The κ-twisted group algebra of  is defined to be the K -vector space K [, κ] with basis {Tγ : γ ∈ } and multiplication rules Tγ Tγ  = κ(γ , γ  )Tγ γ  γ , γ  ∈ .

(42)

Its representations can be considered as projective -representations. Schur showed (see [24, Theorem 53.7]) that there exists a finite central extension ˜ of , such that ˜ • char(K ) does not divide ||, • every irreducible projective -representation over K lifts to an irreducible K -linear ˜ representation of . ˜ namely the image of a minimal idempoThen K [, κ] is a direct summand of K [], ˜ ˜ is semisimple, tent in K [ker( → )]. The condition on char(K ) ensures that K [] so K [, κ] is also semisimple.

4.4 A Partition of the Enhancement of the Unipotent Variety of G Notation 4.18 Let t = (M, (O0 , ε))G ∈ B(Ue (G)). We set Wt := NG (t)/M

and

Wt◦ := NG ◦ (M◦ )/M◦ .

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Theorem 4.19 [15, § 4] Let (O, ρ) ∈ Ue (G). There exists a 2-cocycle κt : Wt /Wt◦ × Wt /Wt◦ → C× and a map, called the cuspidal support map for Ue (G) G : Ue (G) → B(Ue (G)) (which coincides with the map G ◦ defined above in the case when G is connected), such that Ue (G)t , (43) Ue (G) = t∈B(Ue (G))

in which the fiber Ue (G)t of t under the map G is isomorphic to the set of isomorphism classes Irr(C[Wt , κt ]) of irreducible representations of a κt -twisted version C[Wt , κt ] of the group algebra C[Wt ] of the finite group Wt . Notation 4.20 Let t denote the bijection Ue (G)t −→ Irr(C[Wt , κt ]) mentioned in Theorem 4.19. Remark 4.21 When G is connected, the cocycle κt is trivial. When G is disconnected, the cocycle κt is not always trivial. An intuitive way of thinking of Theorem 4.19 is to view the set B(Ue (G)) as a “palette of colors” and the map G as a way to paint the elements of the fiber Ue (G)t of t under G in the same color as t. Moreover, for each color, the subset of elements of Ue (G) with that color has a nice structure (here, that of the isomorphism classes of irreducible representations of a twisted group algebra).

5 Cuspidality for Enhanced L-Parameters: Definition and Conjecture Definition 5.1 An enhanced L-parameter (φ, ρ) ∈ (G)e is called cuspidal if φ is discrete and (u φ , ρ) is a cuspidal enhanced unipotent class in Gλφ in the terminology of Definition 4.11. Recall that an irreducible smooth complex representation of the group G is called supercuspidal if it does not appear in any G-representation induced from a proper Levi subgroup of G. It was proved by Jacquet and by Harish-Chandra that an irreducible representation of G is supercuspidal if and only if its matrix coefficients have compact support modulo the center of G (see, for instance, [20, pp. 34–35]).

Local Langlands and Springer Correspondences

25

Cuspidality Conjecture 5.2 [15, § 6] The cuspidal enhanced Langlands parameters correspond under the LLC to the irreducible supercuspidal representations of G. Conjecture 5.2 is known to be true in the following cases: • for general linear groups and split classical groups (any representation), with F of characteristic equal to 0, see [52], • for inner forms of linear groups and of special linear groups, see [15, Example 6.11], quasi-split unitary groups (any representation) with F of characteristic equal to 0, see [15, Example 6.13], and for Deligne-Lusztig depth-zero representations, [15, Example 6.14], • for the representations with unipotent reduction of the group G of the F-rational points of a connected reductive algebraic group which splits over an unramified extension of F, see [26, Theorem 2] (when G is simple of adjoint type it is a special case of [47, 48]).

6 A Partition of the Set of Enhanced Langlands Parameters We shall define a similar partition of the set of enhanced Langlands parameters by plugging the above construction into the framework of the Langlands correspondence. The idea is very similar as above: Using the data defined in (47), we will construct a set B∨ ( L G) which will play the role of the palette of colors for e ( L G), and we will define a “way to paint” the elements in e ( L G) with this set of colors (i.e., we will construct a cuspidal support map) such that there is a decomposition e ( L G) =



∨

e ( L G)s ,

(44)

s∨ ∈B∨ ( L G) ∨

where, for each s∨ in B∨ ( L G), the subset e ( L G)s of elements with color s∨ is related to a generalized affine Hecke algebras with possibly unequal parameters.

6.1 Inertial Classes for Enhanced Langlands Parameters Definition 6.1 Let T∨ ⊂ G∨ be a torus such that the projection ZL G (T∨ ) → W F is surjective, where ZL G (T∨ ) is the centralizer of T∨ in L G. Then we call ZL G (T∨ ) a Levi L-subgroup of L G. Remark 6.2 In [21], such groups are called Levi subgroups of L G. However, we prefer to stick to the connectedness of Levi subgroups.

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Remark 6.3 Choose a W F -stable pinning for G∨ . This defines the notion of standard Levi subgroups of G∨ . An alternative characterization of the Levi L-subgroups of L G is as follows [15, Lemma 6.2]: Let ZL G (T∨ ) be a Levi L-subgroup of L G as in Definition 6.1. There exists a W F -stable standard Levi subgroup L∨ of G∨ such that ZL G (T∨ ) is G∨ -conjugate to L∨  W F =: L L and ZL G (T∨ ) ∩ G∨ is conjugate to L∨ . Conversely, every G∨ -conjugate of this L L is a Levi L-subgroup of L G. Notation 6.4 In the sequel, we will use the (slightly abusive) notation L L for an arbitrary L-Levi of G. Definition 6.5 Let L L be a Levi L-subgroup of L G. (1) A Langlands parameter for L L is a group homomorphism ϕ : W F → L L satisfying the requirements of Definition 2.12. (2) An enhancement of ϕ is an irreducible representation  of π0 (ZL1 sc∨ (ϕ)), where ∨ is the simply connected cover of the derived group of L∨ :== L L ∩ G∨ . Lsc Then (ϕ, ) is called an enhanced Langlands parameter for L L. The group L∨ acts on the collection of enhanced L-parameters for L L by (19) and we denote the set of L∨ -orbits of enhanced Langlands parameters for L L by e ( L L). Definition 6.6 We say that an enhanced (ϕ, ) for L L is cuspidal if ϕ     L-parameter is discrete for L L and u ϕ := ϕ 1, 01 11 ,  is a cuspidal pair for ZL1 sc∨ (ϕ|W F ). Notation 6.7 If an enhanced Langlands parameter (ϕ, ) for L L is cuspidal, then any element in its L∨ -orbit is cuspidal. We denote the subset of L∨ -orbits of cuspidal enhanced Langlands parameters for L L by e,cusp ( L L). Let L∨ be a Levi subgroup of G∨ , and let Lc∨ denote the pre-image of L ∨ under ∨ ∨ → G∨ . The derived group of Lc∨ is the simply connected cover of Lder , and we Gsc ∨ ∨ ∨ with the inverse image of Lder under Gsc → G∨ . identify Lsc Definition 6.8 A cuspidal datum for L G is a triple ( L L , ϕ, ε) where L L is a Levi L-subgroup of L G, such that (ϕ, ε) is a cuspidal enhanced Langlands parameter for L L. It is relevant for G if ∨ ∨ WF ∨ WF • ε = ζ on Lsc ∩ Z(Gsc ) , where ζ ∈ Irr(Z(Gsc ) ) parametrizes the inner twist ∗ G of G via the Kottwitz isomorphism (9): ∨ ∩ Z(Lc∨ )◦ . • ε = 1 on Lsc

Recall from [33, § 3.3.1] that the group of unramified characters of a Levi subgroup L of G is naturally isomorphic to Z(L∨  I F )◦W F . We consider this as an object on the Galois side of the local Langlands correspondence and we write X nr ( L L) := (ZL∨ I F )◦W F .

(45)

Local Langlands and Springer Correspondences

27

Given (ϕ, ) ∈ e ( L L), and ξ ∈ X nr ( L L), we define (ξ · ϕ, ) ∈ e ( L L) by ξ · ϕ := ϕ on I F × SL2 (C) and (ξ · ϕ)(Fr F ) := ξ˜ ϕ(Fr F ), where ξ˜ ∈ Z◦L ∨ I F represents ξ . Definition 6.9 We denote by s∨ the G∨ -conjugacy class of ( L L , X nr ( L L) · (ϕ, ε)), where L is a Levi subgroup of G, and (ϕ, ε) is a cuspidal enhanced Langlands parameter for L L. We write s∨ := s∨G = [ L L , (ϕ, ε)]G ∨

(46)

and we call s∨ an inertial class for e ( L G). We denote by B∨ ( L G) the set of such s∨ .

6.2 The Cuspidal Support Map for Enhanced Langlands Parameters Let (φ, ρ) ∈ ( L G). We apply the constructions of Sect. 4.2 with G the group Gφ defined in (18). We set [Mφ , (O0 , ε)]Gφ := Gφ (u φ , ρ) ∈ B(Ue (Gφ )),

(47)

where the map Gφ is that of Theorem 4.19. Hence Mφ is a quasi-Levi subgroup of Gφ and (O0 , ε) is a cuspidal enhanced unipotent class in Mφ . For simplicity, we will often refer to the unipotent class O0 in Mφ by a unipotent element v in it. Let Z(Mφ ) denote the center of Mφ . The torus Z(Mφ )◦ commutes with Mφ , so ZG∨ (Z(Mφ )◦ ) is a Levi subgroup of G∨ which contains the image of Mφ in G∨ . 1 ◦ As Z(Mφ )◦ ⊂ ZG ∨ (φ| W F ), the subgroup ZG∨ W F (Z(Mφ ) ) is a Levi L-subgroup sc ∨ of G  W F . As proved in [15, Proposition 7.3], upon replacing (φ, ε) by a G∨ conjugate, there exists a Levi subgroup L of G such that (φ|W F , v, ε) is a cuspidal enhanced Langlands parameter for L, and L

◦ L := L∨  W F = ZL G (ZM ). φ

(48)

Then ( L L , φ|W F , v, ε) is a G-relevant cuspidal datum for L G. Let L  : e ( L G) → B∨ ( L G) be the map defined by L

(φ, ρ) := [ L L , (φ|W F , v, ε)]G∨ .

(49)

Let (φ, ρ) be an enhanced Langlands parameter for G and write [ L L , (φ|W F , v, ε)]G∨ = (φ, ρ). Up to Gφ -conjugacy there exists a unique algebraic group homomorphism γv : SL2 (C) → M◦φ such that L

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  • γv 01 11 = v,   0    z −1 0  γv 0 z for all z ∈ C× , • γv (SL2 (C)) commutes with φ 1, 0z z −1 and the cocharacter

  0   −1  χφ,v : z → φ 1, 0z z −1 · γv z 0 0z

has image in Z(Mφ )◦ , see [15, lemma 7.12]. Let · : W F → R>0 be the group homomorphism with w = q if w( f ) = f q for all f in the algebraic closure of the residue field of F. We define a Langlands parameter φv : W F → ZG∨ W F (Z(Mφ )◦ ) by φv (w, x) := φ(w) · χφ,v (w1/2 ) · γv (x),

where w ∈ W F and x ∈ SL2 (C).

Let L be as in (48). Then ( L L , (φv |W F , v, ε)) is also G-relevant cuspidal datum for L G. Definition 6.10 The cuspidal support of (φ, ρ) is Sc(φ, ρ) := [ L L , φv |W F , v, ε]G∨ ∈ B∨ ( L G) and the map Sc : e ( L G) → B∨ ( L G) is called the cuspidal support map for e ( L G). ∨

Given s∨ ∈ B∨ ( L G), we define e ( L G)s to the fiber of s∨ under the map Sc. We proved in [15] that e ( L G) decomposes as e ( L G) =



∨

e ( L G)s .

(50)

s∨ ∈B∨ ( L G) ∨

In this sense, we may consider e ( L G)s as Bernstein series in the space of enhanced Langlands parameters for L G. Moreover, for a fixed s ∈ B∨ ( L G), the series ∨ e ( L G)s is in bijection with the simple modules of an extended affine Hecke algebra H(s∨ , z) (see [16, 17]).

6.3 A Generalized Springer Correspondence for Enhanced L-Parameters Let s∨ = [ L L , (ϕ, ε)]G∨ ∈ B∨ ( L G). We associate to s∨ the finite group Ws∨ := stabilizer of s∨L∨ in NG∨ ( L L)/L∨ . ∨

(51)

Let (φ, ρ) ∈ e ( L G)s and write as above Gφ (u φ , ρ) =: [Mφ , v, ε]Gφ . Fix a Grelevant datum cuspidal datum ( L L , ϕ, ε) and set v := u ϕ ∈ L L. We write

Local Langlands and Springer Correspondences

29

tφ := [Gφ ∩ Lc∨ , v, ε]Gφ , t◦φ := [Gφ◦ ∩ Lc∨ , v, ε0 ]Gφ◦ .

(52)

The next result may be viewed as a version of the generalized Springer correspondence for enhanced L-parameters instead of enhanced unipotent classes. Proposition 6.11 [15, Proposition 9.1] (a) There is a bijection L

tφ : L  −1 ( L L , φ|W F , v, ε) ←→ Irr(C[Wtφ , κtφ ]) (φ, ρ) → tφ (u φ , ρ) (τ )) → τ, (φ|W F , t−1 φ

where tφ is the bijection defined in Notation 4.20. (b) The canonical bijection t◦φ between G−1◦ (t◦φ ) ⊂ Ue (G ◦ ) and Irr C (Wt◦ ) relates to part (a) by  L tφ (φ, ρ)|Wt◦ = t◦φ (u φ , ρi ), φ

i



where ρ = i ρi is a decomposition into irreducible AGφ◦ (u φ )-subrepresentations. (c) The G∨ -conjugacy class of (φ|W F , u φ , ρi ) is determined by any irreducible C[Wt◦φ ]-subrepresentation of L tφ (φ, ρ).

7 A Galois Version of the ABPS Conjecture 7.1 Twisted Extended Quotients Let  be a group acting on a topological space X , and let be a given function which assigns to each x ∈ X a 2-cocycle x

: x × x → C× , where x = {γ ∈  : γ x = x}.

We assume that γ x and γ∗ x define the same class in H 2 (γ x , C× ), where γ∗ : x → γ x sends α to γ αγ −1 . Let C[x , x ] be the corresponding twisted algebra as defined in Sect. 4.3. We set

X := {(x, E) : x ∈ X, E ∈ Irr C[x ,

x ]}.

and we topologize it by decreeing that a subset of

X is open if and only if its projection to the first coordinate is open in X . We require, for every (γ , x) ∈  × X , a definite algebra isomorphism φγ ,x : C[x ,

x]

→ C[γ x ,

γx]

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such that: • if γ x = x, then φγ ,x is conjugation by an element of C[x , • φγ  ,γ x ◦ φγ ,x = φγ  γ ,x for all γ  , γ ∈ , x ∈ X .

x]

×

;

Then we can define a -action on

X by γ · (x, E) := (γ x, E ◦ φγ−1 ,x ). We form the twisted extended quotient X / . (X// ) :=

Remark 7.1 Furthermore, we note that (X// ) reduces to the extended quotient of the second kind (X// )2 from [11, § 2] if x is trivial for all x ∈ X and φγ ,x is conjugation by γ . The extended quotient of the second kind is an extension of the ordinary quotient in the sense that it keeps track of the duals of the isotropy groups. Namely, in (X// )2 every point x ∈ X/  has been replaced by the set Irr(x ).

7.2 The Bernstein Decomposition of the Category of Smooth Representations Let Rep(G) denote the category of smooth representations of G. For P a parabolic subgroup of G with Levi factor L, we write IGP for the functor of normalized parabolic induction from Rep(L) to Rep(G). Notation 7.2 Let Irr cusp (L) be the set of isomorphism classes of irreducible supercuspidal representations in Rep(L). Let σ ∈ Irr cusp (L). We call (L , σ ) a cuspidal pair, and we consider such pairs up to inertial equivalence. This is the equivalence relation generated by: • unramified twists, (L , σ ) ∼ (L , σ ⊗ χ ) for χ ∈ X nr (L), where X nr (L) is the group of unramified (not necessarily unitary) characters L → C× ; • G-conjugation, (L , σ ) ∼ (gLg −1 , g · σ ) for g ∈ G. We denote a typical inertial equivalence class by s = [L , σ ]G . In particular, s L := [L , σ ] L = {σ ⊗ χ ∈ Irr(L) : χ ∈ X nr (L)}. Bernstein attached to every s a block in the category Rep(G) in the following way. First, if π is an irreducible object in Rep(G), then there is a cuspidal pair (L , σ ) such that π is a composition factor of IGP (σ ), for any parabolic subgroup P with Levi factor L. The conjugacy class of the pair (L , σ ) is uniquely determined by π . It is

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called the supercuspidal support of π . Thus the corresponding inertial class [L , σ ]G is also uniquely determined by π . It is the inertial support of π . We now define Irr(G)s = {π ∈ Irr(G) : π has inertial support s}, Rep(G)s = {π ∈ Rep(G) : every irreducible subquotient of π belongs to Irr(G)s }. We denote the set of all inertial equivalence classes for G by B(G). Theorem 7.3 [18, Proposition 2.10] The category Rep(G) decomposes as Rep(G) =



Rep(G)s .

s∈B(G)

In particular, the space of irreducible G-representations decomposes as the disjoint union Irr(G) = Irr(G)s . s∈B(G)

The group

X nr (L , σ ) := {χ ∈ X nr (L) : σ ⊗ χ ∼ = σ}

is finite. Thus there is a bijection X nr (L)/ X nr (L , σ ) → Irr(L)sL : χ → σ ⊗ χ ,

(53)

which endows Irr(L)sL with the structure of a complex torus. Up to isomorphism, this torus depends only on s, and it is known as the Bernstein torus Ts attached to s. We note that Ts is only an algebraic variety, it is not endowed with a natural multiplication map. In fact, it does not even possess an unambigous “unit”, because in general there is no preferred choice of an element σ ∈ s L . The group W (G, L) := NG (L)/L acts on Irr(L) by ¯ w · π = [w¯ · π : l → π(w¯ −1l w)]

for any lift w¯ ∈ NG (L) of w ∈ W (G, L). (54) Bernstein also associated with s the finite group Ws := {w ∈ W (G, L) : w · Irr(L)sL = Irr(L)sL }.

(55)

It acts naturally on Ts , by automorphisms of algebraic varieties. Closely related to the Bernstein decomposition is the theory of the Bernstein center. By [18, Théorème 2.13], the categorical centre of the Bernstein block Reps (G) is (56) Z(Rep(G)s ) ∼ = O(Ts )Ws = O(Ts /Ws ),

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where O denotes the ring of the regular functions on the given affine variety. Then taking supercuspidal support gives a map sc : Irr(G)s → Ts /Ws ,

(57)

which is surjective and has finite fibers [18, § 3]. Via the map sc (called the cuspidal support map), Irr s (G) can be regarded as a non-separated algebraic variety lying over Ts /Ws .

7.3 The ABPS Conjecture Let s = [L , σ ]G ∈ B(G). Let Ws,t be the stabilizer in Ws of a point t ∈ Ts . The ABPS Conjecture from [7, § 15] and [12, Conjecture 2] in its roughest form asserts that there exists a family of 2-cocycles t

and a bijection

: Ws,t × Ws,t → C×

t ∈ Ts ,

Irr(G)s ←→ (Ts //Ws )

(58)

such that: • it restricts to a bijection between tempered representations and the unitary part of the extended quotient (as explained below); • it is canonical up to permutations within L-packets, that is, for any φ ∈ (G), the image of φ (G) ∩ Irr s (G) is canonically defined (assuming a LLC for G exists). The set Irr cusp (L) is stable under the W (G, L)-action (54). The definitions of Ws and of extended quotients imply that for a fixed Levi subgroup L of G there is a natural bijection

  (Ts //Ws ) → Irr cusp (L)//W (G, L) .

(59)

s=[L ,σ ]G

Let Lev(G) be a set of representatives for the conjugacy classes of Levi subgroups of G. In view of Theorem 7.3, the ABPS Conjecture can also be formulated in terms of a bijection   Irr cusp (L)//W (G, L) . Irr(G) ←→ (60) L∈Lev(G)

In this version, the conjecture asserts that Irr(G) is determined by a much smaller set of data, namely the supercuspidal representations of Levi subgroups L of G, and the actions of the Weyl groups W (G, L) on those.

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It is expected that the group cohomology classes t ∈ H 2 (Ws,t , C× ) reflect the character of Z(G ∨sc )W F which via the Kottwitz isomorphism determines how G is an inner twist of a quasi-split group. In particular should be trivial whenever G is quasi-split. The simplest known example of a nontrivial cocycle involves a nonsplit inner form of SL10 (F) [10, Example 5.5]. That example also shows that it is sometimes necessary to use twisted extended quotients in the ABPS Conjecture (see [13]).

7.4 A Version for Enhanced L-Parameters The main goal of this section is to state an analog of (58) and (59) for enhanced Langlands parameters. Let L ∈ Lev(G). We write e,ζG (G, L) := {(φ, ρ) ∈ e,ζG (G) : Sc(φ, ρ) ∈ cusp (L)}, where e,ζG (G) was defined in (23). Let ϕ : W F → L L be a Langlands parameter for L. Let RϕL the analog for L of the group defined in (12), that is, RϕL := π0 (ZL∨ (ϕ)/Z(L∨ )W F ). There exists a natural ∨ )Z(L ∨c )◦ injection RϕL → Rϕ . We extend the character ζG to a character of Z(Gsc L ∨ ◦ which is trivial on Z(L c ) , and denote by ζG the restriction of the latter character to ∨ Z(Lsc ). Then we write cusp,ζGL (L) := e,ζGL (L) ∩ cusp (L). The following results are proved in [53, Theorem 3.3] for G a split classical group, and in [15, Theorem 9.3] for general G. Theorem 7.4 [15, Theorem 9.3] (a) Let s∨L = [ L L , φ|W F , v, ε]L L be n G-relevant inertial equivalence class for the Levi L-subgroup L L of L G. The maps L t from Proposition 6.11.(a) combine to a bijection   ∨ ∨ e ( L G)s ←→  e ( L L)sL //Ws∨ κ  L (φ, ρ) (φ, ρ), t (u φ , ρ)  →  −1 ( L L , φv , ε, τ ). φv |W F , t (τ ) → (b) The bijection from part (a) has notably the following properties: • It preserves boundedness of (enhanced) L-parameters. • The restriction of τ to Wt◦ canonically determines the (non-enhanced) Lparameter in L qt (τ ).

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(c) The bijections from part (a) give a bijection   e,ζG (G, L) ←→ cusp,ζGL (L)//W (G, L) κ . (d) The maps from part (c) combine to a bijection

e,ζG (G) ←→

  cusp,ζGL (L)//W (G, L) κ .

L∈Lev(G) ∨ (e) Assume that Z(Lsc ) is fixed by W F for every L ∈ Lev(G) (for instance, G is an inner twist of a split group). The union of part (d) for all such ϑ ∈ H 1 (F, G ad ) ∼ =   ∨ WF is a bijection Irr C Z(Gsc )

e ( L G) ←→





ϑ∈H 1 (F,G ad ) L ϑ ∈Lev(G ϑ )

  cusp (L ϑ )//W (G ϑ , L ϑ ) κ .

7.5 A Conjectural Diagram The above result leads to the following conjecture stated in [15]: Conjecture 7.5 There is a commutative diagram of bijections e,ζG (G)

Irr(G) 

 L∈Lev(G)

 Irr cusp (L)//W (G, L) κ

 L∈Lev(G)

  cusp,ζG (L)//W (G, L) κ

in which the arrows are given as follows: • the right-hand side is Theorem 7.4.(d), • the upper horizontal map is a local Langlands correspondence for G, • the lower horizontal map is obtained from local Langlands correspondences for Irr cusp (L) by applying (·//W (G, L))κ , • the left-hand side is the conjectural bijection (60). With this conjecture, one can reduce the problem of finding a LLC for G to that of finding local Langlands correspondences for supercuspidal representations of its Levi subgroups. Conjecture 7.5 is currently known in the following cases: • • • •

inner forms of GLn (F) [9, Theorem 5.3], inner forms of SLn (F) [9, Theorem 5.6], split classical groups [52, § 5.3], when the characteristic of F is 0, principal series representations of split groups [11, § 16].

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Arithmetic of Cuspidal Representations Colin J. Bushnell

Contents 1

2

3

4

5

6

7

Basic Ideas and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Intertwining and Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Level Zero Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Hereditary Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Duality and Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 A Fundamental Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Adjoint and Tame Corestriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Critical Exponent and Simple Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Structure of Simple Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Groups Defined by Simple Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Simple Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types and Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Maximal Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Inertial Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Encounter with the Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Inertial Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Local Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer, Endo-equivalence and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Transfer of Simple Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Endo-Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conductors of Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tame Lifting and Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Tame Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Functoriality Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ramification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tame Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Structure of Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Tame Parameter Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Comparison of Tame Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 43 44 45 48 49 52 53 54 55 57 58 61 62 66 67 67 69 70 73 74 76 80 82 83 85 86 89 89 92 95

C. J. Bushnell (B) Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_2

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Description of the Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Algebraic Induction Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Naïve Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Higher Ramification and the Herbrand Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Ramification and Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Comparison of Ultrametrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Lifting and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Some Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Carayol Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Conformity and the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Galois Representations of Carayol Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Parameter Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96 96 98 100 103 105 105 109 111 113 113 114 116 117 119 123

Abstract Here, F is a non-Archimedean local field of residual characteristic p. We are concerned with the irreducible, cuspidal representations of the general linear groups GL(n, F). A complete classification of these representations has been known for a long time. It is achieved using rather complicated objects, the simple types and simple characters. The methods it requires have been useful more widely, and the general scheme is now known to apply to many more groups, including GL(m, D) (where D is a central F-division algebra), orthogonal groups SO(n), symplectic groups Sp(2n), (both for p not equal to 2) and even a couple of exceptional groups. In some cases, it is known that the common classification conforms to the requirements of Functoriality. The most interesting, and presently the most difficult, instance of Functoriality is the basic connection between the irreducible cuspidal representations of GL(n, F) and the irreducible, n-dimensional representations of the Weil group of F. These notes describe the classification of the cuspidal representations, introducing the results and techniques currently necessary for making this connection more explicit, given that it is known to exist. Let F be a non-Archimedean local field of residual characteristic p. Let WF = WF/F ¯ ¯ be the Weil group of F, formed relative to a chosen separable algebraic closure F/F,  F be the set of equivalence classes of irreducible representations1 of WF . For and let W cuspidal an integer n  1, let A0n (F) be the set of equivalence classes of irreducible   F = n1 A0n (F). The representations of the general linear group GLn (F) and set GL Langlands correspondence thus provides a canonical bijection F,  F −→ W GL π −→ Lπ.

1 When

speaking of a representation, I invariably mean a smooth complex representation.

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 F are classified in terms of explicit data. Following [25], the representations π ∈ GL In these notes, I explore the manner in which this classification translates, via the Langlands correspondence, into information concerning the structure of representations of WF . Ideally, one would like to be able to read off Lπ directly from π and the other way round. I report on the progress made towards this goal in a long-running joint project with Guy Henniart. The notes are effectively in two halves. The first, occupying Sects. 1–3, extracts the necessary background from [25]. The important part for us is the procedure for constructing cuspidal representations, particularly the definition and properties of simple characters. These are complicated objects, and one has to be able to manipulate them freely. On the other hand, the “Exhaustion Theorem”, asserting that the construction yields all cuspidal representations, can be taken as read. The amount of detail provided is therefore heavily skewed to the construction side. However, the strategy of proof followed in [25] dictates the shape of the construction, particularly the theory of simple characters. The basic idea is traceable, in various parts, to Howe [43], Kutzko [50, 51] and Carayol [30], along with [7], but the effective starting point is the combination of [6, 54]. That leads directly to an iterative problem of which simple characters, and then simple types, are the solution. I have summarized this train of thought in Sect. 1 and illustrated it with key examples. The main material of the notes starts with Sect. 4. In the introductory first section, I list properties of the Langlands correspondence that can be interpreted in terms of the classification theory. I examine them successively in Sects. 5–8. To summarize the situation at the end of Sect. 8, let π ∈ A0n (F). According to [17, 25], π contains a simple character θπ , and only one (up to conjugation in G = GLn (F)). The G-normalizer of θπ is an open, compact mod. centre subgroup J of G, the natural representation Λπ of J on the θπ -isotypic subspace of π is irreducible, and π ∼ = c-IndJG Λπ . (The pair (J, Λπ ) is an extended maximal simple type in G.) The  group J has a unique maximal compact subgroup J , and the restriction λπ = Λπ  J F. is irreducible: the pair (J , λπ ) is a maximal simple type in G. Write σ = Lπ ∈ W (1) The conjugacy class of θπ determines the restriction of σ to the wild inertia subgroup of WF , and vice versa, but we have little idea of how this process works. (2) The conjugacy class of the maximal simple type λπ determines the restriction of σ to the inertia subgroup of WF and vice versa. The two are connected by an explicit formula. (3) Starting from σ, one can define π  ∈ A0n (F) by an explicit formula and show that π  is almost (in a rather precise sense) the same as π. In (3), the representation π  is a sort of tamely ramified, internal twist of π, much as in the essentially tame case of [13, 14, 16]. In the general case, we have only partial control of the twisting character or “rectifier”, to use the language of those papers. In Sects. 9 and 10, I give a framework for analysing the relation between simple characters and ramification. I treat the first step of what will have to be an iterative process, mirroring the structure of simple characters.

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Throughout, I give very few proofs. Those in the first three sections are included for purely illustrative purposes. Particularly at the beginning, I concentrate on producing a coherent narrative of the concepts and results that contribute directly to the main purpose. I pay scant attention to the expansive logical structures and background necessary for their proofs. Later on, I tend to follow the literature more closely but proofs remain rare. The main results in Sect. 8 depend on comparing an explicit algebraic induction process with automorphic induction, via some quite elaborate character calculations. I decided to omit them completely, as they do not condense readily. Since I’ve chosen to keep the focus very narrow, it is difficult to say much about any broader context. That context exhibits a coherence quite unexpected at the beginning. The entire scheme of [25] can be transported systematically to an inner form GLm (D) of GLn (F), after a first step in [3]. As reported in Vincent Sécherre’s conference talk, we are (at least) very close to knowing that this transfer implements the Jacquet–Langlands correspondence [68]. Stevens [73] has shown that the cuspidal representations of a classical group can be classified in a scheme transferred from that of [25], although the extra complications are considerable. In a recent development [2], it has emerged that the explicit relation between general linear groups and symplectic groups, implemented by transfer of simple characters, is that demanded by Functoriality. Prerequisites I assume some familiarity with the smooth complex representation theory of locally profinite groups, particularly GLn (F). The requirements here are modest and mostly gathered together, in adequate generality, in [15]. I use standard notation for objects associated with the field F. Thus oF is the discrete valuation ring in F, while pF is the maximal ideal of oF . The residue field kF = oF /pF is a finite field of q = qF elements and characteristic p. Also, UF = o× F is the unit group and, for k  1, UFk = 1 + pkF . Finally, υF : F ×  Z is the normalized valuation. On occasion, μF will be the group of roots of unity in F of order not divisible by p. By convention, all representations to be considered are smooth and complex.

1 Basic Ideas and Examples This section outlines first steps towards the systematic analysis of irreducible cuspidal representations of the group G = GLn (F), along with a couple of examples indicative of the scope of the problem. Recall that “representation” means “smooth complex representation”.

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1.1 Intertwining and Induction For this section only, G may denote the group G(F) of F-points of a connected reductive algebraic group G defined over F: we say that “G is a connected reductive F-group”. We view G as endowed with the locally profinite topology inherited from F. The basic material of this section is all covered in [15]. 1.1.1 Let Z be the centre of G. Let K be an open subgroup of G that is either compact or compact modulo Z. If ρ is an irreducible representation of K and π one of G, say that π contains ρ if the spaces HomK (ρ, π) ∼ = HomK (π, ρ) are nonzero. For i = 1, 2, let Ki be such an open subgroup of G and let ρi be an irreducible representation of Ki . An element g of G intertwines ρ1 with ρ2 if g

HomK1g ∩K2 (ρ1 , ρ2 ) = 0. g

g

Here, ρ1 is the representation x → ρ1 (gxg −1 ) of the group K1 = g −1 K1 g. Observe g  g g that the representations ρ1 K1 ∩ K2 and ρ2  K1 ∩ K2 are semisimple. Write IG (ρ1 , ρ2 ) for the set of g ∈ G that intertwine ρ1 with ρ2 . Thus IG (ρ1 , ρ2 ) is either empty or a union of double cosets K1 xK2 . We abbreviate IG (ρ1 , ρ1 ) = IG (ρ1 ) and say that an element of IG (ρ1 ) intertwines ρ1 . The set IG (ρ1 ) is certainly non-empty, and it is the support of the Hecke algebra H(G, ρ1 ). That function algebra is canonically identified with the algebra of Gendomorphisms of the smooth, compactly induced representation c-IndKG1 ρ1 . If there exists an irreducible smooth representation containing both ρ1 and ρ2 , then IG (ρ1 , ρ2 ) is non-empty. However, IG (ρ1 , ρ2 ) being non-empty does not imply that an irreducible representation of G containing ρ1 also contains ρ2 . 1.1.2 Our overall strategy is based on the following fact. Proposition Let K be an open subgroup of G that is compact modulo the centre of G. Let ρ be an irreducible representation of K. If IG (ρ) = K, then (1) the compactly induced representation π = c-IndKG ρ is irreducible and cuspidal; (2) the space HomK (ρ, π) has dimension one. The intertwining hypothesis implies that the C-algebra End G (π) has dimension one. The proof then requires the fact that an irreducible smooth representation of G is admissible. From there, it follows the case of G = GL2 (F) in [15] 11.4. The proposition provides the only method available for producing families of cuspidal representations in any degree of generality. Currently, it is known to yield all the cuspidal representations of G when G is GLn (F) [25], SLn (F) [26, 27], an inner form GLm (D) of GLn (F) [67], or a classical group when the residual characteristic p of F is not 2 [73]. If G is an arbitrary connected reductive F-group G, Yu [77] uses the technique to construct a series of “tame” cuspidal representations of G. If p is sufficiently large, in a sense depending on G, Kim [48] shows that Yu’s tame series comprises all the irreducible cuspidal representations of G.

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In all of these cases, the argument proceeds by systematically constructing the inducing datum (K, ρ): a mere existence theorem would be much less interesting. For arbitrary G, I am unaware of any compelling reason for even believing in an existence theorem.

1.2 Level Zero Representations We revert to the case G = GLn (F) to recall an example from an extreme end of the spectrum. The main features of the general theory are discernible here, albeit in an attenuated form. 1.2.1 In the group G = GLn (F), take the standard maximal compact subgroup K = GLn (oF ) and its first congruence subgroup K 1 = 1 + pF Mn (oF ). Identify K/K 1 with GLn (kF ). Let Z be the centre of G: thus Z is the group of scalar matrices in G and Z∼ = F × . The group KZ is compact modulo Z and is the G-normalizer of K. Let ρ be an irreducible representation of K that is trivial on K 1 . Thus ρ is the inflation of an irreducible representation ρ¯ of GLn (kF ). Say that ρ is residually cuspidal if ρ¯ is cuspidal in the usual sense. Otherwise, say that ρ is residually split. Proposition (1) If ρ is residually cuspidal , then IG (ρ, ρ) = KZ. (2) If ρ¯ is residually cuspidal and if σ is the inflation to K of an irreducible repre¯ sentation σ¯ of GLn (kF ), then IG (ρ, σ) is non-empty if and only if σ¯ ∼ = ρ. The proof is a straightforward exercise, for which [15] 11.5 provides an adequate hint.  1.2.2 If R is an irreducible representation of KZ, then R  K is irreducible. Conversely, an irreducible representation of K may be extended to a representation of KZ.  Say that an irreducible representation R of KZ is of level zero if R  K 1 is trivial. If (π, V ) is an irreducible representation of G, say that (π, V ) is of level zero if the 1 space V K of K 1 -fixed vectors in V is nonzero. Proposition (Construction  Theorem) Let R be an irreducible representation of KZ, of level zero, such that R  K is residually cuspidal. The representation G πR = c-IndKZ R

is irreducible, cuspidal and of level zero. If R is a representation of KZ with the same properties as R, then πR ∼ = πR if and only if R ∼ = R. ∼ The first assertion here follows from the propositions of 1.1.2 and 1.2.1. If πR = πR , the representations R, R must intertwine in G and 1.2.1 Proposition implies that R∼ = R . Indeed, one can recover R from πR directly: it is the natural representation of KZ on the space of K 1 -fixed vectors in πR .

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Corollary Let π be an irreducible representation of G, of level zero. Suppose π contains an irreducible representation ρ of K that is trivial on K 1 and residually  cuspidal. The representation π is then cuspidal and of the form πR , where R  K ∼ = ρ. 1.2.3 The Construction Theorem of 1.2.2 gives an unambiguous classification of a certain family of irreducible, cuspidal, level zero representations of G in terms of explicit data. Exhaustion Theorem Let (π, V ) be an irreducible representation of G of level 1 zero. If the natural representation of K on V K has an irreducible component that is residually split, then π is not cuspidal. Consequently, all irreducible, level zero, cuspidal representations are given by 1.2.2. The structure of the cuspidal representations of GLn (k), when k is a finite field, is thoroughly understood, as in [34] (recalled in 8.1.1 below). So, this parametrization can reasonably claim to be explicit. The Exhaustion Theorem is much harder to prove and, in line with our policy in these notes, we say no more if it. In [25], it is treated in Sect. 8.3 as an instance of a more complex argument. The main ideas are already present in [44, 76]. Results 1.2.2 Proposition and 1.2.3 Theorem admit generalization to any connected reductive F-group [59, 61].

1.3 Hereditary Orders To deal with cuspidal representations in general, we have to specify a canonical family of compact open subgroups of GLn (F). To do this, it is better to work with an F-vector space V of dimension n, along with the group G = Aut F (V ) and the algebra A = EndF (V ). For background on the hereditary orders that dominate this section, Chapter 9 of [63] and the early pages of [7] contain everything necessary in a moderate number of pages. 1.3.1 Let L be an oF -lattice in V . By definition, L is a finitely generated oF submodule of V that contains an F-basis of V . Equivalently, L is the oF -span of an F-basis of V . In particular, an oF -lattice in V is a compact open subgroup of V . An oF -order in A is a subring a of A that contains 1 and is an oF -lattice in the vector space A. There is a fundamental example. Let L be an oF -lattice in V and set m(L) = {x ∈ A : xL ⊂ L}. This is an oF -order in A, canonically identified with EndoF (L). Identifying V with F n via a basis of L, the order m(L) becomes Mn (oF ). If L is some other lattice in V , there exists g ∈ G such that L = gL, so m(L ) = gm(L)g −1 . Comparing Haar

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measures, we see that m(L ) ⊂ m(L) if and only if m(L ) = m(L). An elementary argument shows that m(L ) = m(L) if and only if L = xL, for some x ∈ F × . Orders of the form m(L) are called the maximal orders in A: there is no non-trivial containment relation between them and, if a is an oF -order in A, there exists an a-stable lattice L in V , whence a ⊂ m(L). 1.3.2 An oF -lattice chain in V is a non-empty set L of oF -lattices in V that is linearly ordered under inclusion and satisfies xL ∈ L for any L ∈ L and x ∈ F × . One can enumerate the elements of L, say L = {Li : i ∈ Z}, so that Li  Li+1 , for all i. In particular, if L ∈ L, then pF L ∈ L and there is a unique integer e = eF (L) such that pF Li = Li+e , for all i. This is the F-period of L. Note that 1  e  n. 1.3.3 There is a list of rings and groups associated with the lattice chain L. Let a = aF (L) = {x ∈ A : xL ⊂ L, L ∈ L}. This is surely an oF -order in A. An order of this form is called a hereditary order. The order a and the lattice chain L determine each other. Indeed, L is the set of all a-lattices in V . If we impose an enumeration {Li }i∈Z on L (as in 1.3.2) and take j ∈ Z, we may set

j

aF (L) = {x ∈ A : xLi ⊂ Li+j , i ∈ Z}. Thus a = a0F (L). The lattice a1F (L) is an ideal of a, in fact the Jacobson radical j rad a of a. Abbreviating a1F (L) = p, one has aF (L) = pj , j ∈ Z: this makes sense for negative j, since p is an invertible fractional ideal of a (one of the definitions of “hereditary”). The period e = e(a) = eF (L) appears in the relation pF a = pe . The unit group Ua = a× is a compact open subgroup of G = Aut F (V ). It has a canonical series of open normal subgroups Uak = 1 + pk , k  1. One also sets Ka = Aut L = {g ∈ G : gL = L} = {g ∈ G : gag −1 = a}. Thus Ka is open in G and compact modulo the centre Z of G. It normalizes each of the groups Ua and Uak , k  1. It is indeed the G-normalizer of any one of them, cf. [7] 1.3.10. In the same vein, one can recover the order a from any one of the groups Ua , Uak .

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Remark Fix an oF -lattice L in V . The set of lattice chains L in V , such that L ∈ L, is in canonical bijection with the set of flags of subspaces of the n-dimensional kF space L/pF L. This leads to a useful block matrix description of hereditary orders and their associated groups: see [25] (2.5.1) for some pictures. 1.3.4 There is a more general context. In the language of [5, 75], these groups Ua are the parahoric subgroups of G and {Uak }k1 is the standard filtration of Ua . It is often necessary to use other filtrations given by lattice sequences in V . A lattice sequence is a map L : i → Li , from Z to the set of oF -lattices in V , such that Li ⊇ Li+1 , and with the periodicity property pF Li = Li+e , for an integer e = e(L)  1. (Note the non-strict containment here.) One defines aj (L) = {x ∈ A : xLi ⊂ Li+j } as before. The image of L is a lattice chain L, and clearly a0 (L) = a(L). However, the groups 1 + aj (L), j  1, provide another filtration of Ua(L) . These non-standard filtrations are instances of the very general filtrations of [60]. The filtrations derived from lattice sequences are, in many cases, known to be the Moy–Prasad filtrations with rational jumps [4, 55]. Lattice sequences are necessary for the systematic analysis of general representations of GLn (F) [29]. They become more important in inner forms of GLn (F) [64–66] and are central to the analysis of classical groups [72]. 1.3.5 We return to the main theme. There are some useful incidence relations. For example, a lattice chain L contains exactly eF (L) lattice chains L with eF (L ) = 1. It is contained in a finite number of lattice chains L satisfying eF (L ) = n (cf. 1.3.3 Remark). Using the obvious notation for the associated orders and their radicals, we have a ⊃ a ⊃ a and p ⊂ p ⊂ p . 1.3.6 There is a special case we need to mention. Say that the lattice chain L = {Li } is uniform if the index (Li :Li+1 ) is independent of i. Equivalently, there exists g ∈ G such that gLi = Li+1 for all i. Any such element g satisfies ga = ag = p, so one says that the associated hereditary order a = aF (L) is principal. In this case, the group Ka is generated by g and Ua . The groups Ka , for principal orders a, are the maximal compact modulo centre subgroups of G, [7] (1.3.2) Theorem. 1.3.7 A certain relative structure arises repeatedly. Let E/F be a finite field extension, and suppose that V is given as an E-vector space. Set B = EndE (V ). If A = EndF (V ) as before, then E is an F-subalgebra of A while B is the centralizer of E in A. Let L be an oE -lattice chain in V and form the hereditary oE -order b = aE (L) in B. The set L is also an oF -lattice chain in V and, if a = aF (L), then b = a ∩ B. Indeed, if p = rad a and q = rad b, then qj = pj ∩ B, j ∈ Z. We have the relation

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eF (a) = eE (b) e(E|F). As E × ⊂ Ka , so E × normalizes a and also the groups Ua , Uak . These properties are reversible. Suppose we are given an oF -lattice chain L in V and a subfield E/F of A. Say that a = aF (L) is E-pure if E × ⊂ Ka . This is equivalent to L being an oE -lattice chain and we are in the previous situation. Example Let E/F be a finite field extension. View E as an F-subalgebra of AE/F = EndF (E). There is then a unique E-pure, hereditary oF -order aE/F in AE/F : it is the j one given by the lattice chain {pE : j ∈ Z}.

1.4 Duality and Strata We keep the notation of Sect. 1.3. We examine a class of characters of the groups Ua1+k , k  0, where a is a hereditary oF -order in A. To do this, we choose a smooth character ψF of F that is of level one, meaning that ψF is non-trivial on oF but trivial on pF . This condition serves mainly to keep the bookkeeping simple: changing it has only trivial consequences. Details of everything in this section can be found in [6, 7, 25]. 1.4.1 Let tr A : A → F be the matrix trace map and set ψA = ψF ◦ tr A . Thus ψA is a non-trivial smooth character of A. Write  A for the Pontrjagin dual of A. Lemma If a, x ∈ A, write ψA ◦ a(x) = ψA (ax). The map A −→  A, a −→ ψA ◦ a, is an isomorphism of topological groups. If L is an oF -lattice in A, the set L⊥ = {a ∈ A : ψA (aL) = 1} is an oF -lattice in A, and ψA induces a topological isomorphism of A/L⊥ with the Pontrjagin dual  L of L. Proposition If a is a hereditary oF -order in A with rad a = p, then  1+m ⊥ p = p−m , m ∈ Z. 1.4.2 For a ∈ A, define a function ψA ∗ a on A by ψA ∗ a(1 + x) = ψA (ax), and take a, p as in 1.4.1 Proposition.

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Proposition Suppose 0  j < k  2j + 1. The isomorphism p1+j /p1+k −→ Ua1+j /Ua1+k , x −→ 1 + x, induces an isomorphism   p−k /p−j −→ Ua1+j /Ua1+k ,  a −→ ψA ∗ a  U 1+j . a

This follows directly from 1.4.1 Proposition. 1.4.3 There is a convenient administrative device for handling 1.4.2 Proposition. Definition A stratum in A is a quadruple s = [a, m, l, a] in which a is a hereditary oF -order in A, with p = rad a, while m > l are integers and a ∈ p−m . A stratum [a, m, l, a ] is deemed equivalent to s if a ≡ a (mod p−l ). If 0  l < m  2l + 1, the equivalence class of the stratum s and the character  χs = ψA ∗ a  Ua1+l determine each other uniquely (by 1.4.2 Proposition). For i = 1, 2, let si = [ai , mi , li , ai ] be a stratum in A and write pi = rad ai . An element g ∈ G is said to formally intertwine s1 with s2 if −l2 1 g −1 (a1 + p−l 1 )g ∩ (a2 + p2 )  = ∅

Proposition Suppose that 0  li < mi  2li + 1, i = 1, 2. An element g of G formally intertwines s1 with s2 if and only if it intertwines χs1 with χs2 . Formal intertwining of strata often provides a practical method for computing intertwining of the associated characters.

1.5 A Fundamental Dichotomy We use the machinery of Sects. 1.3 and 1.4 in a preliminary analysis of the irreducible representations of G = Aut F (V ). 1.5.1 Here, we consider only strata of the form s = [a, m, m − 1, a], m  1. Let rad a = p and set (s) = m/eF (a). Let π be an irreducible representation of G. Define the level (π) of π as the minimum value of m/ea , where a ranges over the hereditary oF -orders in A = EndF (V )

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and m over the non-negative integers such that π contains the trivial character of Ua1+m . Thus (π) = 0 in this sense if and only if π is of level zero as in 1.2.2. Say that π contains the stratum s = [a, m, m − 1, a] if m  1 and π contains the character χs = ψA ∗ a  Uam . Definition The stratum s = [a, m, m − 1, a] is fundamental if the coset a + p1−m contains no nilpotent element of A. Equivalently, s is fundamental if and only if at ∈ / p1−tm for any t  1 [6]. Theorem Let π be an irreducible representation of G such that (π) > 0. Let s = [a, m, m − 1, a] be a stratum contained in π. The following conditions are equivalent. (1) (s) = (π). (2) s is fundamental. If (π) = 0, the π contains no fundamental stratum a with (s) > 0. The result nominates a special family of characters of certain compact open subgroups of G, at least one of which is guaranteed to occur in a given representation π of positive level. Comments The theorem, in this form, for GLn (F) is proved in [6]. In a far-reaching generalization, Moy and Prasad [60] prove an analogue for representations of any connected reductive F-group G. 1.5.2 Let s be a stratum [a, m, m − 1, a], m  1. Abbreviate e = eF (a), set g = gcd(m, e) and choose a prime element of F. The element a0 = ae/g m/g then lies in a, and the equivalence class of s determines the coset a0 + p. If a is given by the lattice chain {Li }i∈Z , then a/p ∼ =

e−1 

EndkF (Li /Li+1 ).

i=0

As an endomorphism of the vector space e−1 i=0 Li /Li+1 , the coset a0 + p has a characteristic polynomial, which is the reduction, modulo pF , of the characteristic polynomial of the F-endomorphism a0 ∈ A = EndF (V ). This polynomial, call it chs (t) ∈ kF [t], is determined by the equivalence class of s. Any monic polynomial of degree n = dimF V can arise here, but s is fundamental if and only if chs (t) = t n (1.5.1 Theorem). If s, s are fundamental strata occurring in the same irreducible representation of G, they must intertwine formally. We deduce: Proposition Let π be an irreducible representation of G with (π) > 0. If s, s are fundamental strata occurring in π, then (s) = (s ) = (π) and chs (t) = chs (t). Definition A fundamental stratum s is called split if chs (t) has at least two distinct irreducible factors. Otherwise, it is non-split.

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Theorem Let π be an irreducible representation of G with (π) > 0 containing a fundamental stratum s. If s is split, then π is not cuspidal. So, if π is cuspidal, it contains a non-split fundamental stratum. The proof of the theorem is an instance of the arguments of [25] 8.2 and is quite elaborate. Observe a parallel with 1.2.3 Theorem. 1.5.3 Since we are only concerned with cuspidal irreducible representations π, we restrict to the case in which π contains a non-split fundamental stratum. There is a generic example of such strata. Example Let E/F be a subfield of A. Let α ∈ E satisfy E = F[α] and ν = −υE (α) > 0. Let a be an E-pure hereditary oF -order in A with rad a = p. In particular, αa = p−m , where m = νe(a)/e(E|F). The quadruple [a, m, m − 1, α] is a non-split fundamental stratum in A. A non-split fundamental stratum need not be of this form. On the other hand, the example itself can be refined. Definition Let α ∈ G. Say that α is minimal over F if (1) the algebra E = F[α] is a field, (2) υ = υE (α) is relatively prime to e = e(E|F) and (3) if is a prime element of F, the coset αe −υ + pE generates the residue field extension kE /kF . A stratum of the form [a, m, m − 1, α] will be called simple if α is minimal over F, the hereditary order a is F[α]-pure and αa = p−m . Theorem 1 Let π be an irreducible representation of G such that (π) > 0. If π contains a non-split fundamental stratum, in particular if π is cuspidal, then it contains a simple stratum s = [a, m, m − 1, α]. Theorem 1 relies on matrix calculations carried through in section 2.5 of [25]. The proof has an important feature. If π contains the non-split fundamental stratum [a, m, m − 1, a], a simple stratum contained in π will take the form [a , m , m − 1, a ] for a possibly different hereditary order a . Indeed, the proof starts by taking a as small as possible. To analyse the irreducible representations π containing a fixed simple stratum s = [a, m, m − 1, α], we will need to know the formal intertwining of s. Theorem 2 Let s = [a, m, m − 1, α] be a simple stratum in A. Let B be the Acentralizer of α. An element g ∈ G intertwines s formally if and only if g ∈ Ua1 B× Ua1 . At this point, we give an example of how one can deduce properties of strata from information about their intertwining. Corollary Let [a, m, m − 1, α] be a simple stratum in A, and suppose that the field extension E = F[α]/F has degree n = dimF V . If β ∈ α + p1−m , then [a, m, m − 1, β] is simple, the field extension F[β]/F has degree n = dimF V and e(F[β]|F) = e(F[α]|F).

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Proof The A-centralizer of α is E itself, so the formal intertwining set of [a, m, m − 1, α] is E × Ua1 and this set is compact modulo centre. On the other hand, the formal intertwining set of the equivalent stratum [a, m, m − 1, β] contains Ua1 C × Ua1 , where C is the A-centralizer of β. We conclude that C × is compact modulo F × , whence [F[β] : F] = n and C = F[β]. Moreover, C × ⊂ E × Ua1 , so e(C|F)  e(E|F) and f (C|F)  f (E|F). Also, υF (det β) = υF (det α). The remaining assertions follow readily.   1.5.4 The strata of 1.5.3 Corollary provide a family of examples complementary to those in Sect. 1.2. This case was first observed by Carayol [30]. Theorem Let s = [a, m, m − 1, α] be a simple stratum in A, and suppose that E = F[α]/F is of degree n. An irreducible representation π of G that contains s is then cuspidal and of the form π = c-IndKG Λ, × 1 where  m K = E Ua and Λ is an irreducible representation of K that contains ψA ∗  α Ua .

This follows directly from 1.5.3  Theorem 2. Observe that any irreducible representation of K containing ψA ∗ α  Uam induces irreducibly to a cuspidal representation of G. Comment We begin to see the emergence of a pattern in the general case, echoing that of the level zero case in Sect. 1.2. We have two families of representations of certain compact open subgroups of G, namely the split fundamental strata and the simple strata. An irreducible representation π of G contains a member of one family, but not both. If it contains a split fundamental stratum, then it is not cuspidal. So, to isolate and analyse the cuspidal representations of G, we need to refine the concept of simple stratum and the associated character. This occupies the following two sections.

2 Simple Characters This section is an account of the simple characters in G = GLn (F). These constitute a canonical family of characters of compact open subgroups of G, so named because their intertwining features simple algebras. The concept of simple character is not amenable to direct definition. One starts from a simple stratum in A = Mn (F), in a sense more general than that of Sect. 1.5. A simple stratum s is a comparatively straightforward object attached to a hereditary order a in A. It defines an open subgroup of the unit group Ua1 and then, after choosing a character ψF as in Sect. 1.4, a finite family C(s, ψF ) of characters of that group: these are the simple characters defined by s. Simple characters are defined inductively, via explicit formulæ following the structure of s. This process involves many choices,

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but one shows them to be ultimately insignificant. The complexity of the process derives from the fact that one cannot recover s from the set C(s). In consequence, the evolution in Chapters 2 and 3 of [25] is indirect and convoluted, with several different ideas being developed in an interwoven pattern. Here, we pay no regard to this logical structure. We aim for a coherent narrative of the important definitions and basic results, occasionally informed by later papers. We have chosen results that give insight into the structure or, as is often the same, are subsequently re-used frequently. The aim is to minimize the need for a reader to return to the detailed arguments of [25]. Throughout this section, V will denote an F-vector space, of finite dimension n, with A = EndF (V ) and G = Aut F (V ).

2.1 Adjoint and Tame Corestriction We introduce a pair of maps A → A. To do this, we need a subfield E/F of A and an element β of G such that E = F[β]. Let B = EndE (V ) be the A-centralizer of E. 2.1.1 The standard adjoint map aβ : A −→ A, x −→ βx − xβ, provides a B, B-bimodule map with kernel B. 2.1.2 There is a canonical family of B, B-bimodule maps A → B that are, in a certain sense, complementary to the adjoint. The space HomB,B (A, B) of bimodule homomorphisms A → B is an E-vector space of dimension one. Any map f ∈ HomB,B (A, B) satisfies f (E) ⊂ E and f (A) = B, provided f = 0. Definition A tame corestriction, on A relative to E/F, is a B, B-bimodule homomorphism s : A → B such that, if a is an E-pure hereditary oF -order in A, then s(a) = a ∩ B. A tame corestriction exists and is unique up to multiplication by a unit of E. The proof proceeds via a duality property. As in Sect. 1.4, let ψF be a character of F of level one, and set ψA = ψF ◦ tr A . Likewise, let ψE be a character of E of level one and put ψB = ψE ◦ tr B . For a ∈ A, the map b → ψA (ab) is a character of B, so there is a unique element s(a) ∈ B such that ψA (ab) = ψB (s(a)b), b ∈ B. This map s has all required properties. The kernel of s is aβ (A), while its image is B = Ker aβ . Tame corestrictions have a finer property relative to hereditary orders.

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Proposition Let a be an E-pure hereditary order in A. Set b = a ∩ B, p = rad a and q = p ∩ B = rad b. If s is a tame corestriction on A, relative to E/F, then s(pr ) = qr , r ∈ Z. Example Suppose that E/F is separable. The non-degenerate, symmetric F-bilinear form (x, y) → tr A (xy) on A then restricts to a non-degenerate form on B. The orthogonal projection π : A → B is a B, B-homomorphism A → B, but it is a tame corestriction if and only if E/F is tamely ramified. Indeed, let d (E|F) be the exponent of the different of E/F and let w ∈ E have valuation d (E|F) − e(E|F) + 1. The map wπ is then a tame corestriction.  Of course, if E/F is not separable, then B ⊂ aβ (A) and tr A  B = 0. There is no projection with which to compare, but the tame corestriction exists nonetheless.

2.2 Critical Exponent and Simple Strata We have already mentioned examples of simple strata in Sect. 1.5. We now give the general definition. 2.2.1 We need a preliminary definition. Definition A stratum [a, m, r, a] in A is pure if (1) the algebra F[a] is a field; (2) the order a is F[a]-pure; (3) aa = p−m , where p = rad a. The property of purity is not invariably preserved under equivalence of strata. 2.2.2 Let [a, m, r, β] be a pure stratum in A and write E = F[β]. Let B denote the A-centralizer of E and set b = a ∩ B, p = rad a. Intertwining properties depend on the behaviour of aβ relative to the lattices pk , k ∈ Z. This can be quite unstable as β varies over its p−r -coset. It is controlled via a critical exponent k0 (β, a) as follows. For k ∈ Z, set

Nk (β, a) = {x ∈ a : aβ (x) ∈ pk }.

As aβ (a) ⊂ p−m , so Nk (β, a) = a for k  −m. On the other hand, a short argument shows that Nk (β, a) ⊂ b + p, for k sufficiently large. We accordingly define k0 (β, a) = max {k : Nk (β, a) ⊂ b + p}. Remark In the case β ∈ F × , we have {0} = aβ (a) ⊂ pk for all k, while b + p = a. One therefore sets k0 (β, a) = −∞ in this case. The exponent k0 (β, a) depends on a in a straightforward way. Let E = F[β], let V1 be a finite-dimensional E-vector space and let a1 be an E-pure hereditary order in

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1 EndF (V1 ). If p1 = rad a1 , then βa1 = pm 1 , where m1 /e(a1 ) = m/e(a). The critical exponent has a similar property:

k0 (β, a1 ) k0 (β, a) = , e(a1 ) e(a)

(2.2.1)

[25] (1.4.13). Definition A simple stratum is a pure stratum [a, m, r, β] such that r < −k0 (β, a). The following result illustrates the rôle played by the lattices Nk (β, a). Recall that g ∈ G is said to formally intertwine the stratum s = [a, m, r, β] if g −1 (β + p−r )g ∩ (β + p−r ) = ∅. We let IG (s) denote the set of such elements g. Proposition Let s = [a, m, r, β] be a simple stratum in A. Let B be the A-centralizer of β, write p = rad a, b = a ∩ B and q = rad b = p ∩ B. If k = k0 (β, a) and N = Nk (β, a), then IG (s) = (1 + q−(r+k) N) B× (1 + q−(r+k) N). Remark The formal intertwining set IG (s) depends only on the equivalence class, call it S, of the stratum s. The last formula describes this set in terms of one simple element s of S. So, the formula gives a loose relation between the simple elements of the equivalence class S. This looseness is intrinsic to the situation. Two simple strata [a, m, r, βi ] may be equivalent while the field extensions F[βi ]/F are not isomorphic: see 2.3.1 below for an extreme example. At this stage, the intertwining formula is the only available method for comparing them. The proof of the proposition requires a certain apparatus but is basically a formal result, not requiring any real knowledge of simple strata. However, it has a consequence from which the structure theory is developed. Using the notation of the proposition, let s : A → B be a tame corestriction with respect to F[β]/F. Corollary For i = 1, 2, let ai ∈ p−r . The strata [b, r, r − 1, s(ai )] in B are equivalent if and only if there exists x ∈ 1 + q−(r+k) N such that the strata [a, m, r − 1, x−1 (β + a1 )x], [a, m, r − 1, β + a2 ] are equivalent. This follows from the formalism of the critical exponent. It indicates the utility of passing to the centralizer via a tame corestriction.

2.3 Structure of Simple Strata We give a procedure for constructing simple strata in A and indicate how, up to equivalence, it yields them all.

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2.3.1 Let [a, m, r, β] be a pure stratum in A. We know that k0 (β, a) is either  −m or = −∞, the latter case corresponding to β ∈ F. We characterize this extreme case. Proposition If [a, m, r, β] is a pure stratum in A, then k0 (β, a)  −m if and only if β is minimal over F. In particular, a pure stratum [a, m, m − 1, β] is simple if and only if β is minimal over F. Example Recall that p is the residual characteristic of F. Let E/F be totally ramified of degree pr , for some r  1. Let a be the unique E-pure hereditary oF -order in A = EndF (E) (as in 1.3.7 Example). If α ∈ E has valuation −1, then α is minimal over F and s = [a, 1, 0, α] is a simple stratum in A. Let E  /F be totally ramified of degree pr . We may embed E  in A so that a is E  -pure. Choose α ∈ E  , of valuation −1, such that det α ≡ det α (mod UF1 ). The stratum [a, 1, 0, α ] is again simple and it is conjugate, under Ka , to a simple stratum equivalent to s. (See [18] section 2 for a full discussion of this example.) 2.3.2 The proposition of 2.3.1 serves as the first step in an iterative procedure for constructing more complex simple strata. Starting from a simple stratum [a, m, r, γ] in A = EndF (V ), set E = F[γ] and let B denote the A-centralizer of γ. Set b = a ∩ B, p = rad a, q = rad b. Let s : A → B be a tame corestriction relative to E/F. Theorem Let [b, r, r − 1, α] be a simple stratum in B and let a ∈ p−r satisfy s(a) ≡ α (mod q1−r ). The stratum [a, m, r − 1, γ + a] is equivalent to a simple stratum [a, m, r − 1, β]. Moreover, (1) k0 (β, a) = max {k0 (α, b), k0 (γ, a)}; (2) [F[β] : F] = [E : F] [E[α] : E] and e(F[β]|F) = e(E|F)e(E[α]|E). That is,

k0 (β, a) =

if α ∈ E, k0 (γ, a) / E. −r > k0 (γ, a) if α ∈

Remarks There are some special cases worthy of comment. (1) If E[α] is a maximal subfield of B, then [a, m, r − 1, γ + a] is a simple stratum in A, not just equivalent to one. Also, F[γ + a] is a maximal subfield of A. (2) If E/F is tamely ramified, one may take s to be the orthogonal projection A → B and a = α. One sees that F[β] = E[α], so F[β] contains E. When E/F is not tamely ramified and [F[β] : F] > [E : F], the element a can never commute with γ since that would imply s(a) ∈ q1−r (cf. 2.1.2 Example). 2.3.3 We now quote a sequence of results from [25] 2.4. Together, these give a complete account of the structure of simple strata and show that they are all obtained from the case of 2.3.1 Proposition via the refinement process of 2.3.2. Theorem 1 Let s = [a, m, r, β] be a pure stratum in A. There exists a simple stratum s0 = [a, m, r, γ] equivalent to s. For any such s0 , e(F[γ]|F) divides e(F[β]|F) and f (F[γ]|F) divides f (F[β]|F).

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We remark that in Theorem 1, if s is not simple, then −k0 (β, a)  r < −k0 (γ, a). Corollary Let s = [a, m, r, β] be a pure stratum in A. Among the pure strata [a, m, r, γ] in A that are equivalent to s, the simple ones are those for which the degree [F[γ] : F] is minimal. Next, we compare the basic invariants of two equivalent simple strata. Theorem 2 For i = 1, 2, let si = [a, m, r, γi ] be a simple stratum in A. If s1 is equivalent to s2 , then (1) k0 (γ1 , a) = k0 (γ2 , a), (2) e(F[γ1 ]|F) = e(F[γ2 ]|F) and f (F[γ1 ]|F) = f (F[γ2 ]|F). (3) Let s1 be a tame corestriction on A relative to F[γ1 ]/F. There exists δ ∈ F[γ1 ] such that s1 (γ1 − γ2 ) ≡ δ (mod p1−r ), where p = rad a. The final result in the sequence shows that the construction in 2.3.2 yields all simple strata, up to equivalence. Theorem 3 Let s = [a, m, r, β] be a pure stratum in A with r = −k0 (β, a). Let [a, m, r, γ] be a simple stratum equivalent to s. Let B denote the A-centralizer of γ, set b = a ∩ B, and let s : A → B be a tame corestriction relative to F[γ]/F. The stratum [b, r, r − 1, s(β − γ)] in B is equivalent to a simple stratum [b, r, r − 1, δ] such that δ ∈ / F[γ]. 2.3.4 We conclude this survey with a fundamental, “intertwining implies conjugacy”, property. This is the first of a sequence of such results, essential to any classification scheme. Theorem For i = 1, 2, let si = [a, m, r, βi ] be a simple stratum in A. Suppose there exists g ∈ G that formally intertwines s1 with s2 . There then exists x ∈ Ua such that [a, m, r, x−1 β1 x] is equivalent to [a, m, r, β2 ].

2.4 Groups Defined by Simple Strata We take a simple stratum in A = EndF (V ), of the form [a, m, 0, β], and attach to it a pair H 1 (β, a) ⊂ J 1 (β, a) of open subgroups of Ua1 . In all cases, we put H k (β, a) = H 1 (β, a) ∩ Uak , J k (β, a) = J 1 (β, a) ∩ Uak ,

k  1.

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2.4.1 The definition of these groups is inductive along β. For the first step, let [a, m, 0, α] be a simple stratum in which α is minimal over F. Thus k0 (α, a) is either −m or −∞. Let B denote the A-centralizer of α and b = a ∩ B. Define H 1 (α, a) = Ub1 Ua1+[m/2] , J 1 (α, a) = Ub1 Ua[m+1/2] , where x → [x] is the greatest integer function. In particular, if α ∈ F then H 1 (α, a) = J 1 (α, a) = Ua1 . 2.4.2 For the general step, let [a, m, 0, β] be a simple stratum in A such that r = −k0 (β, a) < m. Let B denote the A-centralizer of β and put b = a ∩ B. Choose a simple stratum [a, m, r, γ] equivalent to [a, m, r, β] (as in 2.3.3 Theorem 1), and define H 1 (β, a) = Ub1 H 1+[r/2] (γ, a), J 1 (β, a) = Ub1 J [r+1/2] (γ, a). The definition is independent of the choice of γ. Each of these groups is normalized by Kb . We have H 1+k (β, a) = H 1+k (γ, a) when k  [r/2], and similarly for the groups J . The groups H 1 (β, a), J 1 (β, a) depend only on the equivalence class of the simple stratum [a, m, 0, β]. Stronger statements, following from the intertwining properties of simple strata, are given in [25]. They are necessary for the inductive construction but we may omit them.

2.5 Simple Characters Let ψF be a character of F of level one, and use the notation ψA , ψA ∗ a of 1.4.1, 1.4.2. Following another inductive procedure, we define the set of simple characters attached to ψF and a simple stratum in A of the form [a, m, 0, β]. 2.5.1 Let [a, m, 0, β] be a simple stratum in A. Let B be the A-centralizer of β and b = a ∩ B. Definition 1 Suppose that β is minimal over F. Define C(a, β, ψF ) to be the set of 1+[m/2] such that characters θ of H 1 (β, a) = Ub1 Ua 1+[m/2]

, and (1) θ(u)  = ψA ∗ β(u), u ∈ Ua (2) θ  Ub1 factors through the determinant map det B : B× → F[β]× . Definition 2 Suppose that β is not minimal over F and set r = −k0 (β, a). Let [a, m, r, γ] be a simple stratum equivalent to [a, m, r, β]. Define C(a, β, ψF ) to be the set of characters θ of the group H 1 (β, a) = Ub1 H 1+[r/2] (γ, a) such that

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 (1) θ  Ub1 factors through the determinant map det B and (2) there exists ϑ ∈ C(a, γ, ψF ) such that θ(x) = ψA ∗ (β − γ)(x) ϑ(x), x ∈ H 1+[r/2] (γ, a). This definition is equivalent to that in [25], but that equivalence can only be verified at a later stage. In all cases, the set C(a, β, ψF ) is finite and non-empty. Remark The dependence of the definition on the character ψF is straightforward. F has level one, there exists u ∈ UF so that ψF (x) = ψF (ux), x ∈ F. For, if ψF ∈  The quadruple [a, m, 0, u−1 β] is again a simple stratum and C(a, β, ψF ) = C(a, u−1 β, ψF ). There is no harm, therefore, in fixing ψF once for all and omitting it from the notation. In more formal contexts, the following reformulation of the definition is to be preferred. Definition 3 A simple character in G is a smooth character θ of a compact open subgroup K of G with the following property: there exists a simple stratum [a, m, 0, β] in A such that K = H 1 (β, a) and θ ∈ C(a, β). 2.5.2 A lengthy computation yields a concise formula for the intertwining set IG (θ) of a simple character θ. It implies, among other things, that the set C(a, β) depends only on the equivalence class of [a, m, 0, β]. Intertwining Theorem Let [a, m, 0, β] be a simple stratum in A and let θ ∈ C(a, β). If B denotes the A-centralizer of β, then IG (θ) = J 1 B× J 1 , where J 1 = J 1 (β, a). The G-normalizer of θ is NG (θ) = Kb J 1 , where b = a ∩ B. It follows that a is the unique hereditary oF -order in A such that NG (θ) ⊂ Ka . In other words, the simple character θ determines the order a and therefore the parameter m: the character θ is trivial on Ua1+m , and m is the least integer with this property. 2.5.3 There is a more general notion of “truncated” simple character. If [a, m, 0, β] is a simple stratum in A and l is an integer, 0  l < −k0 (β, a), we may define C(a, l, β) as the set of characters of H 1+l (β, a) of the form θ  H 1+l (β, a), θ ∈ C(a, β).  If −k0 (β, a)  l < m, we may equally define C(a, l, β) as the set of restrictions θ  H 1+l (β, a), θ ∈ C(a, β). We have the following connection. Proposition If [a, m, l, γ] is a simple stratum equivalent to [a, m, l, β], then H 1+l (β, a) = H 1+l (γ, a) and C(a, l, β) = C(a, l, γ). Comment The development in [25] works throughout in terms of truncated simple characters, as a matter of necessity. Once the results are complete, one can use the simpler version here.

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2.5.4 A feature of the definition is that a simple stratum gives rise to a finite set of simple characters. This relation has a certain coherence. Proposition 1 For i = 1, 2, let [a, m, 0, βi ] be a simple stratum in A and suppose that H 1 (β1 , a) = H 1 (β2 , a). If C(a, β1 ) ∩ C(a, β2 ) = ∅, then C(a, β1 ) = C(a, β2 ). The proposition applies equally to sets of truncated simple characters. One can recover some properties of the simple stratum [a, m, 0, β] from the set C(a, β) or, therefore, any one of its elements. Proposition 2 For i = 1, 2, let [a, m, 0, βi ] be a simple stratum in A and let l be an integer, 0  l < m. Suppose that H 1+l (β1 , a) = H 1+l (β2 , a) and that C(a, l, β1 ) ∩ C(a, l, β2 ) = ∅. If [a, m, l, γi ] is a simple stratum equivalent to [a, m, l, βi ], then k0 (γ1 , a) = k0 (γ2 , a),   e(F[γ1 ]  F) = e(F[γ2 ]  F),   f (F[γ1 ]  F) = f (F[γ2 ]  F). 2.5.5 Simple characters have a remarkable “intertwining implies conjugacy” property analogous to that for simple strata (2.3.4). Theorem For i = 1, 2, let [a, mi , 0, βi ] be a simple stratum in A and let θi ∈ C(a, βi ). If θ1 intertwines in G with θ2 , then (1) m1 = m2 and (2) there exists x ∈ Ua such that C(a, β2 ) = C(a, x−1 β1 x) and θ2 = θ1x . Observe here that C(a, x−1 β1 x) = C(a, β1 )x . Remark The theorem implies that if an irreducible representation of G contains two simple characters, attached to isomorphic hereditary orders, then those two characters are conjugate in G. If the two underlying hereditary orders are not isomorphic, one can only conclude that the two characters intertwine in G. As it turns out, intertwining of simple characters is a relation every bit as strong as conjugation: see 5.2.2 Complement 3. 2.5.6 Let [a, m, 0, β] be a simple stratum in A and let θ ∈ C(a, β). An irreducible representation π of G that contains θ necessarily contains an irreducible representation θ˜ of J 1 = J 1 (β, a) that contains θ. The intertwining formula of 2.5.2 implies ˜ that θ˜  H 1 (β, a) is a multiple of θ. We now account for all such representations θ. From the definitions, H 1 (β, a) is a normal subgroup of J 1 (β, a) and the quotient is a finite, elementary abelian p-group. Indeed, it is a finite-dimensional vector space over the residue field kF . Proposition 1 Let θ ∈C(a, β). There is a unique irreducible representation η = ηθ  of J 1 (β, a) such that η  H 1 (β, a) contains θ. Further, η  H 1 (β, a) is a multiple of θ and IG (η) = J 1 B× J 1 = IG (θ), where J 1 = J 1 (β, a).

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In particular, θ induces to a multiple of η on J 1 (β, a). The reasons for the form of the proposition are elementary and familiar. Write J for the Fp -vector space J 1 (β, a)/H 1 (β, a). The pairing (x, y) −→ θ[x, y] = θ(xyx−1 y−1 ) on J 1 (β, a) takes its values in the group of p-th roots of unity in C, which we identify with Fp . The pairing induces an alternating bilinear form on the Fp -space J that is non-degenerate. It follows that the finite p-group J 1 (β, a)/Ker(θ) is extra special of class 2 with centre Im θ. The proposition thus comes down to a familiar variation on Heisenberg representations. We sometimes say that ηθ is the 1-Heisenberg representation defined by θ. The intertwining of the representation η has a singular feature of technical importance. Proposition 2 For b ∈ B× , the space Hom(J 1 )b ∩J 1 (η b , η) has dimension 1. In another direction, the pairing on J 1 (β, a) actually only depends on the underlying stratum. If we take elements 1 + x, 1 + y ∈ J 1 (β, a), the value of θ on the commutator [1 + x, 1 + y] is   θ[1 + x, 1 + y] = ψA β(xy − yx) ,

(2.5.1)

[11] 6.1 Proposition. 2.5.7 In order to accommodate representations of level zero, it is convenient to introduce the notion of a trivial simple character. Definition A trivial simple character in A is the trivial character of Ua1 , where a is a hereditary oF -order in A. We can treat these on the same basis as before, agreeing that F[β] means F, m = 0 and H 1 (β, a) = J 1 (β, a) = Ua1 .

3 Types and Classification We complete our narrative account of the construction of the cuspidal representations of GLn (F), accepting from [25] that the process gives them all. We then see that the method we have used is effectively the only one. Throughout, G = GLn (F) and A = Mn (F).

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3.1 Maximal Types Let s = [a, m, 0, β] be a simple stratum in A. Let B denote the A-centralizer of β and set b = B ∩ a. Write E = F[β]. 3.1.1 Let θ ∈ C(a, β) and let η = ηθ be the 1-Heisenberg representation over θ, as in 2.5.6. Write J = J (β, a) = Ub J 1 , J 1 = J 1 (β, a). By the intertwining theorem of 2.5.2, the group J is the Ua -normalizer of the simple character θ. We now have one of the major results of [25].  Theorem There exists a representation κ of J (β, a) such that κ  J 1 (β, a) ∼ = η and IG (κ) = IG (η) = IG (θ) = J 1 B× J 1 . For any such κ and g ∈ G, the space HomJ g ∩J (κg , κ) has dimension at most one. The proof of the main assertion, in [25] 5.2, is quite elaborate. The assertion concerning intertwining spaces follows from 2.5.6 Proposition 2. Definition Let H0 (θ) be the set of equivalence classes of irreducible representations of J (β, a) satisfying the conditions of the theorem. Remarks (1)  There are cases0 in which the group J admits a representation ξ such / H (θ). that ξ  J 1 ∼ = η but ξ ∈ (2) The defining condition on κ is independent of the choice of simple stratum giving rise to θ, since it is expressed in terms of intertwining. (3) In [25], a representation κ satisfying the conditions of the theorem is called a “β-extension” of η. No replacement for this unsatisfactory term has ever been agreed so we avoid the issue by using the notation H0 (θ). The intertwining property of the theorem has a useful consequence. Corollary Let κ ∈ H0 (θ). For i = 1, 2, let σi be an irreducible representation of J that is trivial on J 1 . The representations κ ⊗ σi are irreducible and IG (κ ⊗ σ1 , κ ⊗ σ2 ) = J 1 IB× (σ1 , σ2 )J 1 . The corresponding intertwining spaces have the same dimensions. The representations κ ∈ H0 (θ) have uniqueness properties derived from the Ub /Ub1 ∼ intertwining properties. The inclusion Ub ⊂ J induces an isomorphism =  1 1  J /J . If φ is a character of UE trivial on UE , we form φ ◦ det B Ub and transfer it to a character φ˜ of J , trivial on J 1 . Clearly, if κ ∈ H0 (θ), then φ˜ ⊗ κ ∈ H0 (θ). Proposition (1) If κ, κ ∈ H0 (θ), then κ intertwines with κ in G if and only if κ = κ .

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(2) The pairing (φ, κ) → φ˜ ⊗ κ endows H0 (θ) with the structure of a principal homogeneous space over the group (UE /UE1 ). (3) There is a unique κ0 ∈ H0 (θ) such that the character det κ0 has order a power of p. The action of (UE /UE1 )on H0 (θ) cannot be regarded as canonical, owing to the ill-defined nature of the relation between E and θ. We will remedy this defect in Sect. 7.2 below. The representation κ0 is canonically determined, but is not always the most natural choice. 3.1.2 Since we focus entirely on cuspidal representations, we restrict to a special case. Let s = [a, m, 0, β] be a simple stratum in A. Write E = F[β] and let B denote the A-centralizer of E. Set b = a ∩ B. Say that s is max-simple if b is a maximal oE -order in B or, equivalently, if e(a) = e(E|F). In this case, Ub /Ub1 = J 0 (β, a)/J 1 (β, a) ∼ = GLt (kE ), where [E : F] = n/t. A max-simple character in G is a simple character in G defined by a max-simple stratum in A. We remark that, if [a, m, 0, β] is max-simple, the hereditary order a is principal: if p = rad a, then p = E a, for any prime element E of E. Definition A maximal simple type in G is a pair (J , λ), where J is an open compact subgroup of G and λ is a representation of J , satisfying the following conditions. (1) There is a max-simple stratum [a, m, 0, β] in A such that J = J (β, a). (2) There is a simple character θ ∈ C(a, β) such that λ = κ ⊗ σ, where (a) κ ∈ H0 (θ), and (b) σ is inflated from an irreducible cuspidal representation of the finite general linear group J 0 (β, a)/J 1 (β, a). Note that, by 3.1.1 Corollary, the representation λ is irreducible. Taken together with 1.2.1 Proposition, the same result implies: Proposition Let (J , λ) be a maximal simple type in G and use the notation of the definition. An element g of G intertwines λ if and only if g ∈ J(β, a) = E × J . In particular, the set J(β, a) is the G-normalizer of λ and an open, compact mod. centre subgroup of G. Indeed, J(β, a) is the G-normalizer of the simple character θ contained in λ (2.5.2). The group J = J (β, a) is the unique maximal compact subgroup of J = J(β, a). The quotient J/J is infinite cyclic and J = F[β]× J contains F × J with index e(a). One can equally recognize J 1 (β, a) in canonical terms: it is the unique maximal, normal, pro-p subgroup of J . 3.1.3 Continue in the same situation. Since J/J is infinite cyclic and contains F × J with finite index, the representation λ admits extension to a representation Λ of J. This extension is uniquely determined up to tensoring with a character χ ◦ det G , where χ is an unramified character of F × (cf. 3.1.1 Proposition).

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Lemma Let Λ be a representation of J that extends λ. Let χ be an unramified character of F × and write χΛ for the representation x → χ(det G x)Λ(x), x ∈ J. The following conditions are equivalent. (1) The representations Λ, χΛ intertwine in G. (2) The representations Λ, χΛ are equivalent. (3) The character χ satisfies χn/e(F[β]|F) = 1. This follows readily from the intertwining properties of κ. A pair of this form (J, Λ) will be called an extended maximal simple type in G. We abbreviate this term to EMST . 3.1.4 We summarize the implications of the discussion in 3.1.2, 3.1.3. Construction Theorem If (J, Λ) is an EMST in G, the representation πΛ = c-IndJG Λ is irreducible and cuspidal. There is a complementary uniqueness property. Uniqueness Theorem For i = 1, 2, let (J i , Λi ) be an EMST in G. The following conditions are equivalent. (1) The representations πΛ1 , πΛ2 are equivalent. (2) The representations Λ1 , Λ2 intertwine in G. (3) The representations Λ1 , Λ2 are conjugate in G. Proof The equivalence of (1) and (2) is a formal property of intertwining and compact induction, given that the induced representations c-Ind Λi are both irreducible. Surely (3) implies (1), so we show that (1) implies (3). The representation Λi is given in terms of a max-simple stratum [ai , mi , 0, βi ] in A, in which ai is a principal order of period n/di , where di is the number of unramified characters χ such that χπΛi = πΛi . Statement (1) implies that d1 = d2 whence a1 ∼ = a2 . The simple characters θ1 , θ2 contained in Λ1 , Λ2 intertwine in G, and so are conjugate in G (2.5.5). We may as well assume, therefore,  that a1 = a2 = a say, θ1 = θ2 = θ and β1 = β2 = β. The representations λi = Λi  J (β, a) intertwine in G. The intertwining element g, say, intertwines θ so we may take g ∈ B× , where B is the A-centralizer of β. Write λi = κ ⊗ σi , κ ∈ H0 (θ) as in 3.1.1. The Corollary of 3.1.1 implies that g intertwines σ1 with σ2 and 1.2.1 Proposition implies σ1 ∼ = σ2 . We may as well take σ1 = σ2 whence λ1 = λ2 and Λ2 = χΛ1 , for some unramified character χ of F × . The result now follows from 3.1.3 Lemma.   3.1.5 To complete the picture, we need the following result that combines Theorems (6.2.1) and (8.4.1) of [25]. Exhaustion Theorem If π is an irreducible cuspidal representation of G = GLn (F), then π contains some EMST (J, Λ) in G. In particular, π∼ = πΛ = c-IndJG Λ.

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We give an outline of the strategy of the proof of the Exhaustion Theorem in 5.1.4, after we have recalled some more machinery. With the Exhaustion Theorem in hand, we can extract a consequence of earlier arguments. Corollary Let π1 , π2 be irreducible cuspidal representations of G = GLn (F). The following conditions are equivalent. (1) There is an unramified character χ of F × such that π2 = χπ1 . (2) The representations π1 , π2 contain isomorphic maximal simple types. 3.1.6 If π is an irreducible cuspidal representation of G, containing the EMST (J, Λ), it surely contains the max-simple character θ on which J and Λ are constructed. Any two simple characters obtained in this way are conjugate. However, it is not clear whether π could contain an essentially different simple character θ . Back-engineering of the arguments of [25] Chapter 8 yields the following clarification [17]. Theorem Let π be an irreducible cuspidal representation of G = GLn (F). (1) A simple character contained in π is max-simple. (2) Any two simple characters contained in π are conjugate in G. So, let π contain a simple character θ. Let J be the G-normalizer of θ. The natural representation of J on the θ-isotypic subspace of π is then the EMST Λ for which π = πΛ . Comment Let θ be a simple character in G = GLn (F). If θ is max-simple, we know that θ occurs in an irreducible cuspidal representation of G and also that any irreducible cuspidal representation contains some max-simple character. However, except in the trivial case of n = 1, a max-simple character will occur in a non-cuspidal irreducible representation. More generally, for arbitrary θ, one can describe exactly the irreducible representations of G that contain θ [17]. 3.1.7 To conclude the section, we record some technical developments that will be needed later in the notes. If E/F is a finite field extension, write X1 (E) (resp. X0 (E)) for the group of tamely ramified (resp. unramified) characters of E × . If s is a positive integer, let X0 (E)s be the group of χ ∈ X0 (E) such that χs = 1. Let θ be a max-simple character in G = GLn (F). Let J θ denote the G-normalizer of θ. Let Jθ be the maximal compact subgroup of J θ and Jθ1 the pro-p radical of Jθ . Definition 1 Let X1 (θ) be the group of characters χ of J θ such that (1) χ is trivial on Jθ1 and (2) χ is intertwined every g ∈ IG (θ).

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To describe the group X1 (θ), choose a simple stratum [a, m, 0, β] such that θ ∈ C(a, β). Write E = F[β], let G E be the G-centralizer of E × and let det E : G E → E × be the determinant map. Thus J θ = J(β, a) = G E ∩ J θ · Jθ1 . Let s = n/[E : F]. From [19] 3.1, we have: Lemma Let φ ∈ X1 (E). There exists a unique character φθ ∈ X1 (θ) such that φθ (x) = φ(detE x), x ∈ G E ∩ J θ . The map φ → φθ is a surjective homomorphism X1 (E) → X1 (θ) with kernel X0 (E)s . Let η = ηθ be the unique irreducible representation of Jθ1 that contains θ. Definition 2 Let H(θ) be the set of equivalence classes of representations κ of J θ such that  (1) κ  Jθ1 = η and (2) κ is intertwined by every element of IG (θ). Thus we have a restriction map H(θ) → H0 (θ) (notation of 3.1.1) and the following stronger version of 3.1.1 Proposition, adapted from [19] 3.2. Proposition The group X1 (θ) acts on H(θ) by (χ, κ) → χ ⊗ κ. Relative to this action, H(θ) is a principal homogeneous space over X1 (θ). The restriction map H(θ) → H0 (θ) is surjective.

3.2 Inertial Uniqueness At this stage, it is natural to enquire whether the classification we have just given is the best available. We recall a result of Paskunas indicating that it is, in effect, the only one. 3.2.1 We quote from [62] Theorem 1.3. Theorem Let K be a maximal compact subgroup of G = GLn (F) and let π be an irreducible cuspidal representation of G. There exists at most one irreducible representation  of K such that (a) π contains  and, (b) an irreducible representation π  of G contains  if and only if π  ∼ = χπ, for some unramified character χ of F × .

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The assertion of the theorem is independent of the maximal compact subgroup K, since any two such subgroups are conjugate in G. A given irreducible cuspidal representation π of G contains a maximal simple type (J , λ). The group J is contained in a finite number of maximal compact subgroups K. Choosing one at random, the representation  = c-IndKJ λ is irreducible and satisfies the requirements of the theorem.

4 Encounter with the Langlands Correspondence This section introduces the second part of the notes, concerning connections between the Langlands correspondence and the classification theory of the first three sections. Here, I give a preliminary list of the available techniques, looking quickly at the Galois side for orientation. Before they can be used effectively, most of these techniques require further development of the theory of simple characters. I treat them one at a time in sections to follow. In this one, I give just one application, namely a formula for the Godement–Jacquet local constant of an irreducible cuspidal representation of GLn (F) in terms of the extended maximal simple type it contains.

4.1 The Langlands Correspondence Recall that A0n (F), n  1, is the set ofequivalence classes of irreducible cuspidal  F = n1 A0n (F). Write gr(π) = n to indicate that representations of GLn (F) and GL  F be the set of equivalence classes of irreducible representations π ∈ A0n (F). Let W of WF and let F,  F −→ W GL π −→ Lπ, denote the Langlands correspondence. 4.1.1 We first recall the standard characterization of the Langlands correspondence.  F . Let ψ be a non-trivial character of F and s a complex variable. Let π1 , π2 ∈ GL To the pair (π1 , π2 ) are attached the L-function L(π1 × π2 , s) and the local constant  F . To the semisimple ε(π1 × π2 , s, ψ), as in [47, 71]. On the other side, let σ1 , σ2 ∈ W representation σ1 ⊗ σ2 of WF are attached the Artin L-function L(σ1 ⊗ σ2 , s) and the Langlands–Deligne local constant ε(σ1 ⊗ σ2 , s, ψ) [74] or [15] 29.4.  F with the F → W Theorem The Langlands correspondence is the unique map GL following properties.  F , then dim Lπ = gr(π). (1) If π ∈ GL (2) If χ is a character of F × = GL1 (F), then Lχ is the character of WF attached to χ by local class field theory.

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 F and 1  gr(π1 ) < gr(π2 ), then (3) If π1 , π2 ∈ GL L( Lπ1 ⊗ Lπ2 , s) = L(π1 × π2 , s), ε( Lπ1 ⊗ Lπ2 , s, ψ) = ε(π1 × π2 , s, ψ).

(4.1.1)

F. The correspondence is bijective, and the relations (4.1.1) hold for all π1 , π2 ∈ GL For this version, see [38]. Recent work of Jacquet and Liu [46] shows that the bound in (3) can be replaced by 1  gr(π1 )  21 gr(π2 ). In (2), we often fail to distinguish between χ and Lχ. Observe the following consequences.  F and σ = Lπ, then: Corollary If π ∈ GL ˇ (1) Lπˇ = σ; (2) L(χπ) = Lχ ⊗ σ, for any character χ of F × ; (3) det σ = Lωπ , where ωπ is the central character of π. Recall also that L(π1 × π2 , s) = 1 except when π2 is an unramified twist χπˇ 1 of the contragredient πˇ 1 of π1 . 4.1.2 A major obstruction is that the local constant ε(π × π  , s, ψ) remains resistant to effective calculation: given the amount of information it contains, this is perhaps unsurprising. However, there are a number of other techniques that can be used to investigate the correspondence.  F and let 1 denote the trivial character of GL1 (F) ∼ (1) Take π ∈ GL = F × . The local constant ε(π, s, ψ) = ε(π × 1, s, ψ) is that of Godement and Jacquet [33]. If π contains the extended maximal simple type (J, Λ), there is an explicit formula for ε(π, s, ψ) in terms of Λ (Sect. 4.3). (2) Let πi ∈ A0ni (F), i = 1, 2, and let q = qF be the cardinality of the residue field kF . The local constant has an expression ε(π1 × π2 , s, ψ) = ε(π1 × π2 , 0, ψ) q−(a(π1 ×π2 )+n1 n2 c(ψ))s in which c(ψ) is an integer depending only on ψ. The term a(π1 × π2 ) is an integer, called the Rankin–Selberg exponent of the pair (π1 , π2 ). It is equal to the Artin exponent a( Lπ1 ⊗ Lπ2 ). There is an explicit formula for a(π1 × π2 ) in terms of the simple characters occurring in the factors πi (Sect. 5.3). Continuing with our list, let Atm n (F) be the set of equivalence classes of irreducible  tm  tm tempered representations of GLn (F), and set GL F = n1 An (F). On the other  ss be the set of equivalence classes of finite-dimensional semisimple repside, let W F   ss  F ⊂ GL  tm resentations of WF . Thus GL F and WF ⊂ WF . The Langlands correspontm  ss that we continue to denote by F → W dence extends canonically to a bijection GL F L π → π.

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If K/F is a finite cyclic field extension, there are canonical maps  F −→ GL K , bK/F : GL tm

tm

 tm  tm aK/F : GL K −→ GLF , known as base change and automorphic induction. They correspond respectively to the restriction and induction maps  ss −→ W  ss , ResK/F : W F K  ss −→ W  ss , IndK/F : W K F on representations of the Weil groups. That is,   tm bK/F (π) = Lπ  WK , π ∈ GL F , tm L L  aK/F (ρ) = IndK/F ρ, ρ ∈ GLK . L

In the case of F having characteristic zero, see [1] for base change and [39] for automorphic induction. The extension of both theories to characteristic p is achieved in [40, 41]. The final item in our list of available techniques is: (3) If K/F is finite and tamely ramified (but not necessarily Galois), there are analogues of base change and automorphic induction for simple characters (Sect. 6.2). In some circumstances, one can even identify the effect of base change or automorphic induction on extended maximal simple types, as in [9].

4.2 Inertial Parameters We make a first, straightforward connection. 4.2.1 Let IF be the inertia subgroup of WF and PF the wild inertia subgroup. Let  IF be the set of equivalence classes of irreducible representations of the profinite group PF . The group WF acts on each of these sets by conjugation. IF . Likewise define  There are surjective restriction maps rF  F −−− W −→ WF \ IF

and +

rF  F −−− W −→ WF \ PF .

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 For  example, if σ ∈ WF , then rF (σ) is the WF -orbit of an irreducible component of σ  IF . On the other side, let n  1 and define Tn (F) to be the set of conjugacy classes of extended maximal simple types in GLn (F). Set T(F) = n1 Tn (F). Define Tn0 (F) to be the set of conjugacy classes of maximal simple types in GLn (F). Set T 0 (F) =  0 n1 Tn (F). We have a diagram ≈

T(F) −−−−→ ⏐ ⏐

F W ⏐ ⏐r F WF \ IF

T 0 (F)

in which the horizontal arrow combines the Langlands correspondence and Classification Theorem, while the vertical arrows are restriction maps. Proposition The Langlands correspondence induces a canonical bijection ≈ T 0 (F) −−−−→ WF \ IF

so that the diagram

≈

T(F) −−−−→ ⏐ ⏐

F W ⏐ ⏐

≈ T 0 (F) −−−−→ WF \ IF

commutes. Proof Let X0 (F) denote the group of unramified characters of F × (or of WF ). The  F by (χ, σ) → χ ⊗ σ, and the orbit space X0 (F)\W  F is group X0 (F) acts on W  F by IF . On the other hand, X0 (F) acts on GL canonically identified with WF \ (χ, π) → χπ. We transfer this to an action of X0 (F) on T(F), as in 3.1.3. The orbit space X0 (F)\T(F) is then identified with T 0 (F) (3.1.5 Corollary). Since the  Langlands correspondence respects these actions of X0 (F), we have the result.  We will dedicate several sections to identifying, finally in Sect. 6.3, the automorphic object that corresponds to the space WF \ PF . Once that is done, we give an IF (8.4.3 Theorem). explicit account of the map T 0 (F) → WF \

4.3 Local Constants We deal with item (1) on the list in 4.1.2.

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Let π ∈ A0n (F), n  2, and let ψF be a character of F of level one: in terms of 4.1.2 (2), this is equivalent to c(ψF ) = −1. We give a formula for the Godement– Jacquet local constant ε(π, s, ψF ), leading to a Galois-theoretic interpretation of some properties of the simple character contained in π. We shall exclude the case n = 1 for administrative convenience, although everything is valid there provided π is not unramified: see, for example, [15] section 23 for the exceptional case. The effect of changing the additive character is minor and well-understood, so we stick with the one we have been using. 4.3.1 Let π ∈ A0n (F), n  2, and let (J, Λ) be an EMST contained in π. In particular, Λ is attached to a principal oF -order a in A = Mn (F). If π has level zero, we set m = 0. Otherwise, Λ contains a simple character θ ∈ C(a, β, ψF ), for some simple stratum [a, m, 0, β] in A. Let  be the representation of Ka induced by Λ. Thus  is irreducible, it is trivial on Ua1+m , and m is the least integer with this property. Lemma Let  act on the vector space X , and let c ∈ Ka satisfy ca = p−m . The operator

T(, ψF ) = (cx) ˇ ψA (cx) x∈Ua /Ua1+m

on X ∨ is scalar. Its unique eigenvalue τ (, ψF ) is nonzero and independent of the choice of c. Allowing for some minor changes of convention, this is (2.3.4) Lemma of [7]. In the language of [7], the representation  is non-degenerate, so (3.3.8) Theorem of that paper asserts the following. Theorem Using the preceding notation, ε(π, s, ψF ) = (p−m : a)( 2 −s)/n 1

τ (, ψF ) 1

(a : p1+m ) 2

.

4.3.2 We recast 4.3.1 Theorem in a more usable form. In the notation of 4.1.2, the character ψF satisfies c(ψF ) = −1 so ε(π, s, ψF ) = ε(π, 21 , ψF ) q( 2 −s)(a(π)−n) , π ∈ A0n (F), n  2, 1

where a(π) is the Artin exponent of π. For n  2, one writes sw(π) = a(π) − n, and calls sw(π) the Swan exponent of π. For n = 1, the same applies except when π is an unramified character of F × = GL1 (F). In that case, sw(π) = a(π) = 0. Corollary Let π ∈ A0n (F), n  2. (1) If π has level zero, then a(π) = n and sw(π) = 0.

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(2) Let π have positive level and contain a simple character θ ∈ C(a, β), for some simple stratum [a, m, 0, β] in A = Mn (F). If d (π) is the number of unramified characters χ of F × for which χπ ∼ = π, then sw(π) = md (π).

(4.3.1)

Proof The condition (π) = 0 is equivalent to m = 0, whence follows (1). In part (2), note that a/p is a kF -vector space of dimension e(a)d (π)2 = nd (π). The order   a is principal, so (p−m : a) = qmd (π) , q = |kF |, as required.  F , dim σ  2, the Langlands–Deligne local constant has the form For σ ∈ W ε(σ, s, ψF ) = q( 2 −s)sw(σ) ε(σ, 21 , ψF ). 1

So, if σ = Lπ, with π as in the corollary, the Swan exponent sw(σ) of σ is sw(σ) = md (σ) = md (π),

(4.3.2)

in the obvious notation. 4.3.3 Continue with the notation of 4.3.1. Suppose first that π has level zero. Thus Ka = J,  = Λ and we may as well take a = Mn (oF ). The integer m is zero and, in 4.3.1 Lemma, we may take c = 1. The Gauss sum τ (, ψF ) is the unique eigenvalue of the operator

Λ∨ (x) ψA (x).

x∈Ua /Ua1

Taking traces, ε(π, 21 , ψF ) =

qn

1 dim Λ

tr λ(x−1 ) ψA (x),

(4.3.3)

x∈Ua /Ua1

 where λ = Λ  Ua . The representation λ is inflated from an irreducible cuspidal ¯ The representation λ¯ of GLn (kF ), and (4.3.3) only depends on the representation λ. sum in (4.3.3) reduces to a Kondo Gauss sum [49, 57]. Suppose therefore that π has positive level, and use the notation of 4.3.2 Corollary 2. The natural representation of J = J(β, a) on the θ-isotypic subspace of  is equivalent to Λ. Consequently, ε(π, 21 , ψF ) =

1 1 2

(a : pm+1 ) dim Λ

where J = J (β, a). Equivalently:

x∈J /Ua1+m

tr Λ∨ (βx) ψA (βx),

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Proposition The constant ε(π, 21 , ψF ) is the unique eigenvalue τ (Λ, β, ψF ) of the operator

T(Λ, β, ψF ) = (a : p1+m )− 2

1

Λ∨ (βx) ψA (βx).

(4.3.4)

x∈J /Ua1+m

The formula for the Gauss sum τ (Λ, β, ψF ) can be simplified to various degrees. To give the easiest example, write x = y(1 + z), where y ∈ J /Uam and 1 + z ∈ Uam /Ua1+m . For a fixed y, consider the sum

Λ∨ (βy(1 + z)) ψA (βy(1 + z)) = Λ∨ (βy)ψA (βy)

z

Λ∨ (1 + z) ψA (βyz).

z

For z ∈ pm , the operator Λ∨ (1 + z) is scalar with eigenvalue ψA (−βz). So, the inner sum vanishes if y ∈ / J 1 = J 1 (β, a). Otherwise, it takes the value (Uam : Ua1+m ) = nd (π) . Therefore τ (Λ, β, ψF ) is the unique eigenvalue of the operator q qnd (π) Λ∨ (βy) ψA (βy)

Λ∨ (1 + z) ψA (βz).

(4.3.5)

1+z∈J 1 /Uam

Corollary Let π ∈ A0n (F), n  2, contain a simple character θ ∈ C(a, β, ψF ), for a simple stratum [a, m, 0, β]. If χ is a tamely ramified character of F × , then ε(χπ, s, ψF ) = χ(det β)−1 ε(π, s, ψF ). One can use the same device to further restrict the range of summation in (4.3.5) and so generalize the corollary. See [9] section 6 for some examples.

5 Transfer, Endo-equivalence and Applications This section introduces a fundamental property of “transfer” of simple characters between different hereditary orders. A first version (here in Sect. 5.1) is used in [25] to analyse a single representation and its Jacquet modules. It is the foundation on which the proof of the Exhaustion Theorem is erected: we give a brief outline of its rôle in 5.1.4. In Sect. 5.2, we follow [8] to organize the transfer process into an equivalence relation on the entire corpus of simple characters in all groups GLn (F), n  1. This version provides a framework for comparing representations of different groups and also, in some circumstances, different base fields. In this section, we use F. it in an explicit formula for the Rankin–Selberg exponent a(π1 × π2 ), π1 , π2 ∈ GL

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5.1 Transfer of Simple Characters 5.1.1 Let [a, m, 0, β] be a simple stratum in A = EndF (V ), where V is some finitedimensional F-vector space. Write E = F[β] and let V  be a finite-dimensional E-vector space. Let a be an E-pure hereditary oF -order in A = EndF (V  ). If m = me(a )/e(a), then m is an integer and [a , m , 0, β] is a simple stratum in A (2.2.1). We follow [25] 3.6 to construct a bijection β

≈

ta,a : C(a, β, ψF ) −−−−→ C(a , β, ψF ).

(5.1.1)

This bijection, called a “transfer”, is to be transitive with regard to the hereditary orders a, a and canonical for a fixed choice of β. Example Suppose that β is minimal over F. Let E = F[β], let B be the A-centralizer 1+[m/2] = Ub1 Ua . If θ ∈ C(a, β), there is a unique of E and b = a ∩ B. Thus H 1 (β,  a) 1 1  U θ = χ ◦ det . We write θ = θχ . There is a unique character χ of UE such that B b  θχ ∈ C(a , β) such that θ  Ub1 = χ ◦ det B , in the obvious notation. Here we may β set ta,a (θχ ) = θχ (but this ad hoc approach is inefficient in general). 5.1.2 Revert to a general simple stratum [a, m, 0, β] in A = EndF (V ), and set E = F[β]. We describe the transfer in two special cases from which all others can be deduced. Theorem Let a be an E-pure hereditary oF -order in the algebra A. If θ ∈ C(a, β), there exists a unique θ ∈ C(a , β) such that θ (x) = θ(x), x ∈ H 1 (β, a) ∩ H 1 (β, a ). The map C(a, β) −→ C(a , β), θ −→ θ ,

(5.1.2)

is a bijection. This result is [25] (3.6.1) Theorem. In this case, (5.1.2) provides the transfer β map ta,a . Suppose we have a simple stratum [a, m, 0, β  ] such that C(a, β  ) = C(a, β). Suppose that a is also F[β  ]-pure and that C(a , β  ) = C(a , β). The theorem implies that β β ta,a = ta,a . (Compare 5.1.1 Example here.) 5.1.3 Let aE/F be the unique E-pure, hereditary oF -order in AE/F = EndF (E). With β [a, m, 0, β] as in 5.1.2, we exhibit the map ta,aE/F . Let L be the chain of a-lattices in V . Choose an oE -basis of L, in the following sense. Each L ∈ L is an oE -lattice. Fixing an L at random, it has an oE basis {v1 , v2 , . . . , vd } with the following property: any L ∈ L has an oE -basis

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of the form {x1 v1 , x2 v2 , . . . , xd vd } for some xj ∈ E × . (This is a variation on the standard block matrix picture of a hereditary order.) Write Wj = Evj , so that V = W1 ⊕ W2 ⊕ · · · ⊕ Wd . Let M be the Levi subgroup of G that stabilizes each Wj and choose parabolic subgroups P, P  of G so that P ∩ P  = M . Let N and N  be the unipotent radicals of P and P  respectively. By construction, the set {X ∩ Wj : X ∈ L} is the unique oE -lattice chain in the one-dimensional E-space Wj . Proposition Let θ ∈ C(a, β). (1) The group H 1 = H 1 (β, a) satisfies H1 = H1 ∩ N · H1 ∩ M · H1 ∩ N, the factors in the product being disjoint. (2) The character θ is trivial on the subgroups H 1 ∩ N  and H 1 ∩ N . (3) The group H 1 ∩ M takes the form H ∩M = 1

d 

H 1 ∩ Aut F (Wj ).

j=1

Further, H 1 ∩ Aut F (Wj ) = H 1 (β, aj ), where aj is the unique E-pure hereditary oF -order in EndF(Wj ) . (4) Identify each aj with aE/F via the obvious isomorphism Wj → E. There is then a unique θ0 ∈ C(aE/F , β) such that  θ  H 1 ∩ M = θ0 ⊗ θ0 ⊗ · · · ⊗ θ0 . In this situation, set

β θ = θ0 and taβE/F ,a θ0 = θ. ta,a E/F

If a is an E-pure hereditary oF -order in EndF (V  ), say, we then have β

β

β . ta,a = taE/F ,a ◦ ta,a E/F

5.1.4 By way of an aside, we sketch the manner in which transfer of simple characters intervenes in the proof of the Exhaustion Theorem. Let π be an irreducible smooth representation of G = GLn (F), of positive level. Thus π contains a fundamental stratum (1.5.1 Theorem). If that stratum is split, then π is not cuspidal and we discard it. Otherwise, it contains a simple stratum [a1 , m1 , m1 − 1, a] (by 1.5.3 Theorem 1). So, there exists a simple stratum [a2 , m2 , 0, β] in A = Mn (F), a character θ ∈ C(a2 , β) and an integer t, 0  t < m, such that π contains

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 θ  H 1+t (β, a2 ). Choose this data to minimize t/e(a2 ). Suppose that  t > 0, and choose a simple stratum [a2 , m2 , t, γ] equivalent to [a2 , m2 , t, β]. Thus θ  H 1+t (β,  a2 ) lies in C(a2 , γ). The representation π contains a character of the form θ ψA ∗ a  H t (β, a2 ), for some a ∈ p−t . Let B be the A-centralizer of γ and set b = a2 ∩ B. Consider the stratum [b, t, t − 1, sγ (a)] in B, where sγ is a tame corestriction. If this stratum is not fundamental, one can reduce the relative level t/e(a2 ). Assume therefore that [b, t, t − 1, sγ (a)] is fundamental. If it is split, one shows that π is not cuspidal. If it is non-split, one may adjust the original choice of [a2 , m2 , 0, β] and θ to reduce t/e(a2 ). Overall, π contains some simple character so, at the beginning, we could have taken t = 0. Each of those steps requires a change to the hereditary order a2 and relies on the ability to change θ with it. That done, π contains a simple character θ ∈ C(a, β), for some choice of simple stratum [a, m, 0, β]. The final step in the argument imitates the proof of 1.2.3 Theorem, but has to be conducted “over θ”, again requiring the facility of changing the order a.

5.2 Endo-Equivalence The transfer structure of Sect. 5.1 leads to an equivalence relation on the class of all simple characters. We follow the original treatment in [8], but use a more condensed terminology. 5.2.1 We make an elementary observation. Lemma Let E/F be a finite field extension. If a is a hereditary order in A = EndF (V ), defined by a lattice chain {Li }i∈Z in V , the following are equivalent. (1) There exists an F-embedding ι : E → A such that a is ι(E)-pure. (2) The integer e(a) is divisible by e(E|F) and dimkF Li /Li+1 is divisible by f (E|F), for all i. Suppose these conditions are satisfied, and let ι : E → A be an F-embedding. The order a is then ι (E)-pure if and only if ι = Ad x ◦ ι, for some x ∈ Ka . Say that a is E/F-conformal if the conditions of the lemma are satisfied. 5.2.2 For i = 1, 2, we are given a finite-dimensional F-vector space Vi and a simple character θi in G i = Aut F (Vi ). This θi is invariantly attached to a hereditary oF -order ai in Ai = EndF (Vi ). Relative to our fixed character ψF ∈  F, we choose the following data. 5.2.2.1 Data (1) For i = 1, 2, let [ai , mi , 0, βi ] be a simple stratum in Ai such that θi ∈ C (ai , βi , ψF ).

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(2) Let V be a finite-dimensional F-vector space and let a be a hereditary oF -order in A = EndF (V ) that is F[βi ]/F-conformal for i = 1, 2. (3) Let ιi : F[βi ] → A be an F-embedding such that a is ιi (F[βi ])-pure. Set θiV = taβii ,a θi .

(5.2.1)

We remark that, in (5.2.2.1), the choice of embeddings ιi is immaterial since, for example, any two embeddings ι1 are conjugate under Ka1 . Theorem Suppose there exists a choice of data (5.2.2.1) for which the simple characters θ1V , θ2V are conjugate in G = AutF (V ). The same then holds for all choices of such data. When these conditions holds, we say that θ1 is endo-equivalent to θ2 . The transitivity of the constructions ensures that endo-equivalence is indeed an equivalence relation on the class of all simple characters in all groups AutF (V ), as V ranges over the class of finite-dimensional F-vector spaces. We write E(F) for the set of equivalence classes relative to the relation of endo-equivalence (and tend to use the word “endo-classes” for the elements of the set E(F)). Complements (1) The relation of endo-equivalence does not depend on the choice of character ψF (cf. 2.5.1 Remark). (2) Two simple characters in G = Aut F (V ), attached to the same hereditary order in EndF (V ), are endo-equivalent if and only if they are conjugate in G. (3) Following [17], one knows that two simple characters in G = AutF (V ), attached to arbitrary hereditary orders, intertwine in G if and only if they are endoequivalent. (4) It is convenient to adjoin to E(F) a trivial element 0, which is the endo-class of trivial simple characters. Thus 0 is the class of trivial characters of groups Ua1 , as a ranges over all hereditary oF -orders in all matrix algebras EndF (V ). Terminology Let [a, m, 0, β] be a simple stratum and let θ ∈ C(a, β). If Θ ∈ E(F) is the endo-class of θ, say that θ is a realization of Θ on a (or [a, m, 0, β]). The set of simple characters contained in a representation π ∈ A0n (F) constitutes a single conjugacy class in GLn (F). In particular, all such characters are endoequivalent. We therefore have a canonical map  F −→ E(F), GL π −→ Θ(π).

(5.2.2)

Proposition The map (5.2.2) is surjective. Proof The trivial element 0 is the endo-class simple characters contained in any  F of level zero, so let Θ ∈ E(F) be non-trivial. It is the endo-class of a π ∈ GL

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simple character θ ∈ C(a, β), for a simple stratum [a, m, 0, β] in some EndF (V ). Transferring θ, we may assume dimF V = [F[β] : F]. Thus θ is max-simple and, if Λ is an irreducible representation of J(β, a) containing θ, then Λ is an EMST. The representation πΛ of Aut F (V ) induced by Λ is then irreducible and cuspidal. It   satisfies Θ(πΛ ) = Θ. 5.2.3 An endo-class Θ ∈ E(F) carries a panoply of invariants. Proposition Let Θ ∈ E(F), Θ = 0. Let [a, m, 0, β] be a simple stratum in A = EndF (V ) carrying a realization θ ∈ C(a, β) of Θ. Write E = F[β]. The following quantities depend only on Θ and not on the choice of θ: ςΘ = m/e(a); eΘ = e(E|F); fΘ = f (E|F);

(5.2.3)

deg Θ = [E : F]; kΘ = k0 (β, a)/e(a). These assertions follow readily from properties of simple characters (2.5.4 Proposition 2) and the definition of endo-equivalence. To complete this part of the picture, we set ς0 = 0, e0 = f0 = 1, k0 = −∞.

(5.2.4)

In Sect. 9, we will need some finer invariants of a non-trivial endo-class Θ. We first return to simple strata for an inductive definition. Definition Let s = [a, m, 0, β] be a simple stratum in an algebra EndF (V ). The set j(s) of (normalized) jumps of s is given as follows. (1) If β ∈ F, then j(s) is empty. (2) If β is minimal over F and β ∈ / F, then j(s) = {m/e(a)}. (3) Otherwise, let r = −k0 (β, a), choose a simple stratum [a, m, r, γ] equivalent to [a, m, r, β] and set j(s) = {r/e(a)} ∪ j(s ), where s = [a, m, 0, γ]. Proposition Let Θ ∈ E(F), Θ = 0. Let θ ∈ C(a, β) be a realization of Θ, for a simple stratum s = [a, m, 0, β] in an algebra EndF (V ). The set of rational numbers j(Θ) = j(s) depends only on Θ and not on the choices of θ and s. This follows readily (5.2.3) and the definitions.

(5.2.5)

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The set j(Θ) is the set of jumps of Θ. We could define j(0) = ∅, but the need never arises. 5.2.4 The set E(F) carries a canonical pairing A with non-negative rational values. To define it, let Θ1 , Θ2 ∈ E(F). There is a hereditary order a in some A = EndF (V ), occurring in simple strata [a, mi , 0, βi ], such that C(a, βi ) contains a simple character θi of endo-class Θi . (If, for example, Θ1 = 0, we may take θ1 to be the trivial character of Ua1 .) Let t  0 be the least integer such that the restrictions θi  H 1+t (βi , a) intertwine (and are therefore conjugate) in AutF (V ). Set A(Θ1 , Θ2 ) = t/e(a).

(5.2.6)

Proposition Let Θ1 , Θ2 ∈ E(F). (1) A(Θ1 , Θ2 )  max {ςΘ1 , ςΘ2 }, with equality if ςΘ1 = ςΘ2 . (2) A(Θ1 , Θ2 ) = 0 if and only if Θ1 = Θ2 . (3) If Θ ∈ E(F), then   A(Θ1 , Θ2 )  max A(Θ1 , Θ), A(Θ, Θ2 ) . A proof is given in [21] 5.1 Proposition. Combined with the obvious property A(Θ1 , Θ2 ) = A(Θ2 , Θ1 ), parts (2) and (3) say that A provides an ultrametric on the set E(F). 5.2.5 It is sometimes helpful to use a slightly different language in connection with the ultrametric A. Let  > 0. The binary relation Θ1 ∼ Θ2

⇐⇒

A(Θ1 , Θ2 ) < 

(5.2.7)

is an equivalence relation on E(F). Denote by tc (Θ) the equivalence class of Θ. We refer to tc (Θ) as the -truncation of Θ. Likewise, Θ1  Θ2

⇐⇒

A(Θ1 , Θ2 )  

(5.2.8)

is an equivalence relation, the class of Θ being denoted tc+  (Θ). Proposition Let Θ, Υ ∈ E(F) and  > 0. The following are equivalent. (1) tc (Θ) = tc (Υ ). (2) Let a be a hereditary oF -order in a matrix algebra EndF (V ), admitting realizations θ ∈ C(a, β), υ ∈ C(a, γ) of Θ,  If t is the largest integer  Υ respectively. such that t/e(a) < , the characters θ  H 1+t (β, a), υ  H 1+t (γ, a) are conjugate in AutF (V ). The proposition is a re-statement of the definition of A. The exact analogue holds with tc+  replacing tc and t/e(a)   instead of t/e(a)  .

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5.3 Conductors of Pairs As a first application of the machinery of endo-classes of simple characters, we give an explicit formula for the Rankin–Selberg exponent a(π1 × π2 ) of a pair of  F . It is based on Shahidi’s account [70, 71] of L(π1 × representations π1 , π2 ∈ GL π2 , s) and ε(π1 × π2 , s, ψ) rather than [47]. The proof [23] uses [71] along with representation-theoretic structures [28, 29] not discussed here. 5.3.1 Before stating the result, we need to introduce a certain constant. Let E/F be a finite field extension and let A = EndF (E). Let a be the unique Epure hereditary oF -order in A. Let β ∈ E × satisfy E = F[β] and m = −υE (β) > 0. Assume moreover that the quadruple [a, m, 0, β] is a simple stratum in A. Let aβ denote the adjoint map A → A, x → βx − xβ, and let s : A → E be a tame corestriction relative to E/F (2.1.2). The sequence aβ

s

0 → E −→ A −−−−→ A −−−−→ E → 0 is then exact. There exist oF -lattices l, l in E and m, m in A such that the sequence aβ

sE/F

0 → l −→ m −−−−→ m −−−−→ l → 0 is exact. For Haar measures μE and μA on E and A respectively, the quantity C(β) =

μE (l) μA (m ) μE (l ) μA (m)

is independent of all these choices. If q = |kF |, there is an integer c(β) such that C(β) = qc(β) .

(5.3.1)

We give a method for calculating c(β) in 5.3.4 below.  F . Now set 5.3.2 Let π ∈ A0n (F) ⊂ GL ς(π) = ςΘ , where Θ = Θ(π) is the endo-class of a simple character θπ contained in π. So, if θπ ∈ C(a, β), for a simple stratum [a, m, 0, β] in Mn (F), then ς(π) = m/e(a) = md (π)/n = sw(π)/n,

(5.3.2)

by 4.3.2 Corollary. In the notation of 1.5.1, we have (π) = ς(π) = A(Θπ , 0).

(5.3.3)

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5.3.3 Let πi ∈ A0ni (F), i = 1, 2. Let d (π1 , π2 ) be the number of unramified characters χ of F × such that π2 ∼ = χπ1 . Let Θi be the endo-class of a simple character contained in πi . To give the formula for the Rankin–Selberg exponent, we divide into cases. Theorem 1 Let πi ∈ A0ni (F), i = 1, 2. If A = A(Θ1 , Θ2 ) = max {ς(π1 ), ς(π2 )},

(5.3.4)

a(πˇ 1 × π2 ) = n1 n2 (1 + A) − d (π1 , π2 ).

(5.3.5)

then

We remark here that if A > 0 then d (π1 , π2 ) = 0. To deal with the case where (5.3.4) fails, that is, where 0  A = A(Θ1 , Θ2 ) < max {ς(π1 ), ς(π2 )},

(5.3.6)

we need some further notation. Note that (5.3.6) implies ς(π1 ) = ς(π2 ). Notation Let a be a hereditary oF -order in an algebra A = EndF (V ) admitting realizations of both Θ1 and Θ2 . (1) For i = 1, 2, let [a, mi , 0, βi ] be a simple stratum in A such that C(a, βi ) contains a simple character θi of endo-class Θi : here, m1 = m2 by (5.3.6). (2) Let t  0 be the least integer for which the characters θi  H 1+t (βi , a) intertwine in G = Aut F (V ). (3) Let [a, m1 , t, γ] be a simple stratum equivalent to [a, m1 , t, β1 ]. (4) Set dγ = [F[γ] : F] and e = e(a). Theorem 2 Under the hypothesis (5.3.6) and using the preceding notation, we have  a(πˇ 1 × π2 ) = n1 n2

t c(γ) 1+ 2 + dγ edγ

 − d (π1 , π2 ).

Remarks (1) We have d (π1 , π2 ) = 0 except in certain cases where n1 = n2 and A = 0. (2) The formula becomes a little tidier when it is expressed in terms of the Swan exponent, sw(πˇ 1 × π2 ) = a(πˇ 1 × π2 ) − n1 n2 + d (π1 , π2 ). (3) Suppose that π ∈ A0n (F) contains a simple character θ ∈ C(a, β). We then have sw(πˇ × π) c(β) = 2 , 2 n dβ

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where dβ = [F[β] : F]. One can re-phrase the theorems in terms of the Swan conductor and the ultrametric A. This will be used in Sect. 9. Theorem 3 Let Θ ∈ E(F). There is a unique continuous function Θ (x), x  0, with the following property. If π ∈ A0n (F) satisfies Θ(π) = Θ and ρ ∈ A0m (F), then sw(πˇ × ρ) = Θ (A(Θ, Θ(ρ))). mn

(5.3.7)

The function Θ is strictly increasing, piece-wise linear and convex. See [21] 5.4 Proposition. An explicit formula for the function Θ is given in [21] (4.4.1). In particular, the jumps of Θ , that is, the discontinuities of the derivative Θ (x), x > 0, are the jumps of Θ in the sense of 5.2.3. 5.3.4 As promised in 5.3.1, we shed some light on the integer c(β). Proposition 1 Let E = F[α]/F be a finite field extension and suppose that α is minimal over F. If m = −υE (α), e = e(E|F), f = f (E|F), then c(α) = mf (ef − 1). Proof Write a = aE/F (1.3.7 Example) and let p = rad a. There is an exact sequence aα

s

0 → oE −→ a −−−−→ p−m −−−−→ p−m E →0 in which s : AE/F → E is a tame corestriction. The result then follows from a short calculation.   To compute c(β) in general, one uses a reduction argument. Proposition 2 Let [a, m, 0, β] be a simple stratum in Md (F) for which the field extension E = F[β]/F has degree d . Set e = e(E|F). Let r = −k0 (β, a) and assume r < m. Let [a, m, r, γ] be a simple stratum equivalent to [a, m, r, β]. If d (γ) = [F[γ] : F], then c(γ) c(β) r r = . + + d2 ed d (γ)2 ed (γ) Proof See [12] 3.1 Proposition.

 

6 Tame Lifting and Functoriality Let K/F be a finite, tamely ramified field extension. We construct a canonical relation between the sets E(K), E(F) of endo-classes of simple characters over the fields K, F. This tame lifting relation is closely connected with the operations of base change and automorphic induction recalled in 4.1.2. In the first instance, it yields a Galoistheoretic interpretation of the set E(F) of endo-classes of simple characters over F.

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6.1 Tame Lifting We start with a transparent, albeit ad hoc, construction. The results of this section are taken from [8], mostly section 7. 6.1.1 Let [a, m, 0, β] be a simple stratum in A = EndF (V ) and write E = F[β]. Data We are given a subfield K/F of A such that (1) K commutes with β, (2) the algebra EK = K[β] is a field, and (3) the hereditary oF -order a is EK-pure. Let AK be the centralizer of K in A and set aK = a ∩ AK . Proposition The quadruple [aK , m, 0, β] is a simple stratum in AK satisfying k0 (β, aK )  k0 (β, a).

(6.1.1)  

Proof See [8], 2.4 Theorem. There are cases in which the inequality (6.1.1) is strict.

6.1.2 Following 6.1.1 Proposition, we form the subgroups H 1 (β, aK ), J 1 (β, aK ) of G K = A× K. Proposition There are relations H 1 (β, aK ) = H 1 (β, a) ∩ G K , J 1 (β, aK ) = J 1 (β, a) ∩ G K . Proof See [8], 2.2.1 Proposition.

 

Fix a character ψF of F of level one, as before. Set ψK = ψF ◦ Tr K/F .

(6.1.2)

Thus ψK is a character of K of level one. Theorem If θ ∈ C(a, β, ψF ), the restriction  θK = θ  H 1 (β, aK )

(6.1.3)

lies in C(aK , β, ψK ). This is [8] 7.7 Theorem. The restriction map C(a, β, ψF ) → C(aK , β, ψK ) need not be surjective.

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6.1.3 From the foundation of 6.1.2, we outline a more systematic approach following [8] section 9. Let Θ ∈ E(F), Θ = 0. Let θ ∈ C(a, β, ψF ) be a realization of Θ on a simple stratum [a, m, 0, β] in some algebra A = EndF (V ). Write F[β] = E. Let K/F be a finite, tamely ramified field extension: we do not assume that K/F is Galois. The K-algebra E ⊗F K is semisimple and so the direct sum of its minimal ideals, E ⊗F K = E 1 ⊕ E 2 ⊕ · · · ⊕ E r , for an integer r, 1  r  [K : F]. Each ideal E j is a field extension of K. If β j denotes the projection of β in E j , then E j = K[β j ]. For each j, let V j be a finite-dimensional E j -vector space and let aj be an E j -pure hereditary oF -order in Aj = EndF (V j ). There is an integer mj so that [aj , mj , 0, β j ] is a simple stratum in Aj . Using the F-embedding E → Aj induced by β → β j , j we transfer θ to a simple character θj ∈ C(aj , β j , ψF ). Let AK be the centralizer j j of K in Aj and set aK = aj ∩ AK . Following 6.1.2 Theorem, we get a character j j j j θK ∈ C(aK , β j , ψK ). Define ΘK ∈ E(K) to be the endo-class of θK . j

j

Theorem 1 The endo-classes ΘK , 1  j  r, are distinct. The set {ΘK : 1  j  r} depends only on Θ, not on any of the choices made in the definition. j

Further, the set of ΘK does not depend on the choice of character ψF (cf. 2.5.1 j Remark). The endo-classes ΘK , 1  j  r, are the K/F-lifts of Θ. To complete the picture, let 0F be the trivial element of E(F). The class 0F has a unique K/F-lift, namely the trivial element 0K of E(K). We summarize the main properties of the lifting operation. Theorem 2 Let K/F be a finite, tamely ramified field extension. (1) If Ξ ∈ E(K), there is a unique Ξ F ∈ E(F) of which Ξ is a K/F-lift. The map iK/F : E(K) −→ E(F), Ξ −→ Ξ F , is surjective and, for Θ ∈ E(F), the fibre i−1 K/F (Θ) is the set of K/F-lifts of Θ. (2) If L/K is a finite tame extension, then iL/F = iK/F ◦ iL/K . (3) If the tame extension K/F is Galois, then any two K/F-lifts of Θ are conjugate under Gal(K/F). (4) The map iK/F does not depend on the choice of character ψF . 6.1.4 Example A character χ of UF1 is a simple character in GL1 (F). If not trivial, it is attached to a simple stratum [oF , m, 0, c] in M1 (F) = F. Either way, its endo-class has degree 1. If K/F is a finite, tame extension, then χ ◦ NK/F is a simple character over K, and its endo-class is the unique K/F-lift of the endo-class of χ.

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The other direction is more interesting. Let K/F be a finite tame extension and χ a character of UK1 , of endo-class X say. Suppose that χ does not factor through the norm NK/L , for any intermediate field F ⊂ L  K. The endo-class iK/F X then has degree [K : F]. As in [13], one can write down an explicit representative of iK/F X. It is the simple character attached to (K/F, χ) via the classical construction of [42].

6.2 Functoriality Relations We establish a fundamental relation between tame lifting and base change, as promised in item (3) of the list in 4.1.2. 6.2.1 Let K/F be a finite cyclic extension. As in 4.1.2, let  F −→ GL K , bK/F : GL tm

tm

 tm  tm aK/F : GL K −→ GLF be respectively the maps of base change and automorphic induction.  tm If π ∈ GL F , we may realize π as a Langlands quotient in the standard way, π = π 1  π 2  . . .  πr ,  F . Thus where π1 , π2 , . . . , πr ∈ GL π = Lπ1 ⊕ Lπ2 ⊕ · · · ⊕ Lπr .

L

6.2.2 The relation between the pair of maps bK/F , aK/F and the operation of tame  F , let Θ(π) denote the endolifting is contained in two parallel results. For π ∈ GL class of simple characters contained in π. F. Theorem 1 Let K/F be a finite, tamely ramified, cyclic extension and let π ∈ GL If bK/F π = ρ1  ρ2  . . .  ρs ,  K , then where ρj ∈ GL Θ(π) = iK/F Θ(ρj ), 1  j  s. The K/F-lifts of Θ(π) occur among the Θ(ρj ) with equal multiplicities.

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K. Theorem 2 Let K/F be a finite, tamely ramified, cyclic extension and let ρ ∈ GL If aK/F ρ = π1  π2  . . .  πr ,  F , then where πi ∈ GL Θ(πi ) = iK/F Θ(ρ), 1  i  r. Remark Suppose for a moment that K/F is a finite, Galois, tame extension and let K0 /K be the maximal unramified sub-extension of K/F. We may formally write  F , we again have an expansion bK/F = bK/K0 ◦ bK0 /F . For π ∈ GL bK/F π = ρ1  ρ2  . . .  ρs ,  K , the representations Lρj being the irreducible components of for various ρj ∈ GL L  π WK . A similar comment holds for automorphic induction. The transitivity property of 6.1.3 Theorem 2 shows that the theorems hold unchanged for K/F. See [24] section 2 for a discussion of these matters. 6.2.3 The proofs of the theorems of 6.2.2 are closely entwined and spread over the two papers [8, 12]. The first paper deals with some special cases in which s = 1 = r. One works directly with the character relations that define base change [1] and automorphic induction [39]. The general case is then deduced in [12], without appeal to the character relations. It relies instead on more formal aspects of the Langlands correspondence and the conductor formula of 5.3.3 above. Both [1, 39] assume that the base field F has characteristic zero. This assumption is removed in [41]. The general machinery of simple characters, at the level of detail required for these results, is completely insensitive to characteristic, so everything in [8, 12] (and their derivatives) holds unchanged in positive characteristic.

6.3 Ramification Theorem We explore some implications of 6.2.2, following [12].  F → E(F), π → Θ(π), of (5.2.2). 6.3.1 We return to the canonical surjection GL We compare it with the surjective map  F −→ WF \ PF rF+ : W   F to the WF -conjugacy of an irreducible component of σ  PF . that takes σ ∈ W

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 F satisfy Θ(π) = Θ. If σ = Ramification Theorem Let Θ ∈ E(F) and let π ∈ GL L  F , the conjugacy class LΘ = r + (σ) depends only on Θ and not on the choice π∈W F of π. The map PF , E(F) −→ WF \ Θ −→ LΘ, is a bijection and the diagram π  F −−π−→ GL −−→ ⏐ ⏐ Θ L

F W ⏐ ⏐r + F

E(F) −−−−−−→ WF \ PF Θ→ LΘ

commutes. Proof The core assertion of the theorem amounts to the following.  F and set σi = Lπi . The following conditions Proposition For i = 1, 2, let πi ∈ GL are equivalent. (1) Θ(π1 ) = Θ(π2 ). (2) HomPF (σ1 , σ2 ) = 0. The proposition implies straightaway that the map Θ → LΘ is well-defined and injective. The diagram commutes, whence the map is surjective.   To prove the proposition, we need a lemma. Say that σ ∈ WF is totally wild if  σ PF is irreducible. Equivalently, dim σ is a power of p (the residual characteristic  wr be the set of F) and χ ⊗ σ  σ for any unramified character χ = 1 of WF . Let W F F. of totally wild σ ∈ W  F and suppose that σi = Lπi is totally wild. The Lemma For i = 1, 2, let πi ∈ GL following are equivalent. (1) Θ(π  1 ) = Θ(π 2 ). (2) σ1  PF ∼ = σ2  PF . (3) There is a tamely ramified character φ of WF such that σ2 ∼ = φ ⊗ σ1 . Proof The equivalence of (2) and (3) is immediate while (3) surely implies (1). Assume (1) holds. Since σi is totally wild, πi is a representation of GLmi (F), where mi = pri , for some ri  0. Also χπi  πi for any unramified character χ = 1 of F × . We conclude that deg Θ(πi ) = e(Θ(πi )) = pri , whence (1) implies r1 = r2 = r, say. It follows that π2 ∼ = φπ1 , for some tamely ramified character φ of F × , so (3) holds.  

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To prove the proposition, write σi = IndEi /F τi , where Ei /F is a finite tame exten wr . Let E/F be a finite, tame Galois extension containing E1 and E2 . sion and τi ∈ W Ki Thus   τji , σiE = σi  WE = 1jri

 wr . As E/F is tamely ramified, so for integers r1 , r2 and representations τji ∈ W E PE = PF and  HomPE (τj1 , τk2 ). HomPF (σ1 , σ2 ) = HomPE (σ1E , σ2E ) = j,k

As in 6.2.2 Remark, if K/F is the maximal unramified sub-extension of E/F, then bE/F (πi ) = j ρij ,  E and Lρij = τji . If HomPF (σ1 , σ2 ) = 0, there exist indices j, k such where ρij ∈ GL that HomPE (τj1 , τk2 ) = 0. Applying the lemma, we get Θ(τj1 ) = Θ(τk2 ), whence by 6.2.2 Theorem 1, Θ(π1 ) = iE/F Θ(τj1 ) = iE/F Θ(τk2 ) = Θ(π2 ). The argument reverses, and the proof is complete.

 

6.3.2 If K/F is a finite tame extension, then PK = PF as subgroups of WF , so we have a canonical map PK −→ WF \ PF WK \ taking the WK -conjugacy class of an irreducible representation of PF = PK to its WF -conjugacy class. Theorem If K/F is a finite tame extension, the diagram ≈

E(K) −−−−→ WK \ PK ⏐ ⏐ ⏐ ⏐ iK/F E(F) −−−−→ WF \ PF ≈

commutes. Proof The assertion is transitive with respect to the field extension K/F. It is enough, therefore, to treat the case where K/F is of prime degree.  K such Letξ ∈  PK and write OK (ξ) for the WK -conjugacy class of ξ. Take τ ∈ W L   that τ PK contains ξ and write τ = ρ, for ρ ∈ GLK . The Ramification Theorem gives LΘ(ρ) = OK (ξ).

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Suppose, in the first instance, that K/F is cyclic and set Gal(K/F) = Γ . Consider the representation σ = IndK/F τ . If σ is not irreducible, it is the direct sum of representations χ ⊗ σ1 , where σ1  WK = τ and χ ranges over the characters of WF /WK = Γ . Likewise aK/F ρ =  χπ1 , χ

where Lπ1 = σ1 . By 6.2.1 Theorem 2, Θ(π1 ) = iK/F Θ(ρ). However, rF+ (σ1 ) = OF (ξ), in the obvious notation, and the result follows in this case. The case where σ is irreducible follows the same course. We conclude that the result holds whenever K/F is a Galois extension of any finite degree. There remains the case where K/F is not Galois and hence totally tamely ramified. Let E/F be the normal closure of K/F and let E0 /F be the maximal unramified subextension of E/F. Our last remark shows that the result holds for the extensions E/F and E/K. Chasing diagrams, it holds for K/F.  

7 Tame Parameters In the first part of this section, we review a construction of the irreducible representations of WF in terms of their restriction to the wild inertia group PF . Attached to α∈ PF is a finite tame extension E/F and a spectrum of actions of the group of tame characters of WE on the set of representations containing α. In the second part, we seek parallel structures on the GL-side. In the final part, we connect the two sides. Most of the material derives from [19].

7.1 Structure of Galois Representations F. We give a systematic description of the representations σ ∈ W 7.1.1 Let α ∈  PF . Define ZF (α) to be the group of x ∈ WF such that αx = α. Proposition There is a finite tame extension E/F such that WE = ZF (α). Proof The kernel of α is an open normal subgroup of PF , and so contains a group PF ∩ WK , where K/F is a finite Galois extension. Thus  α extends uniquely to a smooth representation α¯ of the group PF WK such that α¯  WK is trivial. Therepresentation α¯ is then stable under the group PF WK , which therefore stabilizes α¯  PF = α. The group PF WK is of the form WL , for a finite, Galois, tamely ramified extension L/F, so the result follows.   We refer to E/F as the F-centralizer field of α and use the notation E = ZF (α). Note that PE = PF , since E/F is tame.

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7.1.2 Let α ∈  PF . We use the F-centralizer field of α to analyse the representations of WF that contain α. Proposition Let α ∈  PF and write E = ZF (α).

 (1) There exists a unique representation αI ∈  IE such that αI  PF ∼ = α and r (det αI )p = 1 for some r  0.  (2) Let ρ be an irreducible representation of IE such that ρ  PF contains α. There is a unique character φ of IE , trivial on PE = PF , such that ρ ∼ = φ ⊗ αI .

Proof The group IE /PF is pro-cyclic, whence α admits extension to a representation ρ of IE . For the same reason, any extension of α to IE is of the form φ ⊗ ρ, for a r character φ of IE trivial on PE = PF . Taking determinants, det(φ ⊗ ρ) = φp det ρ, r where p = dim α. The character φ has finite order not divisible by p, so φ ⊗ ρ ∼ =ρ r if and only if φ = 1. Moreover, there is a unique choice of φ so that φp det ρ = 1, as required for (1). This also implies, in (2), that ρ is of the form φ ⊗ αI , for a character φ of IE /PE . We can recover φ from the character det(φ ⊗ αI ), so it is uniquely determined.    Corollary (1) There exists a representation τ of WE such that τ  IE = αI .   E also satisfies τ   IE = αI , there is a unique unramified character χ (2) If τ  ∈ W of WE such that τ  = χ ⊗ τ . (3) The induced representation IndE/F τ is irreducible and contains α with multiplicity one. Proof Its uniqueness property implies that αI is stable under conjugation by WE .   That WE /IE is cyclic implies the first assertion and the others follow. In the corollary, one may recover τ from  σ = IndE/F τ as the natural representation of WE on the α-isotypic subspace of σ  PF . 7.1.3 As a minor digression, 7.1.2 Proposition implies a convenient description of IF . the space WF \ Definition Let IP(F) be the set of pairs (α, χ), where α ∈  PF and χ is a character of IZF (α) trivial on PZF (α) = PF . The group WF acts on IP(F) by conjugation. Proposition Let (α, χ) ∈ IP(F) and write E = ZF (α). The representation λ(α,χ) = IndIIFE χ ⊗ αI of IF is irreducible. The map (α, χ) → λ(α,χ) induces a canonical bijection ≈ WF \IP(F) −−−−→ WF \ IF .

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Proof An element x of IF that intertwines χ ⊗ αI must intertwine α and so lie in WE ∩ IF = IE . The induced representation is therefore irreducible and (α, χ) → IF . To show it is surjective, take λ(α,χ) gives a well-defined map WF \IP(F) → WF \  F , choose an irreducible component α of σ  PF and set E = ZF (α). It follows σ∈W  that σ  IE has an irreducible component containing α. That component is of the form χ ⊗ αI , for a character χ of IE /PF . Thus IP(F) maps onto WF \ IF . To show the map is injective, take (α, χ), (β, ψ) ∈ IP(F) and suppose that λ(α,χ) is WF -conjugate to λ(β,ψ) . In particular, α is conjugate to β so we may take β = α. The conjugation between λ(α,χ) and λ(α,ψ) is then implemented by an element of WE . Since α occurs in λ(α,χ) with multiplicity one, it follows that ψ is WE -conjugate to χ, as required.   PF the WF -conjugacy class of α. If 7.1.4 For α ∈  PF , denote by OF (α) ∈ WF \  F such that HomPF (α, σ)  F (α; m) to be the set of σ ∈ W m  1 is an integer, define W  F (α; m) depends on the conjugacy class OF (α) rather has dimension m. The set W than α. With αI as in 7.1.2, consider a representation φ ⊗ αI , where φ is a character of IE /PF , E = ZF (α). The WE -centralizer of φ (or of φ ⊗ αI ) is of the form WK , for a finite unramified extension K/E. ¯ of Let E = ZF (α) and let Em /E be the unramified extension of E, inside F,  degree m. Set Δm = Gal(Em /E). The group Δm acts on the set WEm (α; 1). Let  Em (α; 1), that is, the set  Em (α; 1)Δm -reg be the set of Δm -regular elements of W W δ of τ for which the conjugates τ , δ ∈ Δm , are distinct.  Em (α; 1)Δm -reg , the representation IndEm /F ξ is irreducible and Proposition If ξ ∈ W  lies in WF (α; m). The map  Em (α; 1)Δm -reg −→ W  F (α; m), Δ m \W ξ −→ IndEm /F ξ,

(7.1.1)

is a bijection.  Em (α; 1)Δm -reg . The representation IndEm /E θ is then irreducible and Proof Let θ ∈ W contains α with multiplicity m. The definition of E ensures that σ = IndE/F θ is  F (α; m) and depends only on the Δm -conjugacy class of θ. irreducible. It lies in W  F (α; m), the natural representation θα of On the other hand, starting from σ ∈ W WE on the α-isotypic subspace of σ is irreducible. It induces σ and contains α with  E (α; 1), so θα = ξ ⊗ τ , where multiplicity m. By 7.1.2 Corollary, there exists τ ∈ W ξ is an irreducible, m-dimensional representation of WE trivial on PF . The factor ξ is uniquely determined by θα . The representation ξ is induced from an Δm -regular  Em (α; 1). The map character φ of WEm /IE , so σ is induced from an element of W (7.1.1) is therefore surjective. The uniqueness of ξ implies it is injective.    Em (α; 1). The restric F (α; m), and write σ = IndEm /F τ , τ ∈ W Remark Let σ ∈ W tion of τ to IEm = IE takes the form φ ⊗ αI , for a character φ of IE /PF . The image of σ in WF \ IF is then parametrized by the pair (α, φ) ∈ IP(F).

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¯ let X1 (F  ) (resp. X0 (F  )) be the group 7.1.5 If F  /F is a finite extension inside F, of tamely ramified (resp. unramified) characters of WF  . If s is a positive integer, let X0 (F  )s be the group of χ ∈ X0 (F  ) such that χs = 1. Let α ∈  PF and write ZF (α) = E. Define a pairing  F (α; m) −→ W  F (α; m), X1 (E) × W (ξ, σ) −→ ξ α σ,

(7.1.2)

 F (α; m). Following 7.1.4 Proposition, there is an irreducible as follows. Let σ ∈ W  Em (α; 1)Δm -reg such that σ = IndEm /F μ. We define representation μ ∈ W  χ α σ = IndEm /F μ ⊗ χ  WEm , χ ∈ X1 (E).

(7.1.3)

This action does depend on α, not just on the orbit OF (α). Note that χ α σ = σ if and only if ξ ∈ X0 (E)m . There is a useful special case.  F (α; 1) is a principal homogeneous Proposition Under the action (7.1.3), the set W space over X1 (E). In the general case, take Em /E and Δm = Gal(Em /E) as before. Take τ ∈  Em (α; 1)Δm -reg comprises the represen E (α; 1) (cf. 7.1.2 Corollary). The set W W   tations φ ⊗ τ WEm , where φ ranges over X1 (Em )Δm -reg . Restriction of characters induces an isomorphism X1 (E)/X0 (E)m ∼ = X1 (Em )Δm . The action of X1 (E) on  WF (α; m) follows the natural action of X1 (E) on Δm \X1 (Em )Δm -reg .

7.2 Tame Parameter Fields In this section, we attach to an endo-class Θ ∈ E(F) an isomorphism class of finite tame extensions E/F. We construct a family of actions of the group X1 (E) on the set  F such that Θ(π) = Θ. This is intended to mirror Sect. 7.1, of representations π ∈ GL but the technicalities are more elaborate. 7.2.1 Let Θ ∈ E(F), Θ = 0. Let θ ∈ C(a, β) be a realization of Θ on a simple stratum [a, m, 0, β] in a matrix algebra A = EndF (V ). Any field extension F[β]/F, obtained in this way, is a parameter field for Θ. (If Θ = 0, its only parameter field is F, and there is nothing to say.) Definitions (1) Let Θ ∈ E(F). Say that Θ is totally wild if deg Θ = eΘ = pr , for an integer r  0. (2) Let Θ ∈ E(F) and let K/F be a finite, tamely ramified field extension. Say that K/F is a tame parameter field for Θ if (a) Θ has a totally wild K/F-lift and (b) if F ⊂ K   K, then Θ has no totally wild K  /F-lift.

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Let Θ ∈ E(F) be the endo-class of θ ∈ C(a, β), say. We remark that Θ is totally wild if and only if the field extension F[β]/F is totally wildly ramified. Otherwise, let T /F be the maximal tame sub-extension of F[β]/F and define the simple character θT by restriction, as in (6.1.3). The endo-class ΘT of θT is then a totally wild T /F-lift of Θ. Proposition Let Θ ∈ E(F) be the endo-class of θ ∈ C(a, β). Let T /F be the maximal tamely ramified sub-extension of F[β]/F and let K/F be a finite, tamely ramified field extension. The following are equivalent: (1) Θ has a totally wild K/F-lift; (2) there exists an F-embedding T → K. In particular, T /F is a tame parameter field for Θ and, up to isomorphism, it is the only one. The proof is a straightforward consequence of the definitions. Let Aut(T |F) be the group of F-automorphisms of the field T . The group Aut(T |F) acts on the set E(T ) in a natural way. The definitions imply: Corollary The set of totally wild T /F-lifts of Θ is a principal homogeneous space over the group Aut(T |F). 7.2.2 An endo-class Θ determines its tame parameter field up to isomorphism over F, but a specific realization of Θ gives a specific tame extension of F. Reconciliation of these divergent uniqueness properties has a degree of subtlety. We follow [19] in examining this phenomenon only for max-simple characters, that being all we need. We can remove some potential ambiguity at the start. Lemma For i = 1, 2, let [a, m, 0, βi ] be a max-simple stratum in Mn (F) and suppose that C(a, β1 ) = C(a, β2 ). Let Ti /F be the maximal tame sub-extension of F[βi ]/F. There is a unique F-isomorphism T1 → T2 of the form x → j −1 xj, where j ∈ J 1 (β1 , a) = J 1 (β2 , a). The proof is in [19] 2.6. In the notation of the lemma, say that Ti /F is an internal tame parameter field for θ ∈ C(a, βi ). Using the restriction construction of 6.1.2, we form the simple character θTi over Ti . The isomorphism T1 → T2 of the lemma fixes θ and carries θT1 to θT2 . If ι : E → E  is an isomorphism of local fields, denote by ι∗ the induced bijection E(E) → E(E  ). Proposition Let Θ ∈ E(F) have tame parameter field E/F. Let θ be a max-simple realization of Θ in some G = GLn (F) with internal tame parameter field T /F, and define θT as in (6.1.3). Let Ξ ∈ E(E) be a totally wild E/F-lift of Θ. There is a unique F-isomorphism ι : E → T such that θT has endo-class ι∗ Ξ . Proof This combines the lemma with 7.2.1 Corollary.

 

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7.2.3 We use these considerations to normalize certain character actions. We first give the internal version of the construction. Let θ be a max-simple character in some group G = GLn (F). Let J θ denote the G-normalizer of θ. Let Jθ be the maximal compact subgroup of J θ and Jθ1 the pro-p radical of Jθ . Let X1 (θ) be the character group defined in 3.1.7. Proposition Let T /F be an internal tame parameter field for θ and let G T be the G-centralizer of T × . Define s = n/deg Θ, where Θ is the endo-class of θ. (1) Let φ ∈ X1 (T ). There exists a unique character φθ ∈ X1 (θ) such that φθ (x) = φ(detT x), x ∈ G T ∩ J θ . (2) The map φ → φθ is a surjective homomorphism X1 (E) → X1 (θ) with kernel X0 (T )s . Proof Suppose θ is attached to the max-simple stratum [a, m, 0, β] in which T /F is the maximal tame sub-extension of F[β]/F. Write P = F[β]. Composition with the norm NP/T is an isomorphism X1 (T ) → X1 (P), carrying X0 (T )s to X0 (P)s , so the proposition follows from 3.1.7 Lemma.   By 7.2.2 Lemma, this construction is effectively independent of the choice of T . We remark that, if θ, θ ∈ C(a, β), then φθ = φθ . 7.2.4 We externalize the construction. To recap, θ ∈ C(a, β) is a max-simple character, P = F[β] and T /F is the maximal tame sub-extension of P/F. Let G T be the G-centralizer of T × . Define H(θ) as in 3.1.7. The endo-class of θ is Θ ∈ E(F). The character θT is the restriction of θ to H 1 (β, a) ∩ G T . Let ΘT ∈ E(T ) be the endo-class of θT . Let E/F be a tame parameter field for the endo-class Θ, and let Ξ ∈ E(E) be a totally wild E/F-lift of Θ. From 7.2.2 Proposition, there is an F-isomorphism ι : E → T such that ΘT = ι∗ Ξ . The map ι induces an isomorphism ι∗ : X1 (E) → X1 (T ). Write φ Ξ κ = ι∗ φ ⊗ κ, φ ∈ X1 (E), κ ∈ H(θ).

(7.2.1)

This defines an action of X1 (E), or of X1 (E)/X0 (E)s , on H(θ). It does depend on the choice of Ξ . 7.2.5 We re-formulate 7.2.4 in terms of representations of general linear groups. Let Θ ∈ E(F), let s  1 be an integer and set n = s deg Θ. Define  F (Θ; s) = {π ∈ A0n (F) : Θ(π) = Θ}. GL  F (Θ; s), let E/F be a tame parameter field for Θ and let Ξ be a totally Let π ∈ GL wild E/F-lift of Θ. Choose a max-simple realization θ of Θ in G = GLn (F). Let (J θ , Λ) be the EMST in G that contains θ. In particular, π = c-IndJGθ Λ.

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Take κ ∈ H(θ), and let λ be the natural representation of J θ on the space of ˇ Thus Λ = κ ⊗ λ. If φ ∈ X1 (E), we may form the repreJθ1 -fixed vectors in Λ ⊗ κ. sentation φ Ξ κ ∈ H(θ) and then   φ Ξ π = c-Ind (φ Ξ κ) ⊗ λ . This gives a canonical action  F (Θ; s) −→ GL  F (Θ; s), X1 (E) × GL (φ, π) −→ φ Ξ π,

(7.2.2)

 F (Θ; s). In particular, GL  F (Θ; 1) is a principal homogeneous space of X1 (E) on GL over X1 (E)/X0 (E)s . We note that when Θ = 0, we have E = F and (7.2.2) reduces to the standard twisting action  F (0; s). (φ, π) −→ φπ, φ ∈ X1 (F), π ∈ GL  F (0; s) is the set of π ∈ A0s (F) of level zero. In this case, GL

7.3 Comparison of Tame Parameters We relate the two preceding lines of thought. 7.3.1 Recall that, if α ∈  PF , then OF (α) is the WF -conjugacy class of α. Tame Parameter Theorem Let α ∈  PF and set ZF (α) = E. If Θ ∈ E(F) satisfies L Θ = OF (α), then (1) deg Θ = [E : F] dim α and (2) E/F is a tame parameter field for Θ. Proof Start with the case E = F. In this case, α extends to a representation σ of WF . The extension has dimension pr = dim α, r  0, and is totally ramified: if χ = 1 is an unramified character of WF , then χ ⊗ σ  σ. If σ = Lπ, where π ∈ A0pr (F), then π has the same property: χπ  π for any unramified character χ = 1 of F × . We deduce that the endo-class Θ = Θ(π) is totally wild, in that deg Θ = eΘ = pr . It follows that F provides a tame parameter field for Θ(π). Since OF (α) = LΘ(π), we are done in this case. We turn to the general case E = F. Lemma Let OF (α) = LΘ, Θ ∈ E(F), and let T /F be a tame parameter field for Θ. There exists an F-embedding T → E and deg Θ = [T : F] dim α. Proof By 6.3.2 Theorem, we have Θ = iE/F ΘE , where ΘE ∈ E(E) satisfies L ΘE = OE (α). By the first case above, the endo-class ΘE is totally wild of degree pr = dim α.  

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Set d = deg Θ = [T : F] dim α and take π ∈ A0d (F) with Θ(π) = Θ. The representation σ = Lπ thus has dimension d . However, dim σ is divisible by [E : F] dim α, so [E : F] = [T : F] and the lemma implies E ∼   = T , as required. Corollary Let m be a positive integer, let Θ ∈ E(F) and let α ∈ LΘ. The Langlands correspondence induces a bijection ≈  F (α; m).  F (Θ; m) −−− GL −→ W

This is an immediate consequence of the theorem. If E = ZF (α), each of the sets  F (α; m) carries a collection of actions of the group X1 (E). We show  F (Θ; m), W GL in the next section that the Langlands correspondence also matches these actions.

8 Description of the Langlands Correspondence Starting from our “black box” correspondence E(F) → WF \ PF , we use an explicit method to define a canonical bijection F,  F −→ W GL π −→ Nπ, that we call the na¨ive correspondence. We estimate the difference between the naïve correspondence and the Langlands correspondence. The material of this section is almost entirely taken from [19].

8.1 Complements By way of preparation, we re-examine the concept of (extended) maximal simple type in light of the ideas of Sect. 7.2. 8.1.1 Recall the Green parametrization [34, 56] of the irreducible cuspidal representations of GLn (k), where k is a finite field. Let l/k be a field extension of degree n and set Γ = Gal(l/k). Embed l in Mn (k) as a k-subalgebra: any two such embeddings are conjugate, so the choice will be immaterial. Let χ be a Γ -regular character of l × . There is then a unique irreducible cuspidal representation λχ of GLn (k) such that tr λχ (ζ) = (−1)n−1

γ∈Γ

χ(ζ γ ),

(8.1.1)

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for all Γ -regular elements ζ of l × . The map χ → λχ is a bijection between the set of Γ -orbits of Γ -regular characters χ of l × and the set of equivalence classes of irreducible cuspidal representations of GLn (k). The representation λχ has central  character χ  k × . 8.1.2 Returning to our standard situation, let E/F be unramified of degree n and set Γ = Gal(E/F). Let μE be the group of roots of unity in E of order not divisible by p. Reduction modulo pE induces an isomorphism μE → k× E. Let X1 (E)Γ -reg be the set of Γ -regular characters χ ∈ X1 (E). Let 1 denote the trivial character of PF . There are then canonical bijections  F (1; n), Γ \X1 (E)Γ -reg −→ W χ −→ σχ ,

 F (0; n), Γ \X1 (E)Γ -reg −→ GL χ −→ πχ .

The first map is given by σχ = IndE/F χ. For the second, take a maximal order m in A = Mn (F) and embed E in A so that E × ⊂ Km = F × Um . (Any two such configurations are conjugate in G = GLn (F).) Such a map induces an embedding 1 ∼ that of μE in Um /Um = GLn (kF ). Define an EMST (Km , ΛFχ ) in G by deeming   1 and that ΛFχ  F × be ΛFχ  Um be the inflation of the representation λχ|μE of Um /Um  a multiple of χ  F × . The representation πχ is then G ΛFχ . πχ = c-IndK m

Proposition If χ is a Γ -regular character of E × , then Lπχ = σχ , where χ (x) = χ(x) (−1)(n−1)υE (x) , x ∈ E × .  F (1; n) is clas F (0; n) → W This description of the Langlands correspondence GL sical. It is a first exercise in the application of automorphic induction, as in [3, 41]. Remark Let P/F be totally wildly ramified. With the notation otherwise as before,  F (0; n) → define χP = χ ◦ NPE/E . The map Λχ → ΛPχ = ΛχP is then a bijection GL  GLP (0; n), corresponding to restriction of representations from WF to WP . Observe also that, if ζ is a Γ -regular element of μE , it is also a Γ -regular element of μPE and tr ΛχP (ζ) = tr Λχ (ζ [P:F] ). 8.1.3 It is necessary to re-phrase the discussion of extended maximal simple types (3.1.2, 3.1.3) in terms of tame parameter fields. Let θ be a max-simple character in G = GLn (F), say θ ∈ C(a, β). Write P = F[β] and let T /F be the maximal tame sub-extension of P/F. Let G P be the G-centralizer of P × and aP the a-centralizer of P. Recall that T(θ) is the set of equivalence classes of extended maximal simple types in G containing θ. Put d = [P : F] and n = ds. We use the bijection

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 T (0; s) −→ GL  P (0; s) GL of 8.1.2 Remark, induced by the inclusion T → P. Let T(0; aP ) be the set of equivalence classes of EMST’s in G P of level zero, based on the maximal oP -order aP . Likewise define T(0; mT ), for a maximal oT -order mT in Ms (T ). Re-phrasing 8.1.2 Remark, we get a canonical bijection T(0; mT ) −→ T(0; aP ), Λ −→ ΛP .

(8.1.2)

The definition of EMST amounts to a canonical pairing T(0; aP ) × H(θ) −→ T(θ), (Λ, κ) −→ Λ ⊗ κ. (To form the tensor product here, we extend Λ, by triviality on J 1 (θ), to a representation of J(θ).) Combining with (8.1.2), we get a pairing T(0; mT ) × H(θ) −→ T(θ), (Λ, κ) −→ ΛP ⊗ κ = Λ ΘT κ.

(8.1.3)

Here, ΘT is the T /F-lift of the endo-class Θ of θ defined by the inclusion of T in P. The pairing is compatible with the ΘT -actions of X1 (T ) on T(0; mT ) and H(θ).

8.2 Algebraic Induction Maps Let E/F be a finite tame extension. We construct a family of canonical inductionlike maps from certain EMST’s over E to EMST’s over F, refining the map iE/F : E(E) → E(F). The material of this section is taken from section 5 of [19], to which we refer for the copious detail. 8.2.1 Fix a max-simple character θ in G = GLn (F). Thus θ ∈ C(a, β), for a maxsimple stratum [a, m, 0, β] in Mn (F). Let P = F[β] and let E/F be the maximal tame sub-extension of P/F. As usual, θE is the restriction of θ to H 1 (β, a) ∩ G E , where G E is the G-centralizer of E × . Let Θ, ΘE be the endo-classes of θ, θE respectively. The class ΘE is totally wild. In this subsection, we concentrate on the case where E/F is totally ramified and deg Θ = [P : F] = n. Write J = J(θ) = J(β, a) and J E = J(θE ). Again, T(θ) is the set of equivalence classes of EMST’s in G that contain θ, and similarly over E. Theorem Let Λ ∈ T(θ). There exists a unique ΛE ∈ T(θE ) such that tr Λ(x) = E/F tr ΛE (x),

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for all x ∈ J E such that υF (det G x) is relatively prime to n and a constant E/F = ±1. The map T(θ) −→ T(θE ), Λ −→ ΛE ,

(8.2.1)

is a bijection and an isomorphism of X1 (E)-spaces. This result is 5.3 Proposition of [19]. It is based on a construction from the representation theory of finite groups, known as the “Glauberman correspondence”: see Glauberman’s original paper [32] or the development in Isaacs’ book [45]. We translate in terms of representations of general linear groups. Corollary Let Θ ∈ E(F) satisfy deg Θ = eΘ and let E/F be a tame parameter field for Θ. Let ΘE be a totally wild E/F-lift of Θ. The inverse of the map (8.2.1) induces a canonical bijection ≈  E (ΘE ; 1) −−−  F (Θ; 1) indE/F : GL −→ GL

(8.2.2)

satisfying ind E/F (φρ) = φ ΘE ind E/F (ρ),  E (ΘE ; 1). for φ ∈ X1 (E) and ρ ∈ GL  E (ΘE ; 1) and GL  F (Θ; 1) are principal homogeneous spaces over Recall that GL X1 (E) and observe the parallel with the definition of the operation α in 7.1.5. Remark Let F ⊂ E  ⊂ E. The map indE/F factors through indE/E  . In the context of the theorem, we have tr Λ(x) = E/F tr ΛE (x) = E/F −1 E/E  tr ΛE  (x), where ΛE  = indE/E  ΛE . We may write Λ = indE  /F ΛE  , giving a relation indE/F = indE  /F ◦ indE/E  . 8.2.2 There is also an unramified induction map, for which we need a more elaborate notational scheme. We follow that of [19] 5.4 for convenience of reference. Notation (1) Θ ∈ E(F) has degree d and θ ∈ C(a, β) is a max-simple realization of Θ in G = GLn (F), where n = ds, s  1. (2) P0 = F[β] and E0 /F is the maximal tame sub-extension of P0 /F. (3) P/P0 is an unramified extension of degree s, such that P × ⊂ J(β, a). (4) E/F is the maximal tame sub-extension of P/F and K/F is the maximal unramified sub-extension of E/F. (5) Δ = Gal(P/P0 ) = Gal(E/E0 ) = Gal(K/K ∩ E0 ).

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We remark that any two choices of P/P0 are conjugate by an element of Ua that commutes with P0 . Define the simple character θK over K as in (6.1.3), and likewise θE . The set H(θK ) is a principal homogeneous space over X1 (E) and hence, by restriction, a space over X1 (E0 ). The heart of the matter is the following lemma. Lemma 1 There is a canonical X1 (E0 )-map K/F : H(θ) −→ H(θK ).

(8.2.3)

It is injective and its image is the space H(θK )Δ of Δ-fixed points in H(θK ). The construction of K/F  is an elaborate exercise in the use of the Glauberman correspondence. If det κ  J (β, a) has p-power order, the same is true for K/F (κ). In other words, K/F preserves the “standard” elements of the spaces H0 , as in 3.1.1 Proposition (3). It is also necessary to consider the set H(θK )Δ-reg of Δ-regular elements of H(θK ). Lemma 2 A representation κ ∈ H(θK ) is Δ-regular if and only if there exists κ0 ∈ H(θ) and ξ ∈ X1 (E)Δ-reg such that κ = ξ ΘE K/F (κ0 ). The -product here is taken relative to the endo-class ΘE of the simple character θE . The simple character θK lies in C(aK , β), where aK is the a-centralizer of K. The extension K[β] = P/K is totally ramified so T(θK ) = H(θK ). Let Φ = ϕ ΘE K/F (κ) ∈ T(θK )Δ-reg . The character ϕ ∈ X1 (E) defines an EMST Λϕ , of level zero, over the field E0 = E Δ . Set indK/F Φ = Λϕ ΘE κ.

(8.2.4)

Theorem The map (8.2.4) induces a canonical bijection ≈

indK/F : Δ\T(θK )Δ-reg −−−−→ T(θ),

(8.2.5)

and hence a canonical bijection  ≈  K (ΘK ; 1)Δ-reg −−−  F (Θ; s). Δ GL −→ GL

(8.2.6)

8.3 Naïve Correspondence The algebraic induction maps (8.2.2), (8.2.6) enable the construction of an approximate version of the Langlands correspondence.  F be totally wild, that is, deg Θ = eΘ is a power of p. Thus 8.3.1 Let π ∈ GL  F (Θ; 1), where the endo-class Θ ∈ E(F) is totally wild. It follows that the π ∈ GL

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conjugacy class LΘ ∈ WF \ PF has exactly one element, α say. Defining Nπ = Lπ, we get a canonical bijection  F (α; 1),  F (Θ; 1) −→ W GL π −→ Nπ = Lπ.

(8.3.1)

This map commutes with isomorphisms of the base field F and respects twisting with characters χ of F × , N (χπ) = Lχ ⊗ Nπ. Since the tame parameter field of Θ is F, this says that the map also preserves the -actions of X1 (F). Remark The map (8.3.1) is almost completely determined by the relation Θ = OF (α).

L

(8.3.2)

 F (Θ; 1) and σ = LΘ, then First, if π ∈ GL ωπ = det σ.

L

(8.3.3)

Let dim σ = deg Θ = pr , for some r  1. The relations (8.3.2), (8.3.3) determine π up to twisting with an unramified character of F × of order dividing pr . Following [10], there exists a totally ramified representation π  ∈ A0ps (F), with 0  s < r, such that sw(πˇ  × π) is not divisible by p. This distinguishes the various π satisfying the two relations. Further, if sw(π) is not divisible by p, one may take π  to be the trivial character of GL1 (F): this follows from 4.3.3 Corollary.  F is totally ramified. That is, π ∈ 8.3.2 Consider next the case where π ∈ GL  F (Θ; 1), where eΘ = deg Θ. Let E/F be a tame parameter field for Θ, and choose GL PF =  PE a totally wild E/F-lift ΘE of Θ. Embed E in F¯ in some way, and define α ∈   E (ΘE ; 1) by the condition LΘE = {α}. Following (8.2.2), there is a unique ρ ∈ GL such that π = indE/F ρ. The representation ρ is totally wild, so we may set N

π = N(indE/F ρ) = IndE/F Nρ.

This gives a bijection  F (α; 1),  F (Θ; 1) −→ W GL π −→ Nπ.

(8.3.4)

It satisfies N

(χ ΘE π) = χ α Nπ, χ ∈ X1 (E),

(8.3.5)

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and commutes with isomorphisms of the base field.  F (Θ; s), for an arbitrary Θ ∈ E(F) and an integer s  1. 8.3.3 Finally, let π ∈ GL ¯ Let ΘE be Let E/F be a tame parameter field for Θ, viewed as a subfield of F. a totally wild E/F-lift of Θ and define α ∈  PF by the condition LΘE = {α}. Let Es /E be unramified of degree s, and put Δ = Gal(Es /E). Let K/F be the maximal unramified sub-extension of Es /F and identify Δ with Gal(K/K ∩ E).  K (ΘK ; 1)Δ-reg such that π = indK/F ρ. The repAs in (8.2.6), there exists ρ ∈ GL resentation ρ is uniquely determined up to conjugation by Δ. The endo-class ΘK is iEs /K ΘEs , where ΘEs is the unique Es /E-lift of ΘE .  K (α; 1)Δ-reg , so we may define The representation Nρ lies in W N

π = IndK/F Nρ.

We therefore have a bijection  F (α; s),  F (Θ; s) −→ W GL π −→ Nπ

(8.3.6)

with the property N

(φ ΘE π) = φ α Nπ.

(8.3.7)

8.3.4 Altogether, we have produced a canonical bijection F,  F −→ W GL π −→ Nπ. It commutes with isomorphisms of the base field and twisting by characters of F × . However, the determinant character det Nπ is not in general the same as the central character ωπ . The naïve correspondence and the Langlands correspondence behave differently with respect to automorphisms of the coefficient field C. On the other hand, the naïve correspondence has the “higher twisting” property: Theorem Let Θ ∈ E(F) and let E/F be a tame parameter field for Θ. Let Ξ ∈ E(E) PE , then be a totally wild E/F-lift of Θ. If α = LΞ ∈ WE \ N

 F (Θ; s), s  1. (φ Ξ π) = Lφ α Nπ, π ∈ GL

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8.4 Main Theorems We compare the naïve correspondence π → Nπ with the Langlands correspondence π → Lπ and then deduce some consequences for the Langlands correspondence itself. 8.4.1 The correspondences are related as follows. Comparison Theorem Let Θ ∈ E(F) and let s  1 be an integer. Let E/F be a ¯ and let ΘE be a totally tame parameter field for Θ, identified with a subfield of F, L  wild E/F-lift of Θ. Define α ∈ PF by the condition ΘE = {α}. There is a character μ = μFα,s ∈ X1 (E), uniquely determined modulo X0 (E)s , such that  F (Θ; s). π = μ α Nπ, π ∈ GL

L

In the essentially tame case dim α = 1, the character μFα,s is worked out completely in [13, 14, 16]. The answer is somewhat complex, although μ (modulo X0 (E)s ) has order at most 4. The general case is similar in many respects, although more complex in others. We discuss it briefly in 8.4.4 below. Remark Let  be a prime number,  = p. The Comparison Theorem has a direct application to the theory of -modular representations. In brief, it implies that two representations π1 , π2 ∈ A0n (F), with central characters of finite order, are congruent modulo  if and only if the representations Lπi are congruent mod. . See [20] for a full and self-contained account. 8.4.2 Comparing the theorem with the higher twisting property of the naïve correspondence in 8.3.4 Theorem, we get the following corollary. We retain the notation of the Comparison Theorem. Homogeneity Theorem The Langlands correspondence satisfies (φ ΘE π) = φ α Lπ,

L

 F (Θ; s) and φ ∈ X1 (E). for π ∈ GL 8.4.3 We determine exactly how the Langlands correspondence behaves relative to maximal simple types (or inertial Galois parameters). We use the same notation as in the Comparison Theorem. In particular, π ∈  F (Θ; s) ∩ A0n (F), where n = s deg Θ. Thus π contains a simple character θ of GL endo-class Θ. We may assume E is a tame parameter field for θ in the internal sense and let ΘE be the endo-class of θE . Write σ = Lπ. Let αI be the unique representation of the inertia group IE such that   αI PF = α and det αI has p-power order (7.1.2 Proposition). The representation σ  IE thus has an irreducible component χ ⊗ αI , where χ is a character of IE /PF . The WE -stabilizer of χ is WEs , where Es /E is unramified of degree s. The character

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χ determines a maximal simple type λχ in GLs (E), as in 8.1.2. Using the obvious, restricted, analogue of the notation (8.1.3), we get a maximal simple type λχ ΘE κ in GLn (F), for any κ ∈ H0 (θ). Types Theorem Let Δs = Gal(Es /E). (1) There is a unique representation κ1 ∈ H0 (θ) such that tr κ1 is constant on the set of Δs -regular elements of μEs . (2) Let δ be the character of J (θ)/J 1 (θ) that has order 2 when e(E|F) is even, and is trivial otherwise. (3) The representation π contains the maximal simple type λχ ΘE δκ1 . We remark that the factor δ is trivial when p = 2, because E/F is tamely ramified. The inertial parameter (7.1.3) of σ is the class of (α, χ) in WF \IP(F). 8.4.4 The character μFα,s is quite complex in structure. The restriction of μ = μFα,s to units is readily computable, as encapsulated here in the Types Theorem. Using the notation of 8.3.3, the contribution from the unramified step K/F is relatively straightforward. It is of order 2 (mod. X0 (E0 )s ), and its value at a suitable prime element is discussed in [19] 10.7 Corollary. The totally ramified step seems quite difficult. Using the notation of 8.3.2, one has to work inductively, along the group Γ of F-automorphisms of E. If Γ is trivial, the existence of a character μ = μFα,1 , with the required properties, is relatively straightforward [19] 8.3. However, it is not clear how to compute it: certainly the base change method of the essentially tame case is not completely effective. If Γ is not trivial, let l be the largest prime divisor of |Γ | and let L/F be the unique sub-extension of E/F of degree l. Inductively, one can assume that μLα,1 is known. The contribution from L/F is then given by an automorphic induction calculation leading to a formula discussed in [19] 8.9. 8.4.5 This totally ramified case has an interesting feature. Again in the notation of  F (Θ; 1) and write π = indE/F ρ, where ρ ∈ GL  E (ΘE ; 1). We view π 8.3.2, let π ∈ GL × ∼ as a representation of a group G = GLn (F) containing E , and ρ as a representation of the G-centralizer G E of E × . With Γ as before, one has [19] 8.1 tr π(h) = E/F

tr ργ (h),

(8.4.1)

γ∈Γ

valid for all elements h of G E that are elliptic regular in G and such that υF (det G h) is relatively prime to n. Here, E/F = ±1 is the constant sign coming from the Glauberman correspondence, as in 8.2.1. The characters appearing here are uniquely determined by their values on the specified set of elements h.  F (α; 1). Write σ = IndE/F τ . Thus σ ∈ W  F (α; 1) Put another way, let τ = Lρ ∈ W N L     and σ = π = π , for some π ∈ GLF (Θ; 1). Indeed, π = μ  π, μ = μFα,1 , so tr μ−1  π  (h) = E/F

γ∈Γ

tr ργ (h),

(8.4.2)

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for elements h as before. Equally, tr π  (h) = E/F

tr (μρ)γ (h).

(8.4.3)

γ∈Γ

In the case where E/F is cyclic, we have π  = aE/F ρ and we have retrieved the automorphic induction equation in a novel version. In addition, (8.4.3) is valid when E/F is not cyclic, or even Galois.

9 Higher Ramification and the Herbrand Function Following the Types Theorem of 8.4.3, we know how the structure of a cuspidal  F on the standard  F is reflected in the behaviour of σ = Lπ ∈ W representation π ∈ GL subgroups PF ⊂ IF ⊂ WF . We look to next steps. The group PF carries a canonical filtration by ramification subgroups, here denoted RF (x), x ∈ R, x  0. The behaviour of the restriction σ  RF (x) provides penetrating insight into the structure of σ. In this section, we follow [21] to show how, in principle, it is predicted by the endo-class Θ(π) of simple characters contained in π.

9.1 Ramification and Conductors This section is a review of ramification theory and its connections with conductors. We start from the standard ramification theory, for finite Galois extensions, as in [69]. 9.1.1 Let E/F be a finite Galois extension, and set Gal(E/F) = Γ . If x  −1 is a real number, let Γx be the associated ramification subgroup of Γ , in the lower numbering, and Γ x that in the upper numbering. The two filtrations are connected by Γy = Γ ϕE/F (y) , Γ x = ΓψE/F (x) , where ϕE/F , ψE/F are the Herbrand functions for E/F. Both functions are piecewise linear with ϕE/F concave and ψE/F convex. The Herbrand functions are defined for x  −1, but we focus entirely on the range x  0. If K/F is a Galois sub-extension of E/F, then ψE/F = ψE/K ◦ ψK/F . If, on the other hand, E/F is a finite separable extension, not necessarily Galois, there is a finite Galois extension E  /F containing E. The expression ψE/F = ϕE  /E ◦ ψE  /F

(9.1.1)

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defines a continuous, strictly increasing function ψE/F . Proposition Let E/F be a finite, separable extension. (1) The definition (9.1.1) of ψE/F does not depend on the choice of E  .  (x) (2) The function ψE/F (x) is convex in the region x  0, and its derivative ψE/F has only finitely many discontinuities. (3) If F ⊂ K ⊂ E, then ψE/F = ψE/K ◦ ψK/F .

(9.1.2)

For the proof and some more detail, see [22] 1.2. The definition of ψE/F via (9.1.1) is the same as that of [31].  (x), in the region x > 0, are called the jumps of the The discontinuities of ψE/F extension E/F. We note a particularly simple case. Example If K/F is tamely ramified, then ψK/F (x) = e(K|F)x, x  0. 9.1.2 For x ∈ R, x  0, let

x RF (x) = lim ΓE/F , ← − E/F

where E/F ranges over the finite Galois extensions in F¯ and ΓE/F = Gal(E/F). Thus RF (x) is a compact subgroup of RF (0) = IF ,normalized by WF . In parallel, let R+ F (x) be the closure in RF (x) of the group y>x RF (y). This is a compact normal subgroup of WF . In particular, R+ (0) is the wild inertia group PF . For x > 0, F (x)  = R (x) if and only if x is rational: see [21] 2.4 Corollary. The quotient R+ F F (x) is a compact abelian group of exponent p. RF (x)/R+ F The groups RF (x) are the ramification subgroups of WF , often denoted WxF . This traditional notation is unusable in the context of representation theory. 9.1.3 We insert a couple of technical results taken from [22] 1.6. We do not refer to them until late in the next section, but they influence the course of the intervening arguments. If E/F is a finite separable extension, define the wild exponent wE/F of E/F by wE/F = dE/F + 1 − e(E|F),

(9.1.3) d

where dE/F is the differential exponent of E/F. That is, the different of E/F is pEE/F . If E/F is not separable, one may take wE/F = ∞. Proposition 1 Let E/F be a separable, totally wildly ramified extension of degree pr . If j∞ = j∞ (E|F) is the largest jump of E/F, then ψE/F (x) = pr x − wE/F , x  j∞ .

(9.1.4)

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The proposition is equivalent to the standard result [69] IV Proposition 4 connecting the Artin conductor with ramification groups. Corollary If E/F is totally wildly ramified of degree pr , and j∞ = j∞ (E|F), then (pr − 1)j∞  wE/F  pr−1 (p − 1)j∞ . One has (pr − 1)j∞ = wE/F if and only if j∞ is the unique jump of E/F. Inseparable extensions. If E/F is purely inseparable of degree n, one may define ψE/F (x) = nx. The definition can then be extended to arbitrary finite extensions via the transitivity relation (9.1.2). If E/F is not separable, the proposition and corollary remain true with j∞ = wE/F = ∞. Inseparable extensions play no rôle on the Galois side. On the GL-side, a parameter field of a simple character need not be separable over F but we will need to use the classical Herbrand functions in that context. The problem can be avoided via an approximation argument, but this ad hoc approach is simpler. The largest jump appears in another rôle. Proposition 2 If E/F is a finite separable extension, then RF (x) ∩ WE = RE (ψE/F (x)), x  0, and similarly for the groups R+ . Moreover, R+ F (x) ⊂ WE if and only if x  j∞ (E|F).  F . For x  0, the group 9.1.4 Let σ be an irreducible representation of WF , σ ∈ W  RF (x) is compact, so σ  RF (x) is semisimple. Any two irreducible components of the restriction are WF -conjugate and occur with the same multiplicity. The same applies with R+ F (x) replacing RF (x). The definition of ramification groups implies the existence of x > 0 such that RF (x) ⊂ Ker σ. We may accordingly define the slope ς(σ) of σ by ς(σ) = inf {x > 0 : RF (x) ⊂ Ker σ}.

(9.1.5)

In particular, ς(σ)  0. One has ς(σ) = sw(σ)/ dim σ,

(9.1.6)

by [36] Théorème 3.5.  F . If ς = ς(σ), then σ is trivial on R+ (ς). If ς = 0, then Proposition Let σ ∈ W F  σ  RF (ς) is a direct sum of non-trivial characters. The point here is that, for x  0, the commutator group [RF (x), RF (x)] is con tained in R+ F (x). If ς(σ) = 0, then σ RF (0) is a direct sum of WF -conjugate characters, but they may be trivial.

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Remark Let π ∈ A0n (F) and let Θ be the endo-class of a simple character contained in π. If σ = Lπ, then ς(σ) = sw(π)/n = ςΘ ,

(9.1.7)

in the notation of (5.2.3).  F , define 9.1.5 For σ, τ ∈ W (σ, τ ) = inf {x > 0 : HomRF (x) (σ, τ ) = 0}.  +   The pairing  is symmetric. If δ = (σ, τ ) > 0, then σ  R+ F (δ) and τ RF (δ) have an irreducible component in common, while HomRF (δ) (σ, τ ) = 0. Consequently,  takes non-negative rational values. It satisfies an ultrametric inequality, (σ1 , σ3 )  max {(σ1 , σ2 ), (σ2 , σ3 )}, F. for all σi ∈ W  F , depends only on the restrictions of σ and τ to The value (σ, τ ), σ, τ ∈ W + PF = RF (0). It is therefore often more convenient to regard  as a pairing on the  F → WF \ PF . If rF+ denotes the canonical map W PF as in 6.3.1, the orbit space WF \ two versions are related by (σ, τ ) = (rF+ (σ), rF+ (τ )). The second version has the property (ζ, ξ) = 0

⇐⇒

ζ = ξ,

PF . That is,  defines an ultrametric on WF \ PF . for ζ, ξ ∈ WF \ 9.1.6 Following [35] and [21] section 3, the ultrametric pairing  leads to a formula for the conductor of a tensor product.  F . There is a unique continuous function Σσ (x), x  0, Proposition 1 Let σ ∈ W such that   sw(σˇ ⊗ τ ) F. = Σσ (σ, τ ) , τ ∈ W dim σ dim τ

(9.1.8)

The decomposition function Σσ is given as follows. Let V be the representation space of σ. Thus σˇ acts on the space Vˇ = HomC (V, C), and the space X = Vˇ ⊗ V carries the representation σˇ ⊗ σ of WF . For δ  0, let X (δ) be the space of R+ F (δ)fixed points in X . Let X  (δ) be the unique R+ F (δ)-complement of X (δ) in X . Each of the spaces X (δ), X  (δ) is WF -stable so, in particular, the conductor sw X  (δ) is defined. The function Σσ is then

Arithmetic of Cuspidal Representations

  Σσ (δ) = δ dim X (δ) + sw X  (δ) (dim σ)2 , δ  0.

109

(9.1.9)

Therefore: Proposition 2 The continuous function Σσ is strictly increasing, piecewise linear and convex. The derivative Σσ (x) has only finitely many discontinuities in the region x > 0. The discontinuities of Σσ are the jumps of σ: they all lie in the interval 0 < x < ς(σ). We remark that if x is not a jump of σ, then Σσ (x) = dim EndRF (x) (σ)/(dim σ)2 .

(9.1.10)

PF . It is This formula (9.1.9) shows that Σσ depends only on the orbit rF+ (σ) ∈ WF \ sometimes more convenient to use the notation Σσ = Σξ , where ξ = rF+ (σ).  F in representation-theoretic terms. 9.1.7 One may characterize the jumps of σ ∈ W  F , let  > 0 and let σ be an irreducible component of Proposition Let σ ∈ W  σ  RF (). Let  be the group of characters of RF ()/R+ F (). The following are equivalent. (1) The function Σσ (x) is continuous at x = . (2) The representation σ is not WF -conjugate to χ ⊗ σ , for any χ ∈  , χ = 1.  In particular, if σ  R+ F () is not irreducible then  is a jump of σ. This result is 8.1 Proposition of [21]. Note that σ can be irreducible on R+ F () even when  is a jump.

9.2 Comparison of Ultrametrics The Ramification Theorem of 6.3.1 asserts that the Langlands correspondence spePF . These sets carry respectively the cializes to a canonical bijection E(F) → WF \ ultrametrics A, , but the Langlands correspondence does not give an isometry PF , ). If Θ ∈ E(F) and δ > 0, the δ-neighbourhood of Θ (in (E(F), A) → (WF \ the obvious sense) is mapped to the -neighbourhood of LΘ, for a certain  > 0. However,  depends on both δ and Θ. The relation takes the form δ = Θ (), for a certain “Herbrand function” Θ attached to Θ. In the rest of this section, we review the definition and main properties of this function. 9.2.1 To Θ ∈ E(F) we have attached two functions, namely the structure function Θ (5.3.3 Theorem 3) and the decomposition function Σ LΘ (9.1.6). Define the Herbrand function Θ (x), x  0, by Θ = −1 Θ ◦ Σ LΘ .

(9.2.1)

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Proposition (1) The function Θ is continuous, strictly increasing and piecewise linear. (2) It satisfies Θ (0) = 0, while Θ (x) = x for x  ςΘ . (3) The derivative Θ is continuous except at a finite number of points y, all of which satisfy 0 < y < ςΘ . Proof Part (1) is clear. In (2), one has Θ (x) = x for x  ςΘ , either from the definition in [21] or as a consequence of 5.3.3 Theorem 1. If π ∈ A0n (F) satisfies Θ(π) = Θ and σ = Lσ, (9.1.9) implies Σσ (x) = x for x  ςσ = ςΘ . This proves the first assertion. With this same notation, Θ (0) = sw(πˇ × π)/n2 = sw(σˇ ⊗ σ)/n2 = Σσ (0), as the result follows. Part (3) follows from the corresponding properties of Θ and   Σσ . Over the range 0  x < ∞, the Herbrand function Θ is convex if and only if it is linear (see 9.4.2 below). In the interesting range 0  x  ςΘ , it is convex only in special cases (cf. 10.1.2 below). Going forward, we tend to view the structure function Θ as known, on the basis that we usually start from a simple character. We use the Herbrand function as a means of calculating the decomposition function Σσ . 9.2.2 We make a fundamental connection. Higher Ramification Theorem If Θ, Υ ∈ E(F), then A(Θ, Υ ) = Θ (( LΘ, LΥ )). In particular, if  > 0 and δ = Θ (), then ( LΘ, LΥ ) < 

⇐⇒

A(Θ, Υ ) < δ,

( Θ, Υ )  

⇐⇒

A(Θ, Υ )  δ,

L

L

(9.2.2)

for any Υ ∈ E(F). Proof Choose π ∈ A0m (F) and ρ ∈ A0n (F) so that Θ(π) = Θ and Θ(ρ) = Υ . According to 5.3.3 Theorem 3, sw(πˇ × ρ) = Θ (A(Θ, Υ )). mn Setting σ = Lπ and τ = Lρ, (9.1.8) says that sw(σˇ ⊗ τ ) = Σσ ((σ, τ )). mn

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In the alternate notation, Σσ ((σ, τ )) = Σ LΘ (( LΘ, LΥ )), while the Langlands correspondence has the property sw(σˇ ⊗ τ ) sw(πˇ × ρ) = . mn mn The first assertion follows. The function Θ is strictly increasing, whence follows the second.   To interpret (9.2.2) in more concrete the proof and choose  terms, use the notation of L L an irreducible component σ of σ  RF (). The condition ( Θ, Υ ) = (σ, τ ) <    is equivalent to σ being a component of τ RF (). On the other hand, in the notation of 5.2.5, A(Θ, Υ ) < δ is equivalent to tcδ (Θ) = tcδ (Ξ ). In other words, it is equivalent to π, ρ containing the same δ-truncated endo-class of simple characters. A similar comment applies to the non-strict inequality. 9.2.3 There is a useful technical consequence of the theorem. Corollary Let Θ, Υ ∈ E(F). If x  A(Θ, Υ ), then Θ−1 (x) = Υ−1 (x). Equivalently, Θ (y) = Υ (y) for y  (LΘ, LΥ ). Proof Let δ > A(Θ, Υ ) and set  = Θ−1 (δ). The ultrametric inequality then implies that an endo-class Ξ ∈ E(F) satisfies A(Ξ, Υ ) < δ if and only if A(Ξ, Θ) < δ. The second condition is equivalent to (LΞ, LΘ) <  by the theorem, while the first is equivalent to (LΥ, LΞ ) < Υ−1 (δ). On the other hand,   (LΥ, LΞ )  max (LΥ, LΘ), (LΘ, LΞ ) < . It follows that Υ−1 (δ)   = Θ−1 (δ) for δ > A(Θ, Υ ). By symmetry, Υ−1 (δ) = Θ−1 (δ), δ > A(Θ, Υ ). By continuity, the relation holds for δ  A(Θ, Υ ).

 

9.3 Lifting and Interpolation The Herbrand function Θ , Θ ∈ E(F), has a tidy property relative to tame lifting that is not shared by its factors Θ , Σ LΘ . This leads to a method of calculating Θ directly from Θ, without reference to the Galois side. 9.3.1 The basic lifting property is the following (7.1 Proposition of [21]). Proposition Let Θ ∈ E(F). Let K/F be a finite tame extension and set e = e(K|F). If Θ K is a K/F-lift of Θ, then Θ K (x) = eΘ (x/e), x  0.

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The proposition is a detailed refinement of the Tame Parameter Theorem of 7.3.1. 9.3.2 We recall an elementary structure. Let Θ ∈ E(F) and let φ be a character of F × . If θ is a realization of Θ, on a simple stratum [a, m, 0, β] in a matrix algebra A, the character φθ : x −→ φ(detA x) θ(x), x ∈ H 1 (β, a), is a simple character that is either trivial or attached to a simple stratum [a, m , 0, β + c], for an integer m and some c ∈ F: see the appendix to [27]. The endo-class of φθ depends only on φ  UF1 and Θ and so is denoted φΘ. We note: Lemma If φ is a character of F × and Θ ∈ E(F), then φΘ = Θ . 9.3.3 Recall that Θ ∈ E(F) is called totally wild if deg Θ = eΘ = pr , for some r  0. If θ ∈ C(a, β) is a realization of Θ on a simple stratum [a, m, 0, β], then Θ is totally wild if and only if the field extension F[β]/F is totally wildly ramified. For a general Θ ∈ E(F), it follows from the Definition in 6.1.3 that there exists a finite tame extension K/F such that Θ has a totally wild K/F-lift. Indeed, we may choose K so that every K/F-lift of Θ is totally wild. On the other hand, if Θ is totally wild and K/F is a finite tame extension, then Θ has a unique K/F-lift and that lift is totally wild. The effect of 9.3.1. Proposition is that we need only compute Θ for totally wild endo-classes Θ. If K/F is a finite tame extension, let AK be the canonical ultrametric on E(K). Interpolation Theorem Let Θ ∈ E(F) be totally wild. The function Θ has the following properties. (1) It is continuous, strictly increasing and piecewise linear. (2) The derivative Θ is continuous except at a finite number of points. (3) There is a finite set D of positive real numbers such that (a) if K/F is a finite tame extension, with e = e(K|F), and (b) if χ is a character of K × such that e−1 sw(χ) ∈ / D, then AK (Θ K , χΘ K ) = eΘ (e−1 sw(χ))

(9.3.1)

where Θ K is the unique K/F-lift of Θ. These properties determine the function Θ uniquely. The Interpolation Theorem is proved in section 7 of [21]. The theorem is quite difficult to use as a tool for computing the Herbrand function over its entire range. However, it provides essential information on part of that range, notably in [22] 2.6 and 5.3.

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9.4 Some Preliminary Estimates We record a couple of properties of the Herbrand function that are accessible without detailed analysis. 9.4.1 Let Θ ∈ E(F). Following part (2) of 9.2.1 Proposition, the function Θ (x) is only interesting in the region 0 < x < ςΘ . We describe its derivatives towards the ends of this interval. By 9.3.1 Proposition, it is enough to treat the case where Θ is totally wild. In the case deg Θ = 1, there is nothing to do as Θ (x) = x, for all x. So, let Θ ∈ E(F) be totally wild, of degree pr , r  1. Thus ςΘ = mp−r , for an integer m which we write as m = apt , where t  0 and a is not divisible by p. Replacing Θ by χΘ, for an appropriate character χ of F × , we may assume that t < r: since χΘ = Θ (9.3.2 Lemma), this changes nothing. Using this notation, 7.6 Proposition of [21] asserts: Proposition There exist ,  > 0 such that Θ (x) = p−r , 0 < x < , Θ (x) = pr−t , m −  < x < m. 9.4.2 At the other extreme, let Θ ∈ E(F) be essentially tame: in other words, eΘ is not divisible by p. In particular, there is a finite tame extension K/F such that Θ has a K/F-lift of degree 1. Proposition Let Θ ∈ E(F). The following are equivalent. (1) Θ is essentially tame. (2) Θ (x) = x, for all x  0. This follows from the Propositions of 9.3.1 and 9.4.1.

10 Carayol Representations In this section, we look into the problem of computing the Herbrand function Θ of an endo-class Θ ∈ E(F), and uncovering what it tells us about the Langlands correspondence. From 9.3.2 Proposition, we know it is enough to work only with endo-classes Θ that are totally wild, in that deg Θ = eΘ = pr . In view of the hierarchical nature of simple characters, one has to start with the case of endo-classes given by a minimal element. In the notation of (5.2.3), this amounts to kΘ = −ςΘ . An endo-class Θ satisfying all of these conditions, and of degree > 1, will be called totally wild of Carayol type. We denote the set of all such endo-classes by EC (F). Here, we summarize the results of [22] leading to a complete description of Θ , Θ ∈ EC (F). The corresponding Galois representations have a particularly simple and transparent form, and come in explicit “families” almost, but not completely,

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parallel to the families C into which simple characters are organized. This discrepancy is already visible in the case deg Θ = p = 2 of Kutzko’s classic works [52], [53] summarized in [15]. The general case is, at least conceptually, no worse.

10.1 Symmetry We present a fundamental property of Θ , where Θ is totally wild of Carayol type. 10.1.1 We assemble notation and terminology. Let EC (F) be the set of endo-classes Θ ∈ E(F) that are totally wild, of Carayol type and degree pr , for some r  1. Equivalently, Θ is the endo-class of a simple character θ ∈ C(a, α), for a simple stratum [a, m, 0, α] in Mpr (F) such that F[α]/F is totally ramified of degree pr , and m = −υF[α] (α) is not divisible by p. Lemma Let Θ ∈ EC (F). If deg Θ = pr and ςΘ = m/pr , then (1) (2) (3) (4)

Θ (0) = p−2r m(pr − 1); Θ (x) = x, for x  ςΘ ; Θ (x) = p−r , for 0 < x < ςΘ ; Θ is convex in the region 0 < x < ςΘ .

Proof Parts (1–3) are the definition [21] (4.4.1) of Θ in this case, and (4) follows.     F . Say that σ is totally wild if σ  PF is irreducible. Equivalently, dim σ is Let σ ∈ W a power of p and χ ⊗ σ  σ for any non-trivial unramified character χ of WF . Write F.  wr for the set of totally wild σ ∈ W W F  F is totally wild if and only if r + (σ) = LΘ, for a totally A representation σ ∈ W F wild endo-class Θ such that deg Θ = dim σ. Moreover, Θ ∈ EC (F) if and only if sw(σ) is not divisible by p (and dim σ > 1). We say that such representations σ are of Carayol type. 10.1.2 Endo-classes Θ ∈ EC (F) have a striking and characteristic property. Theorem Let Θ ∈ EC (F). The Herbrand function Θ satisfies ςΘ − x = Θ (ςΘ − Θ (x)), 0  x  ςΘ .

(10.1.1)

The functional equation (10.1.1) has a more intuitive interpretation. Corollary In the region 0  x  ςΘ , the graph y = Θ (x) is symmetric with respect to the line x + y = ςΘ . We give a flavour of the proof of the theorem in 10.1.3 below. The corollary simply re-states the theorem. Before passing on, we note that the theorem has a converse.

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Complement Let Υ ∈ E(F) be totally wild of degree pr , r  1, and suppose that ςΥ  ςχΥ for all characters χ of F × . The function Υ has the property (10.1.1), namely, ςΥ − x = Υ (ςΥ − Υ (x)), 0  x  ςΥ , if and only if Υ ∈ EC (F). Proof As in [22] 3.1, the result follows from 9.4.1 Proposition.

 

10.1.3 We sketch the key steps in the proof of 10.1.2 Theorem.  wr be of dimension pr , for some r  1, and let j be a jump of Σσ . Define Let σ ∈ W F the height h = hσ (j) of j by ph = lim Σσ (j + )/Σσ (j − ). →0

 Thus h is a positive integer and j hσ (j) = 2r, the sum being taken over all jumps of Σσ . It is often convenient to think of a point at which Σσ is smooth as being a jump of height 0.  wr is absolutely wild if it factors through a finite Galois group Say that σ ∈ W F Gal(K/F), where K/F is totally wildly ramified. To describe Σσ in general, it is enough to treat the case where σ is absolutely wild. For the first step of the proof, we assume only that σ is absolutely wild. Let a be the least jump of Σσ . There is then a character χ of WF , with sw(χ) = a, such that χ ⊗ σ ∼ = σ [22] 3.3 Lemma 1. If Ker χ = WK , there is an absolutely wild  wr such that σ = IndK/F τ . If k is a jump of τ , there is a jump representation τ ∈ W K j of σ such that k = ψ K/F (j). Moreover, hτ (k)  hσ (j). However, a = ψK/F (a) and hτ (a) < hσ (a). Since k hτ (k) = 2(r − 1), there are only two possibilities, namely (1) hτ (a) = hσ (a) − 1 and there exists a unique jump c > a of Σσ such that hτ (ψK/F (c)) < hσ (c), or (2) hτ (a) = hσ (a) − 2 while hτ (ψK/F (j)) = hσ (j) for all jumps j = a of Σσ . In case (1), we have hτ (ψK/F (c)) = hσ (c) − 1. Assume now that σ is also of Carayol type. The functions Θ and Σσ then have the same jumps. The representation τ is either of Carayol type or of dimension 1. One proves that c is the largest jump of Σσ and that c = (sw(σ) − a)/pr .

(10.1.2)

In case (1), this is the first step of an inductive argument: if dim σ = p, the relation between a and c tells us Θ completely and it satisfies the requirements of the theorem. In case (2), we see that a is the only jump of Σσ and has the value sw(σ)/(1 + pr ). The graph of Θ is made of two line segments and is easy to identify. The proof of (10.1.2) has an interesting feature: it relies on the case of the conductor formula given by 5.3.3 Remark (3) and 5.3.4 Proposition 1.

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10.1.4 In preparation for the next section, we introduce a family of special functions, as in [22] 4.2. Definition Let E/F be totally ramified of degree pr , r  1, and let ς = m/pr , where m is a positive integer not divisible by p. Define × (x) = p−r ψE/F (x), (E/F,ς) + (x) = ς − ϕE/F (pr (ς − x)), (E/F,ς)

(E/F,ς) (x) =

2

0  x  ς.

× + max {(E/F,ς) (x), (E/F,ς) (x)},

The function 2(E/F,ς) is the bi-Herbrand function of the datum (E/F, ς). Proposition The function 2(E/F,ς) is continuous, strictly increasing, piecewise linear and convex. The graph y = 2(E/F,ς) (x) is symmetric with respect to the line x + y = ς. × Indeed, the graph y = 2(E/F,ς) (x) is y = (E/F,ς) (x) in the triangular region x + y  + ς. If x + y  ς, it is y = (E/F,ς) (x).

10.2 Conformity and the Main Theorem Fix a simple stratum [a, m, 0, α] in Mpr (F) as in 10.1.1, and set E = F[α]. Here, the canonical map C(a, α) → E(F) is injective: this follows on comparing the intertwining theorems of 2.2.2 and 2.5.2. Let C(a, α) denote its image. Thus C(a, α) ⊂ EC (F), and the map C(a, α) → C(a, α) is a bijection. The function Θ → Θ is not necessarily constant on C(a, α) . We deal with this by introducing a dissection of the set C(a, α). 10.2.1 Let wα be the wild exponent of E/F, as in 9.1.3. Write lα = max {0, m − wα }, λα = [lα /2].

(10.2.1)

The function ψA ∗ α of 1.4.2 restricts to a character of the group UE1+λα . Say that θ ∈ C(a, α) conforms to α if θ agrees with ψA ∗ α on UE1+λα . Let C∗ (a, α) be the set of θ ∈ C(a, α) that conform to α. Write C∗ (a, α) for the set of endo-classes of elements of C∗ (a, α). Proposition (1) The set C∗ (a, α) is not empty. (2) Let θ ∈ C(a, α). There exists a simple stratum [a, m, 0, α ] such that C(a, α ) = C(a, α) and θ ∈ C∗ (a, α ). Part (1) here is obvious. Part (2) is proved in [22] 7.1.

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Remark When m  2wE/F , there is a wide choice of fields E  = F[α ] for which C(a, α ) = C(a, α). The various choices of E  give rise to different values of wE  /F and different Herbrand functions ψE  /F : this accounts for the need to work with C∗ (a, α). When m > 2wE/F , all choices of α give rise to the same Herbrand function. These phenomena are analysed in [22] section 6. 10.2.2 We describe the Herbrand functions Θ , Θ ∈ C∗ (a, α) , in terms of the bi-Herbrand functions of 10.1.4. Theorem Let Θ ∈ EC (F). Let [a, m, 0, α] be a simple stratum in Mpr (F) such that Θ ∈ C∗ (a, α) , and write E = F[α]. The Herbrand function Θ is given by Θ (x) = 2(E/F,ςΘ ) (x), 0  x  ςΘ . The proof of the theorem is intricate and occupies sections 4–7 of [22]. It starts from the almost Galois-theoretic 10.1.2 Theorem, but then works entirely on the GL-side. We give no details. Other results in [22] section 7 indicate how Θ varies as Θ ranges over C(a, α) , although the picture is not complete. There is a direct consequence, useful for constructing examples. Corollary Let E/F be a totally ramified field extension of degree pr , r  1, and let m be a positive integer not divisible by p. There exists Θ ∈ EC (F), with degree pr and ςΘ = m/pr , such that Θ (x) = 2 (E/F,m/pr ) (x), 0  x  m/pr . Proof Let α ∈ E satisfy υE (α) = −m. In particular, E = E[α]. Let a be the unique E-pure hereditary oF -order in EndF (E). Thus [a, m, 0, α] is a simple stratum. There exists θ ∈ C(a, α) that conforms to α. The theorem implies that the endo-class Θ of θ has the required property.   10.2.3 Suppose for the moment that p = 2. One can then refine the notion of conformity. The function   ψA ∗ α : 1 + x −→ ψA α(x − x2 /2)

2

(10.2.2)

[(1+l )/2]

defines a character of the group UE α . Let C∗∗ (a, α) be the set of θ ∈ C(a, α) agreeing with 2ψA ∗ α on that group. The exact analogue of 10.2.1 Proposition holds for the set C∗∗ (a, α): this approach is used by Mœglin in [58]. The extra precision can be useful here: see 10.4.5.3 below.

10.3 Galois Representations of Carayol Type  wr be of Carayol type and dimension pr , r  1. Thus r + (σ) = LΘ, for Let σ ∈ W F F some Θ ∈ EC (F). Developing ideas from the proof of 10.1.2 Theorem, we write

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  down the structure of σ  RF (x), σ  R+ F (x), for all x > 0. For a complete account of this material, see section 9 of [22].  wr be of Carayol type and dimension pr , r  1. Thus r + (σ) = LΘ, 10.3.1 Let σ ∈ W F F for some Θ ∈ EC (F): to save notation, let us write σ = Θ . Define cσ by the equation cσ + σ (cσ ) = ς(σ). Thus cσ is the x-coordinate of the intersection of the graph y = σ (x) with the line of symmetry x + y = ς(σ). The symmetry property of 10.1.2 Corollary induces an involution, j → j¯ say, on the set of jumps j of σ . We may therefore number them j1 < j2 < · · · < jn < (cσ ) < j¯n < . . . j¯1 , for some n  0, where the term cσ is included when cσ is a jump. Proposition Suppose Θ ∈ C∗ (a, α) . The point cσ is a jump of Θ = σ if and only if j∞ (F[α]|F)  cσ . Proof If cσ is not a jump, the symmetry property (10.1.1) implies that Θ (cσ ) = 1. Together, 9.1.3 Proposition 1 and 10.2.2 Theorem imply cσ > j∞ (F[α]|F). The argument reverses to give the result.   10.3.2 Using the same notation, we write down the ramification structure of σ.  Theorem 1 (1) The restriction σ  R+ F (cσ ) is a direct sum of characters. (c ) occurring in σ, let WLξ be the WF -normalizer (2) Let ξ be a character of R+ F σ of ξ and let ρξ = ρξ (σ) be the natural representation of WLξ on the ξ-isotypic subspace of σ  R+ F (cσ ). (a) The field extension Lξ /F is totally wildly ramified. (b) The representation ρξ of WLξ is irreducible and totally wild. (c) σ = IndLξ /F ρξ (σ). (3) If cσ is not a jump, then ρξ is a character. (4) If cσ is a jump, then dim ρξ = ps , where s = hσ (cσ )/2. The representation ρξ ∈  wr is of Carayol type and has a unique jump, at ψLξ /F (cσ ). W Lξ (5) In both cases, σ (x) = p−r ψLξ /F (x), 0  x  cσ .

(10.3.1)

 Note that the conjugacy class of Lξ /F is determined by σ. In all cases, σ  RF (j¯k ) is a direct sum of characters, 1  k  n. When cσ is a jump, the representation  wr has interesting structure, on which we comment in 10.3.3. We remark that ρξ ∈ W Lξ (10.3.1) determines σ completely, because of the symmetry property of 10.1.2.  Theorem 2 For 1  k  n, the restriction σ  R+ F (jk ) is a multiplicity-free represen tation. Its irreducible components restrict to the isotypic components of σ  RF (j¯k ).  If cσ is a jump, then σ  RF (cσ ) is multiplicity-free.

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10.3.3 The representation ρξ of 10.3.2 has a very special form. We summarize the  wr , say τ : WF → GLn (C), main points from [22]. Starting in generality, let τ ∈ W F and let τ¯ : WF → PGLn (C) be the associated projective representation. Let Ker τ¯ = WK . The finite Galois extension K/F is the centric field of τ . The tame centric field T /F is the maximal tame sub-extension of K/F. The representation τ is called Hcyclic if the finite p-group τ¯ (PF ) is elementary abelian. It follows that the finite p-group τ (PF ) is extra special of class 2. One says that τ is H-singular if ψK/F has a single jump.  wr be of Carayol type and have a single jump. The represenProposition Let τ ∈ W F tation τ is then H-singular. The representation ρξ of 10.3.2 is therefore H-singular.

10.4 Parameter Fields We compare the theorems of 10.3.2 with the main theorem of 10.2.2. The material here is extracted from section 10 of [22]. 10.4.1 We re-set the notation to more closely match the focus of the section. Fix a simple stratum [a, m, 0, α] in Mpr (F) such that E = F[α]/F is totally ramified of degree pr and m is not divisible by p. Abbreviate wα = wE/F and set lα = max {0, m − wα }, λα = [lα /2]. Write

and

Thus

 ∗F (α) = {π ∈ A0pr (F) : Θ(π) ∈ C∗ (a, α) }, GL  F : dim σ = pr and r + (σ) ∈  ∗ (α) = {σ ∈ W W F F  ∗ (α) = W F

L

C∗ (a, α) }.

  ∗  F (α) . GL

L

All endo-classes Θ ∈ C∗ (a, α) have the same Herbrand function, namely Θ = (E/F,m/pr ) . We now denote this by α . In the convention of 10.3.1, we thus have  ∗ (α). Likewise, define cα by the equation α = σ , σ ∈ W F

2

cα + α (cα ) = m/pr .  F , that is, We view the ultrametric A as a pairing on GL F. A(π1 , π2 ) = A(Θ(π1 ), Θ(π2 )), πi ∈ GL

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 ∗F (α), then Lemma (1) If π0 ∈ GL  ∗F (α) = {π ∈ A0pr (F) : A(π, π0 )  λα /pr }. GL  ∗ (α), then (2) If σ0 ∈ W F  F : (σ, σ0 )  α , dim σ = pr },  ∗ (α) = {σ ∈ W W F where α (α ) = λα /pr . Proof Part (1) is the definition of C∗ (a, α) whence part (2) follows from the Higher Ramification Theorem of 9.2.2.   10.4.2 There is a simple result to bear in mind as we go forward. If K/F is a finite extension and c a non-negative integer, let Xc (K) be the group of characters χ of K × such that sw(χ)  c.  ∗F (α) and set G = GLpr (F). Thus π = c-IndJG Λ, where Λ is an irreLet π ∈ GL [(1+m)/2] containing some θ ∈ C∗ (a, α). Surely ducible representation of J = E × Ua [(1 + m)/2] > λα so, if χ ∈ Xλα (E), we may view χ as a character of J trivial on [(1+m)/2] Ua . The representation χ ⊗ Λ of J is an extended maximal simple type in G, containing some θ ∈ C∗ (a, α). Define χ α π = c-IndJG χ ⊗ Λ.

(10.4.1)

Working through the definitions, we find: Proposition Relative to the pairing  ∗F (α) −→ GL  ∗F (α), Xλα (E) × GL (χ, π) −→ χ α π,  ∗F (α) is a principal homogeneous space over Xλα (E). the set GL  ∗ (α). Let j∞ (α) = j∞ (E|F) be 10.4.3 We make a preliminary analysis of the set W F the largest jump of the field extension E/F. (Remember that j∞ (α) = ∞ if E/F happens to be inseparable.) Theorem (1) There exists a character ξ of R+ F (cα ), unique up to conjugation by  ∗ (α). WF , that occurs in every representation σ ∈ W F (2) Fix a character ξ as in (1). Assume that j∞ (α) > cα or that lα is odd. There is a unique irreducible representation ρξ of RF (cα ) that contains ξ and occurs in  ∗ (α). every σ ∈ W F Proof We first clarify the relation between some of the numerical parameters.

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Lemma 1 (1) If j∞ (α)  cα , then cα = (m + wα )/2pr . (2) If j∞ (α) > cα , then cα < (m + wα )/2pr . (3) If lα = 0, then j∞ (α) > cα . Proof Suppose first that j∞ (α)  cα . The definition 10.1.4 of the bi-Herbrand function and 10.2.2 Theorem imply that α (cα ) = p−r ψE/F (cα ) and, by 9.1.3 Proposition 1, ψE/F (cα ) = pr cα − wα . However, the definition of cα is that cα + α (cα ) = m/pr whence assertion (1) follows. In (2), the line y = x − p−r wα lies strictly below the graph y = α (x) and the assertion follows on drawing a picture. In (3), assume for a contradiction that jα  cα , whence cα = (m + wα )/2pr . The hypothesis lα = 0 implies m  wα , whence cα = (m + wα )/2pr  wα /pr < j∞ (α) by 9.1.3 Corollary. This gives the desired contradiction.

 

We translate Lemma 1 in terms of ultrametrics.  ∗ (α) and let r + (σi ) = LΘi , Θi ∈ C∗ (a, α) . We Lemma 2 For i = 1, 2, let σi ∈ W F F then have (σ1 , σ2 )  cα .

(10.4.2)

Moreover, (σ1 , σ2 ) = cα if and only if all of the following conditions are satisfied: (1) A(Θ1 , Θ2 ) = lα /2pr , (2) lα is even, and (3) j∞ (α)  cα . Proof Define α by α (α ) = λα /pr . By 10.4.1 Lemma, we have (σ1 , σ2 )/leα , with equality if and only if A(Θ1 , Θ2 ) = λα /pr . Suppose that j∞ (α)  cα . So, by Lemma 1, cα = (m + wα )/2pr and therefore α (cα ) = lα /2pr . That is, α  cα , with equality if and only if lα is even. All assertions are proved in this case. Suppose now that j∞ (α) > cα . The line y = x − p−r wα lies strictly below the graph y = α (x). The three lines y = lα /2pr , y = x − p−r wα and x + y = ςα all meet at x = (m + wα )/2pr . By Lemma 1, (m + wα )/2pr > cα , so α (cα ) > lα /2pr .   It follows that cα > α  (σ1 , σ2 ), as required. Remark Condition (2) in Lemma 2 is actually redundant, since pr A(Θ1 , Θ2 ) is an integer, for any Θi ∈ C(a, α) . We prove the theorem. Part (1) follows directly from (10.4.2). The assertion of  ∗ (α). Part (2) of the theorem (2) holds if and only if (σ1 , σ2 ) < cα for all σi ∈ W F therefore follows from Lemma 2.  

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Comments We comment further on the conclusions of the theorem, in connection with 10.3.2 Theorem 1. (1) If j∞ (α) < cα , the representation ρξ is a character. (2) In the case j∞ (α) > cα , the representation ρξ is the restriction to RF (cσ ) of the ρξ occurring in 10.3.2. The same comment applies if lα is odd and j∞ (α) = cα . (3) Suppose that j∞ (α) = cα and that lα is even (and so positive, by Lemma 2(3)).  ∗ (α), the natural representation ρξ (σ) of RF (cα ) on the ξ-isotypic For σ ∈ W F subspace of σ is again irreducible of dimension ps , s  1. In this case, it does vary with σ. 10.4.4 We seek an analogue of 10.4.2 Proposition for Galois representations. With no restriction on the parameters at first, fix a character ξ of R+ F (cα ) that occurs in  ∗ (α). The WF -stabilizer of ξ is of the form WL , for a totally wildly ramified σ∈W F extension L/F. Let σξ be the natural representation of WL on the ξ-isotypic subspace  L. of σ, and let L be the canonical ultrametric pairing on W Lemma 1  ∗ (α)} = λα . max {L (σξ , σξ ) : σ, σ  ∈ W F

(10.4.3)

Proof By 10.4.1 Lemma, max{A(Θ, Θ  ) : Θ, Θ  ∈ C (a, α) } = λα /pr . So,   max (σ, σ  ) : σ, σ  ∈ G (α) = α−1 (λα /pr ) = ϕL/F (λα ),  

by (10.3.1). The relation (10.4.3) now follows from (9.1.4).  ∗ (α) and let χ ∈ Xλα (L). Form At the same level of generality, let σ ∈ W F χ α σ = IndL/F χ ⊗ σξ .

(10.4.4)

 ∗ (α). Lemma 2 The representation χ α σ of (10.4.4) is irreducible and lies in W F  + Proof The restriction χ ⊗ σξ  RF (cα ) is a multiple of ξ, so the irreducibility assertion follows from the definition of L. The second assertion follows from Lemma 1 and 10.4.1 Lemma.   Theorem Suppose that either j∞ (α) = cα or that lα is odd. Relative to the pairing  ∗ (α) −→ W  ∗ (α), Xλα (L) × W F F (χ, σ) −→ χ α σ,  ∗ (α) is a principal homogeneous space. the set W F Proof See 10.4 Proposition of [22].

 

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10.4.5 Comments 10.4.5.1 The case of j∞ (α) = cα and lα even positive does arise in practice. Let p = pr = 2. There is a ramified quadratic extension E/F with wE/F odd. Take α ∈ E of valuation −m, where m = 3wE/F . This gives j∞ (α) = wE/F = cα , while lα = 2wE/F > 0.  ∗ (α) 10.4.5.2 Staying in the same case j∞ (α) = cα with lα even positive, let σ ∈ W F and let ρξ (σ) be the natural representation of WL on the ξ-isotypic subspace of σ. Thus ρξ (σ) is H-singular and may vary with σ. However, the representations ρξ (σ),  ∗ (α), all have the same tame centric field T /L. This generalizes Kutzko’s σ∈W F celebrated “cubic polynomial theorem” that lies at the heart of his proof of the Langlands Conjecture for GL(2). See [14] section 45 for an exposition in the present language. The phenomenon is also visible in [18]. On the other hand, the centric field may vary with σ: see [22] 10.5 Theorem. 10.4.5.3 Suppose now that p is odd. One can carry out the same analysis using C∗∗ (a, α) in place of C∗ (a, α). Improvising notation in the obvious way, (10.4.2) is replaced by  ∗∗ (α), (σ1 , σ2 ) < cα , σi ∈ W F and the exceptional case of 10.4.3 Lemma 2 has been removed. In 10.4.3 Theorem, part (2) holds without restriction one either j∞ (α) or lα . Likewise 10.4.4 Theorem, except that λα is replaced by λα =



if lα is odd, [lα /2], (lα /2) − 1, if lα is even.

10.4.5.4 Suppose that p  5 and work in dimension p. Mœglin [58] shows that L = E provided the extensions E/F, L/F are cyclic. When p = 2, one knows that they need not be the same.

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Root Data with Group Actions Jeffrey D. Adler and Joshua M. Lansky

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Restrictions of Root Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Automorphisms of Root Data and Automorphisms of Reductive Groups . . . . . . . . . . . . . 4 Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Remarks on Cohomological Parametrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128 130 134 136 141 143

Abstract Suppose k is a field, G is a connected reductive algebraic k-group, T is a maximal k-torus in G, and  is a finite group that acts on (G, T ). From the above, one obtains a root datum  on which Gal(k) ×  acts. Provided that  preserves a positive system in , not necessarily invariant under Gal(k), we construct an inverse to this process. That is, given a root datum on which Gal(k) ×  acts appropriately, we show how to construct a pair (G, T ), on which  acts as above. Although the pair (G, T ) and the action of  are canonical only up to an equivalence relation, we construct a particular pair for which G is k-quasisplit and  fixes a Gal(k)-stable pinning of G. Using these choices, we can define a notion of taking “-fixed points” at the level of equivalence classes, and this process is compatible with a general “restriction” process for root data with -action. Keywords Reductive algebraic group · Root datum · Quasi-semisimple automorphisms 2010 Mathematics Subject Classification Primary 20G15 · 20G40 · Secondary 20C33. J. D. Adler (B) · J. M. Lansky Department of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA e-mail: [email protected] J. M. Lansky e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_3

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1 Introduction Let k be a field with separable closure k sep . Let  be a finite group. Suppose  is a (reduced) based root datum on which the absolute Galois group Gal(k) acts. Then it is well known ([10, Theorem 6.2.7]) that there exists a connected, reductive, k-quasisplit k-group G, uniquely determined up to k-isomorphism, such that the root datum of G (with respect to a maximal k-torus contained in a Borel k-subgroup) is isomorphic to  and carries the same action of Gal(k). We generalize this result in two directions. (A) Suppose G is a connected reductive k-group, and T is an arbitrary maximal torus. Then, the non-based root datum (G, T ) carries an action of Gal(k). We show that one can reverse this process. That is, given a root datum  with an action of Gal(k), one can obtain a pair (G, T ) that gives rise to . In general, the pair (G, T ) need not be uniquely determined up to k-isomorphism. However, we can always choose G to be k-quasisplit, and all possibilities for G must be k-inner forms of each other. (B) Now suppose that  acts on a pair (G, T ) as above via k-automorphisms. Then  acts on the root datum (G, T ), and the actions of  and Gal(k) commute. We show that one can reverse this process under mild conditions. That is, suppose that  is a root datum with an action of Gal(k) × . Assume that  (but not necessarily Gal(k)) preserves a base. Then, one can obtain a pair (G, T ) as above, carrying an action of  via k-automorphisms. That is, under appropriate conditions, we can lift an action of  from a root datum to a reductive group. Moreover, one can choose G to be k-quasisplit and can choose the action of  to preserve a pinning. The above are all contained in our main result, Theorem 1. In order to state it more precisely, let us consider the collection of abstract root data  that carry an action of Gal(k) ×  such that the action of  stabilizes a base for . We consider two data  and   with such actions to be equivalent if there is a Gal(k) × -equivariant isomorphism  −→   . Let R denote the set of equivalence classes of reduced data with such actions. Let G be a connected reductive k-group and T ⊆ G a maximal k-torus. Suppose there exists some Borel subgroup B ⊆ G (not necessarily defined over k) containing T , and a homomorphism ϕ from  to the group Autk (G, B, T ) of k-automorphisms of G stabilizing T and B. Suppose G  , T  , and ϕ are defined similarly. We say that the triples (G, T, ϕ) and (G  , T  , ϕ ) are equivalent if there exists an isomorphism ν : G −→ G  whose restriction gives a -equivariant k-isomorphism T −→ T  . (In this situation, ν must be an inner twisting by [8, §3.2].) Let T be the set of equivalence classes of such triples (G, T, ϕ). A triple (G, T, ϕ) as above naturally determines a root datum with appropriate actions of Gal(k) and , hence an element of R . It is easily seen that if (G  , T  , ϕ ) and (G, T, ϕ) are equivalent, then they determine the same class in R . Hence, we have a natural map r : T −→ R . Our main result is the following:

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Theorem 1 The map r : T −→ R is a bijection. In Sect. 5, we give cohomological descriptions of R and the inverse of the map r , and thus obtain a cohomological description of T . In [1], we introduce the notion of a parascopic group G¯ for a pair (G, ), where  is a finite group acting via k-automorphisms (in a certain specified way) on a maximal k-torus of the reductive k-group G. This notion axiomatizes and generalizes the relationship between G and the connected part (G  )◦ of the group of -fixed points in the case where the action of  extends to one on G. When G¯ is parascopic for (G, ), we construct a canonical lifting of semisimple stable conjugacy classes from G¯ ∧ to G ∧ , k-groups that are in duality with G¯ and G, respectively. Theorem 1 is a key tool in our further study of this lifting in [2]. In particular, in many settings, this result allows one to factor such a lifting into a composition of simpler ones which are much more readily understood. We also prove a generalization of a second well-known result. Suppose  is a finite group acting algebraically on a connected reductive group G, fixing a Boreltorus pair (B, T ) in G and a pinning for (G, B, T ). Then, the root system of the connected part G¯ := (G  )◦ of the group of fixed points is obtained as follows. The set of restrictions of roots of G from T to T¯ := (T  )◦ is a root system, not necessarily reduced, but there is a preferred way to choose a maximal reduced subsystem. The above is well known, but in Theorem 7 and Lemma 28 we describe the root datum, not just the root system, of G¯ with respect to T¯ . ¯ T¯ ) is compatible Theorem 2 says that the process of passing from (G, T ) to (G, with the bijection of Theorem 1. To state this result more precisely, suppose that the triple (G, T, ϕ) represents an element of T . Then we know [1, Proposition 3.5] ¯ Thus, that G¯ is a connected reductive k-group, and T¯ is a maximal k-torus in G. if we let “1” represent the map from the trivial group 1 to Aut(G), then the triple ¯ T¯ , 1) represents an element of T1 . The equivalence class of (G, T, ϕ) does not (G, ¯ Nonetheless, ¯ T¯ , 1), or even the k sep -isomorphism class of G. determine that of (G, we can obtain a well-defined map T −→ T1 as follows: From Remark 25, we will see that every class in T contains a triple (G, T, ϕ) such G is k-quasisplit and ϕ fixes a Gal(k)-invariant pinning. Use this choice of triple to define G¯ and T¯ , and it is straightforward to show that our choices determine G¯ and T¯ up to k-isomorphism. Suppose that the root datum  represents an element of R . We will see in Sect. 2 ¯ that has that the action of  on  allows us to construct a “restricted” root datum  ¯  . We thus obtain a map R −→ R1 . a preferred choice of reduced subdatum  Theorem 2 Our maps T −→ T1 and R −→ R1 above are compatible with the maps of Theorem 1, in the sense that the following diagram commutes: T

r

R

T1

r1

R1

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We prove both theorems in Sect. 4. Some of the results in this paper have recently appeared elsewhere, and others are generalizations of existing results. For example, as indicated above, [10] presents Statement (A) in the case where G is quasisplit over k and T is contained in a Borel ksubgroup of G. Corollary 15 has appeared as [4, Lemma 4.2(2)]. Lemma 28 overlaps to some extent with [4, Proposition 4.1], but our proof is quite different. Theorem 7 is similar to [4, Lemma 4.2(1)] and [5, Theorem A], except that we describe the full restricted root datum, and our proof does not rely on Steinberg [11].

2 Restrictions of Root Data Let  = (X ∗ , , X ∗ , ∨ ) be a root datum. (We do not assume that  is reduced.) Let  denote a finite group of automorphisms of . We assume that there exists a -stable set  of simple roots in . Let V ∗ = X ∗ ⊗ Q and V∗ = X ∗ ⊗ Q. Let i ∗ denote the quotient map from V ∗ to its space V¯ ∗ := V∗ of -coinvariants. From [1, §2], there is an embedding ι : V¯ ∗ −→ V ∗ with image V ∗  given by ι(v) ¯ =

1  γv, || γ∈

¯ denote the images of X ∗ where v is any preimage in V ∗ of v¯ ∈ V¯ ∗ . Let X¯ ∗ and  ∗ ∗ ∗ and  under i . Then, X¯ is X  modulo torsion, where X ∗ denotes the module of -coinvariants of X ∗ . It is straightforward to see that X¯ ∗ and X¯ ∗ := X ∗ , and thus ¯ := ιx, ¯ where ¯ λ ¯ i ∗ λ , V¯ ∗ and V¯∗ := V∗ , are in duality via the pairing given by x, i ∗ : X¯ ∗ −→ X ∗ is the inclusion map. With respect to these pairings, i ∗ is the transpose of i ∗ . For each β ∈ , let wβ denote the automorphism of X ∗ defined by wβ (x) = x − x, β ∨ β. Let W denote the Weyl group of , i.e., the (finite) subgroup of Aut(X ∗ ) generated by the wβ . Then,  acts naturally on W , and W acts on X ∗ . The group W  of -fixed elements of W acts on both V¯ ∗ and X¯ ∗ via the rule w(i ∗ x) := i ∗ (w(x)) for w ∈ W  and x ∈ X ∗ . Equivalently, for x¯ ∈ X¯ ∗ and w ∈ W  , we have ι(w(x)) ¯ = w(ιx). ¯ Lemma 3 (cf. [11, §1.32(a)]) The natural action of W  on X¯ ∗ is faithful. Proof Let w be a nontrivial element of W  . Then, there exists a positive root β ∈  such that w(β) is negative. Since  stabilizes , it follows that w(γ · β) = γ · (wβ) is also negative for every γ ∈ . Thus, ι(w(i ∗ β)) is a linear combination of roots in   in which all of the coefficients are nonpositive, so w(i ∗ β) = i ∗ β. Notation 4 For each root β ∈ , define a -orbit β in  as in [1, §5]. That is, let β =  · β if this is an orthogonal set. Otherwise, for each θ ∈  · β, there exists a

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unique root θ = θ in  · β such that θ and θ are not orthogonal. Moreover, θ + θ is a root in  and does not belong to  · β. In this case, let β = {θ + θ | θ ∈  · β}. Remark 5 Thus, in all cases, β is an orthogonal -orbit of roots. ¯ then i ∗ −1 (α) is a -orbit of roots in . Lemma 6 If α ∈ , Proof This argument is similar to but more general than that given in the proof of [9, Lemma 10.3.2(ii)]. Suppose β ∈  and i ∗ (β) = α. Then, clearly i ∗ θ = α for any θ ∈  · β. Now suppose β  ∈ , β  = β, and i ∗ β  = α. Since ι(i ∗ (β  − β)) = 0 and since  preserves , when β  − β is written as a linear combination of simple roots, the / . Since β  cannot be a multiple coefficients must sum to 0. In particular, β  − β ∈ of β, we have that β  , β ∨ ≤ 0 by standard results about root systems. Similarly, β  , θ∨ ≤ 0 for all θ = β  in  · β. Therefore, 

β  − β,



      ∨   ∗  = i (β − β), θ∨ = β  − β, i ∗ θ θ∨ ,

θ∈·β

θ∈·β

θ∈·β

  and since i ∗ (β  − β) = 0, this pairing vanishes. Thus θ∈·β β  , θ∨ = θ∈·β β, θ∨ = 2 or 1, depending on whether or not  · β is orthogonal. (This follows from the properties of root orbits discussed in [1, §5].) Since β  , θ∨ ≤ 0 for all θ = β  in  · β, it follows that β  ∈  · β.  ¯ define For each α ∈ , α∨ =

| · β|  ∨ ξ ∈ X¯ ∗ , |β | ξ∈β

where β is any element of  such that i ∗ β = α. The element of X¯ ∗ defined by the above formula is independent of the particular choice of β by Lemma 6. Note that ¯ ∨ = {α∨ | α ∈ }. ¯ α∨ does indeed lie in X¯ ∗ since | · β|/|β | = 1 or 2. Let  ¯ := ( X¯ ∗ , , ¯ X¯ ∗ ,  ¯ ∨ ) is a root datum. Theorem 7 With the above notation,  Remark 8 If  comes equipped with an action of Gal(k), and the action of  com¯ mutes with that of Gal(k), then it is clear that the action of Gal(k) preserves . Proof of Theorem 7 According to [8, §1.1], it suffices to show that • • •

X¯ ∗ and X¯ ∗ are in duality (which we have already observed), ¯ and α, α∨ = 2 for all α ∈ , The automorphisms wα of X¯ ∗ of the form ¯ = x¯ − x, ¯ α∨ α wα (x)

¯ (for α ∈ )

¯ and generate a finite subgroup of Aut( X¯ ∗ ). stabilize 

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¯ Choose β ∈  such that i ∗ β = α, and choose ξ0 ∈ β . Then, we have Let α ∈ . α, α∨ = ια, i ∗ α∨  1  | · β|  ∨  θ, ξ = | · β| |β | θ∈·β ξ∈β   1  = θ, ξ∨ |β | θ∈·β ξ∈β 1     ∨ = ξ, ξ |β |  ξ ∈β

(by the definition of β )

θ∈β

= ξ0 , ξ0∨

(by Remark 5)

= 2, as desired. ¯ Then Now let x¯ ∈ X¯ ∗ , and choose x ∈ X ∗ such that i ∗ x = x. x, ¯ α∨ = x, i ∗ α∨  | · β|   ξ∨ = x, |β |

(by Remark 5) (9)

ξ∈β

| · β|  = x, ξ ∨ . |β | ξ∈β

It follows that ¯ = x¯ − x, ¯ α∨ α wα (x) = i ∗ x − x, ¯ α∨ i ∗ β x, ¯ α∨ ∗   i = i∗x − θ | · β| θ∈·β x, ¯ α∨ ∗    = i∗x − i ξ | · β| ξ  ∈β  1  = i∗x − x, ξ ∨ i ∗ ξ |β |  ξ∈β ξ ∈β  1   = i∗x − x, ξ ∨ i ∗ ξ . |β |  ξ∈β

ξ ∈β

But by Remark 5, for any ξ ∈ β , i∗

 1  ξ  = i ∗ ξ, |β |  ξ ∈β

(by Lemma 6) (by the definition of β ) (by (9))

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so we have

133

 

 ¯ = i∗ x − x, ξ ∨ ξ . wα (x)

(10)

ξ∈β

Also by Remark 5, the reflections wξ ∈ W for ξ ∈ β all commute with one another. If w denotes their product, then x−



x, ξ ∨ ξ = w(x),

ξ∈β

so by (10), we have

wα (x) ¯ = i ∗ (w(x)).

(11)

¯ and β  ∈  satisfies i ∗ β  = α , then In particular, if α ∈ , ¯ wα (α ) = i ∗ (w(β  )) ∈ i ∗ () = , ¯ is stable under the action of wα , as desired. so  ¯ ⊂ Aut( X¯ ∗ ) is finite. To acIt remains to show that the group W¯ := wα | α ∈  ¯ complish this, we show that W embeds naturally in the finite group W  . By Lemma 3, there is a natural injection W  −→ Aut( X¯ ∗ ). To construct an embedding W¯ −→ W  , it is therefore enough to show that the image of this injection contains W¯ . Thus, given w¯ ∈ W¯ , we will show that there ¯ It suffices to prove the exists w ∈ W  whose action on X¯ ∗ coincides with that of w. ¯ existence of w only in the case in which w¯ is a reflection wα through a root α ∈ . ∗ In this case, let w = ξ∈β wξ , where β ∈  is such that i β = α. It follows from Remark 5 that w ∈ W  , and it follows from (11) that for any x ∈ X ∗ , wα (i ∗ x) = i ∗ (w(x)) = w(i ∗ x). This establishes the existence of the desired embedding.



¯ constructed above, unless Remark 12 If  is reduced, then so is the root system   has an irreducible factor of type A2n whose stabilizer in  acts upon it nontrivially. To see this, it is easy to reduce to the case where  is irreducible and  is cyclic (see [1, Proposition 3.5]). The result then follows from [6, §1.3]. Remark 13 There is a way to choose a maximal reduced subsystem that we will later see is preferred. (See Lemma 28.) Specficially, take only the nondivisible (resp. ¯ according as char k is not two (resp. two). nonmultipliable) roots of  Lemma 14 The map i ∗ induces a bijection between the set of -invariant positive ¯ systems in  and the set of positive systems in .

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¯ = i ∗ () ⊆ . ¯ Then there Proof Let  ⊆  be a -invariant positive system. Let  is some vector v ∈ V∗ such that for every root β ∈ , we have that β, v = 0, and β, v > 0 if and only if β ∈ . Since  is -invariant, we may replace v by  ¯ ¯ γ∈ γv, and thus assume that v is -invariant, and so lies in V∗ . Suppose α ∈ . Then α = i ∗ β for some β ∈ , so α, v = β, v . Thus, α, v = 0, and α, v > 0 ¯ This shows that  ¯ is a positive system in . ¯ if and only if α ∈ . ¯ Then ¯ ⊆ ¯ is a positive system, and let  = i ∗ −1 . Conversely, suppose that  ¯ we have that α, v there is some vector v¯ ∈ V¯∗ such that for every root α ∈ , ¯ = 0, ¯ For every root β ∈ , we have β, i ∗ v = and α, v ¯ > 0 if and only if α ∈ . i ∗ β, v , which is never zero, and is positive if and only if β ∈ . Thus,  ⊂  is a  positive system. Since i ∗ v is -invariant, so is .  ¯ Then, the embedding of W¯ into W in Corollary 15 Let W¯ be the Weyl group of . the proof of Theorem 7 is an isomorphism. ¯ and W  Proof Since W¯ acts simply transitively on the set of positive systems in , acts simply transitively on the set of -invariant positive systems in , the result follows from Lemma 14. 

3 Automorphisms of Root Data and Automorphisms of Reductive Groups Let  = (X ∗ , , X ∗ , ∨ ) be a root datum on which a group  acts via automorphisms. Choosing a root basis  of , we obtain a corresponding based root datum ˙ Then,  ˙ also carries an action of . Namely, for σ ∈ , there exists a unique . element c(σ) in the Weyl group W () of  such that σ() = c(σ)(). If we define ˙ given by σ to be the automorphism of  σ χ = c(σ)−1 (σχ)

(16)

˙ is given by σ → σ . for χ ∈ X ∗ , then the action of  on  ˙ ˙ as well as on Since  acts on  and , it acts naturally on Aut() and Aut(), ˙ ˙ the Weyl groups W () ⊂ Aut() and W () ⊂ Aut(). Just as the actions of  on ˙ and on  differ, so the actions of  on W () ˙ and on W () differ, even though the  ˙ let (σ, w) → σ (w) Weyl groups themselves are equal. For σ ∈  and w ∈ W (), ˙ denote the action of  on W (). Then, we have σw = c(σ) (σ w) c(σ)−1 . ˙ ˙ is a cocycle in Z 1 (, W ()). One can check readily that map c :  −→ W () We now turn our attention to based root data arising from reductive algebraic groups. Let G be a connected reductive k-group, B a Borel subgroup of G, and ˙ T ⊆ B a maximal torus of G. Let (G, B, T ) denote the corresponding based root datum.

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Any map ϑ ∈ Aut(G) determines an obvious isomorphism ˙ ˙ B, T ) −→ (G, ϑ(B), ϑ(T )). ϑ∗ : (G, There is a natural homomorphism ˙ π : Aut(G) −→ Aut((G, B, T )) defined as follows. For ϑ ∈ Aut(G), choose gϑ ∈ G(k sep ) such that Int(gϑ ) takes ϑ(B) to B, and ϑ(T ) to T . Then Int(gϑ ) ◦ ϑ stabilizes B and T , and we let π(ϑ) be ˙ B, T ) (which is, in fact, independent of the automorphism (Int(gϑ ) ◦ ϑ)∗ of (G, the choice of gϑ ). Now suppose that T is defined over k. Then, an element σ ∈ Gal(k) naturally ˙ B, T ) determines an automorphism of (G, T ) hence an automorphism σ of (G, ˙ as defined in (16). We thus obtain an action of Gal(k) on (G, B, T ), hence one on ˙ Aut((G, B, T )) as above. These actions are independent of the particular choice of B and T in the sense that if g ∈ G(k sep ) and σ ∈ Gal(k), then we have σ ◦ Int(g)∗ = Int(g)∗ ◦ σ ,

(17)

˙ B, T ) and on where we use the notation σ to denote both the action of σ on (G, g g ˙ (G, B, T ). There is a well-known exact sequence π

˙ 1 −→ Inn(G) −→ Aut(G) −→ Aut((G, B, T )) −→ 1.

(18)

We note that the homomorphisms in (18) are Gal(k)-equivariant. Remark 19 Let  be the set of simple roots for (G, B, T ). Let {X α }α∈ ⊂ Lie(G)(k sep ) be a pinning. It is well known [8, Corollary 2.14] that {X α } deter˙ mines a unique splitting ψ of (18). Namely, if f ∈ Aut((G, B, T )), define ψ( f ) to be the automorphism of G such that • ψ( f ) stabilizes B and T , • the restriction of ψ( f ) to T is determined by the automorphism of X ∗ (T ) given by f , and • ψ( f )(X α ) = X f (α) . Thus, Im(ψ) lies in the subgroup Aut(G, B, T, {X α }) of Aut(G) consisting of automorphisms that stabilize B, T , and the pinning {X α }. If B and T are defined over k, and {X α } is Gal(k)-stable, it follows from [3, §3.10] that ψ is Gal(k)-equivariant.

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4 Proofs of Theorems Proof of Theorem 1 Consider an abstract root datum  = (X ∗ , , X ∗ , ∨ ) with ˙ be the an action of Gal(k) × . Suppose that  is a -stable base for . Let  corresponding based root datum. As discussed in Sect. 3, the action of Gal(k) ×  ˙ Since  is -stable, the actions of  on  on  determines one of Gal(k) ×  on . ˙ coincide. In the notation of (16) with  = Gal(k), the elements c(σ) ∈ W () ˙ and  ˙ must lie in W () ˙  since this action that arise from the action of Gal(k) on  ˙  is a cocycle in commutes with that of . Therefore, the map c : Gal(k) −→ W () 1  ˙ ). We note that the Gal(k) × -isomorphism class of  ˙ depends only Z (k, W () on that of . By [10, Theorem 6.2.7], there exists a triple (G, B0 , T0 ), unique up to kisomorphism, consisting of a k-quasisplit connected reductive group G, a Borel ksubgroup B0 of G, and a maximal k-torus T0 of B0 , such that the associated based root ˙ We will identify  ˙ and (G, ˙ ˙ B0 , T0 ) datum (G, B0 , T0 ) is Gal(k)-isomorphic to . via such an isomorphism. Let {X α } be a Gal(k)-stable pinning for G relative to B0 and T0 . The action of  ˙ determines a homomorphism φ :  −→ Aut(). ˙ Let ϕ be the composition on  φ ψ ˙ = Aut((G, ˙ B0 , T0 )) −→ Aut(G, B0 , T0 , {X α }), ϕ :  −→ Aut()

˙ where ψ : Aut((G, B0 , T0 )) −→ Aut(G, B0 , T0 , {X α }) is the homomorphism from Remark 19.  ◦  ϕ() ◦ . By Lemma 28, G¯ is a k-quasisplit reductive Let G¯ = G ϕ() and T¯0 = T0 ¯ and group, T¯0 a maximal k-torus of G, ¯ T¯0 ). ˙  = W (G, T0 )ϕ() = W (G, W () ¯ T¯0 )). Thus we may view c as a cocycle in Z 1 (k, W (G, ¯ sep ) such that for all σ ∈ Gal(k), By [7, Theorem 1.1], there is some g ∈ G(k ¯ T¯0 ) g −1 σ(g) lies in the normalizer NG¯ (T¯0 )(k sep ), and the image of g −1 σ(g) in W (G, is equal to c(σ). Let T = g T0 and B = g B0 . Since g is ϕ()-fixed, T is a ϕ()-stable maximal k-torus of G, and B is a ϕ()-stable Borel subgroup of G containing T . We have therefore associated to  a triple (G, T, ϕ) of the required type. Suppose we vary the arbitrary choices made in the above construction of (G, T, ϕ). That is, suppose we choose ˙ • another root datum   that is Gal(k) × -isomorphic to , a based root datum     1   ˙ with underlying datum  and base  , and hence a cocycle c in Z (k, W ( ) ); • another triple of k-groups (G  , B0 , T0 ) k-isomorphic to (G, B0 , T0 ) and an identi˙  ; and ˙  , B0 , T0 ) with  fication of (G • a Gal(k)-stable pinning {X α } of G  relative to B0 and T0 , along with the associated ˙  , B0 , T0 )) −→ Aut(G  , B0 , T0 , {X α }) from Remark 19. map ψ  : Aut((G

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137

We will show that these choices lead to a triple (G  , T  , ϕ ) that is equivalent to (G, T, ϕ). By assumption, there exists a Gal(k) × -isomorphism λ :  −→   . The unique element w ∈ W (  ) mapping λ() to  must lie in W (  ) since λ() and  are ˙ −→  ˙  induced by w is equivariant with respect to the both -stable. The map λ() actions of Gal(k) on these based root data. It follows that the composition λ˙ = wλ ˙ −→  ˙  . Via conjugation, λ˙ induces -equivariant is a Gal(k) × -isomorphism   ˙ −→ Z 1 (k, W ( ˙  )). ˙ −→ W ( ˙ ) and τ : Z 1 (k, W ()) isomorphisms W () −1 1   ˙ ˙ ) ), cohomologous to The map τ (c) : σ → λc(σ) λ˙ is a cocycle in Z (k, W ( c ; more precisely, one can check that for σ ∈ Gal(k), ˙ λ˙ −1 )σ  (w) = w −1 (τ (c)(σ))σ  (w), c (σ) = w −1 (λc(σ)

(20)

˙  ). where σ  denotes the result of the action of σ on w, viewed as an element of W (    Use a particular k-isomorphism between (G , B0 , T0 ) and (G, B0 , T0 ) to identify these triples. Following the above construction, we obtain ◦   a homomorphism ϕ :  −→ Aut(G, B0 , T0 , {X α }), as well as an element g  ∈ G ϕ () and a k-torus  T  = g T0 , analogous to g and T , respectively. ˙ B0 , T0 )) that produces a commutative We have a unique element κ of Aut k ((G, square of Gal(k)-equivariant maps ˙  λ˙

˙ 

˙ (G, B0 , T0 )

(21)

κ

˙ (G, B0 , T0 )

Here, the horizontal arrows are the identifications chosen in the respective constructions of ϕ and ϕ . We therefore obtain a diagram ˙ Aut()

˙ Aut((G, B0 , T0 ))

˙ ) Aut(

˙ Aut((G, B0 , T0 ))

(22)



in which the square on the right is induced by (21) (and hence commutes), the vertical maps are given, respectively, by conjugation by λ˙ and κ, the maps out of  are given ˙ and  ˙  , and the triangle commutes by the -equivariance by the actions of  on  ˙ of λ. Identifying c and c , respectively, with cocycles in Z 1 (k, W (G, T0 )ϕ() ) and  1 Z (k, W (G, T0 )ϕ () ) as in the above construction, it follows from (22) and (20) that

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c (σ) = w −1 (κ ◦ c(σ) ◦ κ−1 )σ(w),

(23)

˙ ) where σ(w) here denotes the result of σ ∈ Gal(k) acting on the element w ∈ W ( via the identification of this group with the concrete Weyl group W (G, T0 ) in (22). Let n ∈ NG¯ (T¯0 )(k sep ) be a representative for w and set μ = ψ(κ) ∈ Autk (G, B0 , T0 ). Then by (23), g  −1 σ(g  ) and n −1 μ(g −1 σ(g))σ(n) have the same image in W (G, T0 ). Rearranging terms and letting h = g  n −1 μ(g)−1 , we obtain that σ(h) and h have the same image modulo σ(μ(g)n)

T0 = σ(μ(g)n T0 ) = σ(μ(g) T0 ) = σ(μ(g T0 )) = σ(μ(T )) = μ(T ).

(24)

Let ν = Int(h) ◦ μ. Since −1

ν(T ) = Int(h)(μ(T )) = Int(g  n −1 μ(g)−1 )(μ(T )) = Int(g  n −1 )(μ(g T )) 

= Int(g  n −1 )(μ(T0 )) = Int(g  n −1 )(T0 ) = g T0 = T  , it follows from (24) that ν gives a k-isomorphism T −→ T  . To show that (G  , T  , ϕ ) is equivalent to (G, T, ϕ), it remains to show that ν is -equivariant. It follows from the construction of ϕ that π ◦ ϕ is equal to the compo˙ −→ Aut((G, ˙ sition  −→ Aut() B0 , T0 )) appearing in (22). Similarly, π ◦ ϕ is ˙ ˙  ) −→ Aut((G, B0 , T0 )). Thus equal to the analogous composition  −→ Aut( for any γ ∈ , π(ϕ (γ)) = κ ◦ π(ϕ(γ)) ◦ κ−1 . Applying ψ to this equality and noting that ψ ◦ π ◦ ϕ = ϕ by construction, we obtain ψ(π(ϕ (γ))) = μ ◦ ϕ(γ) ◦ μ−1 . ˙ B0 , T0 )). Note that by definition, ψ( f ) and ψ  ( f ) agree on T0 for any f ∈ Aut((G, Therefore, as automorphisms of T0 , we have ϕ (γ) = ψ  (π(ϕ (γ))) = ψ(π(ϕ (γ))) = μ ◦ ϕ(γ) ◦ μ−1 . It follows that, as maps on T , ϕ (γ) ◦ ν = ϕ (γ) ◦ Int(h) ◦ μ = ϕ (γ) ◦ Int(g  n −1 μ(g)−1 ) ◦ μ = Int(g  ) ◦ ϕ (γ) ◦ Int(μ(g)n)−1 ◦ μ

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139

= Int(g  ) ◦ μ ◦ ϕ(γ) ◦ μ−1 ◦ Int(μ(g)n)−1 ◦ μ = Int(g  ) ◦ μ ◦ ϕ(γ) ◦ Int(gμ−1 (n))−1 = Int(g  ) ◦ μ ◦ Int(gμ−1 (n))−1 ◦ ϕ(γ), ¯ sep ) and Int(μ−1 (n)) ∈ where the last equality above comes from the fact that g ∈ G(k ϕ()  . Thus ϕ (γ) ◦ ν is equal to W (G, T0 ) Int(g  n −1 μ(g)−1 ) ◦ μ ◦ ϕ(γ) = ν ◦ ϕ(γ), showing that ν is -equivariant. Therefore, (G  , T  , ϕ ) is equivalent to (G, T, ϕ), and our construction induces a well-defined map s : R −→ T . We now show that r ◦ s is the identity map on R . Let  be a root datum representing some class in R , and let (G, T, ϕ) be a triple representing the image of the class of  under s . We need to show that (G, T ) is Gal(k) × -isomorphic to . We will make free use of the notation developed in the construction of s . ˙ −→ (G, ˙ The Gal(k)-equivariant isomorphism of based root data  B0 , T0 ) chosen in the definition of s is -equivariant by construction (where the action of  ˙ and (G, ˙ ˙ B0 , T0 ) on (G, B0 , T0 ) is induced by ϕ). We may therefore identify  as based root data with Gal(k) × -action via this isomorphism. This allows us to identify  and (G, T0 ) as root data with -action (but not necessarily with Gal(k)˙ and  differ in general). action since the actions of Gal(k) on  ¯ sep ) chosen in the definition of s . The map Recall the element g ∈ G(k Int(g)∗ :  = (G, T0 ) −→ (G, T ) is -equivariant since g is ϕ()-fixed. Furthermore, Int(g)∗ is Gal(k)-equivariant since for σ ∈ Gal(k) and χ ∈ X ∗ (T0 ),   Int(g)∗ (σχ) = Int(g)∗ c(σ)(σ χ) = gg

−1

σ(g)

(σ χ)

= σ(g) (σ χ) = σ(g χ) = σ(Int(g)∗ (χ)). Thus (G, T ) is Gal(k) × -isomorphic to , as desired. Finally, we show that s ◦ r is the identity map on T . Let (G, T, ϕ) represent a class in T , and let (G  , T  , ϕ ) represent the image of this class under s ◦ r . Since r ◦ (s ◦ r ) = (r ◦ s ) ◦ r = r , it follows that there is a Gal(k) ×  isomorphism (G, T ) −→ (G  , T  ). By [8, Theorem 2.9], this isomorphism is induced by an isomorphism ν : G −→ G  that restricts to a -equivariant k-isomorphism  T −→ T  . Thus (G, T, ϕ) and (G  , T  , ϕ ) are equivalent. Remark 25 Observe that in the definition of the map s above, the triple (G, T, ϕ) is constructed in such a way that G is k-quasisplit and ϕ fixes a Gal(k)-invariant pinning of G. Thus, since s ◦ r is the identity map on T , we see that every equivalence class in T contains such a triple.

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Moreover, suppose that (G, T, ϕ) is a triple of this kind. Applying the construction of s ◦ r to this triple, we see that the triple we obtain is precisely (G, T, ϕ), provided that we make appropriate choices. Remark 26 Recall that in the proof, it is shown that if (G, T, ϕ) and (G  , T  , ϕ ) are two triples that arise by applying the s construction to a root datum , then (G, T, ϕ) and (G  , T  , ϕ ) are equivalent. We note that the equivalence ν constructed in this case is of a special kind. Namely, ν is of the form Int(h) ◦ μ, where h ∈ G  (k sep ) and μ is a k-isomorphism from G to G  . Now suppose that (G, T, ϕ) and (G  , T  , ϕ ) are arbitrary equivalent triples with the properties that G and G  are k-quasisplit and ϕ and ϕ fix Gal(k)-invariant pinnings for G and G  , respectively. Then combining the first part of this remark with Remark 25, it follows that there is an equivalence ν between (G, T, ϕ) and (G  , T  , ϕ ) of the above special form. Remark 27 Suppose that G  is k-quasisplit and T  is a maximal k-torus of G  . Suppose that the finite group  acts via Gal(k)-equivariant automorphisms on (G  , T  ) preserving a base. Then, the equivalence class of (G  , T  ) lies in R . Applying the construction in the definition of s to (G  , T  ), we obtain a triple (G, T, ϕ) where G is k-quasisplit. Since (G  , T  ) and (G, T ) are Gal(k)-isomorphic, G  can be taken to equal G. Moreover, if g ∈ G(k sep ) is chosen such that T  = g T0 , then the cocycle c used to define T can be taken to be the image of σ → g −1 σ(g) in Z 1 (k, W (G, T0 ) ). In particular, it follows from [1, Proposition 6.1] that T  is stably conjugate to T and that the Gal(k) × -equivariant isomorphism between (G, T  ) and (G, T ) can be given by an inner automorphism of G. (Of course, one could also construct another such triple involving a torus not necessarily stably conjugate to T  by taking the image of (G, T, ϕ) under a rational automorphism of G.) We now turn our attention to Theorem 2. Let G be a quasisplit connected reductive k-group, B a Borel subgroup of G, and T ⊆ B a maximal k-torus of G. Let {X α } be a Gal(k)-stable pinning of G with respect to (B, T ). Suppose a group  acts on G via k-automorphisms, preserving B, T , and {X α }. Then  acts on the based root ˙ datum (G, B, T ), and we will freely use the notation of Sect. 2 in the following. ¯ X¯ ∗ ,  ¯ ∨ ) be the restricted root datum associated ¯ = ( X¯ ∗ , , In particular, we let  ¯  by to the action of  on (G, B, T ) by Theorem 7. Construct a new root datum   ¯ of  ¯ by a maximal reduced subsystem  ¯ as in Remark replacing the root system  13, and do likewise with the coroot system. By [1, Proposition 3.5], G¯ := (G  )◦ is a reductive k-group, and T¯ := (T  )◦ is a ¯ We may identify X¯ ∗ with X ∗ (T¯ ). Under this identification, maximal k-torus of G. the restriction βres of a root β ∈ (G, T ) to T¯ corresponds to i ∗ β ∈ X¯ ∗ . Lemma 28 Using the above notation, and under the above identification of X¯ ∗ with ¯ T¯ ) =  ¯ T¯ ) = W (G, T ) . ¯  , and W (G, X ∗ (T¯ ), we have (G, ¯ T¯ ) is W (G, ¯ T¯ ), and W () ¯ = W ( ¯  ), and Proof Since the Weyl group of (G,  ¯ Corollary 15 implies that W () = W (G, T ) , the claim about the Weyl groups will follow from the claim about root data.

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To prove this, we reduce to the well-known case where  is cyclic. It is clear from ¯ T¯ ), it suffices to show  ¯ T¯ ). ¯  = (G, ¯  = (G, the constructions that to show that  This statement follows for G if it holds for a central quotient of G. Therefore, we may assume that (over k sep ) G is a direct product of simple groups. We can also reduce to the case where  acts transitively on the factors of G. As in the proof of [1, Proposition 3.5], we may identify the factors of G with each other, and replace  by a group S × 1 , where S acts by permuting the factors in our product decomposition of G, and 1 preserves each factor and acts in the same way on each, such that • • • •

the action of S × 1 preserves {X α }, G¯ = (G S×1 )◦ , T¯ = (T S×1 )◦ , and ¯ hence  ¯  , does not change when we replace the action of  by that of S × 1 . ,

Working in stages, we may assume that  is simple. Thus, either  acts by permutation of the factors of G, or G has a single factor and thus is simple. In the former case, our result is trivial, so assume that G is simple. Then, G has a connected Dynkin diagram, whose automorphism group is solvable. Since  embeds in this automorphism group,  must be solvable hence cyclic. It is stated without proof in [6, §1.1] ¯ T¯ ) = W (G, T ) ). We include a ¯ T¯ ) =  ¯  (and W (G, that in the cyclic case, (G, proof below. It follows from [11, §8.2(2 )] that since  fixes a pinning (i.e., for each simple root β ∈ (G, T ), we have that cβ = 1 in the terminology loc. cit.), for each β ∈ (G, T ) ¯ T¯ ). Since, by definition,  ¯  , βres belongs to (G, ¯  ⊆ i ∗ ((G, T )), such that i ∗ β ∈  ∗ ∗ ¯ and since βres corresponds to i β under our identification of X (T ) with X¯ ∗ , this ¯ T¯ ). On the other hand, by [1, Proposition 3.5(iv)], every root ¯  ⊆ (G, shows that  ¯ ¯ ¯ T¯ ) ⊆ i ∗ ((G, T )) = . ¯ in (G, T ) is the restriction of a root in (G, T ), so (G,   ¯ ¯ ¯ ¯ ¯ But  is a maximal reduced subsystem of , so (G, T ) =  , which concludes the proof.  Proof of Theorem 2 Consider a class in T . From Remark 25, we can represent this class by a triple (G, T, ϕ), where G is k-quasisplit and the action ϕ of  on G fixes a Gal(k)-invariant pinning. Let G¯ = (G ϕ() )◦ and T¯ = (T ϕ() )◦ . In order to ¯  contained in the prove our result, we must show that the reduced root datum  ¯ T¯ ). This is the content ¯ of Theorem 7 is equivalent to (G, restricted root datum  of Lemma 28. 

5 Remarks on Cohomological Parametrizations In this section, we describe partitions of R and T into blocks, each of which can be parametrized cohomologically. ˙ 0 be a based root datum with Gal(k) ×  action, i.e., the distinguished base Let  ˙ 0 . Let  be a root datum is Gal(k) × -stable. Let 0 be the root datum underlying  with Gal(k) × -action such that

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•  preserves a base of ; and • with respect to some (hence any) choice of -invariant base of , the associated ˙ (with action σ → σ as in (16)) is Gal(k) × -isomorphic based root datum  ˙ 0. to  The collection of such root data  is closed under Gal(k) × -isomorphism (the ˙ relation of equivalence from Sect. 1), and we let R0 denote the set of equivalence classes of data with the above properties. ˙ implementing the above equiva˙ 0 −→  Let ν be a Gal(k) × -isomorphism  lence. For σ ∈ Gal(k), let σ(ν) be the map σ ◦ ν ◦ σ −1 , viewed as an isomorphism 0 −→ . In particular, the action of σ on  here is via the original action of Gal(k), ˙ Evidently, the map ν −1 σ(ν) lies in the group Aut  (0 ) not the action σ → σ on . of -equivariant automorphisms of 0 . It is not difficult to show that in fact, ν −1 σ(ν) lies in the subgroup W (0 ) of Aut  (0 ) and, moreover, that σ → ν −1 σ(ν) determines a cocycle c in Z 1 (k, W (0 ) ). Altering the choice of  in its class or the choice of ν has the effect of replacing c by σ → κ−1 ◦ c(σ) ◦ σ(κ) for some κ ∈ Aut (0 ). Hence, the class of  determines an element of

 ˙ H0 := Im H 1 (k, W (0 ) ) −→ H 1 (k, Aut  (0 )) . It is readily seen that, conversely, an element of this image determines a root datum  as above, unique up to Gal(k) × -isomorphism. Thus, we have defined a one-to-one ˙ ˙ correspondence a : R0 −→ H0 . ˙ We now describe a corresponding parametrization for blocks in T . Define T0  ˙ 0  to be s R , i.e., the set of classes in T containing triples whose associated based ˙ 0. root data are Gal(k) × -isomorphic to  Recall that the definition of s in Theorem 1 involves the choice of a triple (G, B0 , T0 ) consisting of a k-quasisplit connected reductive group G, a Borel ksubgroup B0 of G, and a maximal k-torus T0 of B0 . We may identify the based root ˙ 0 . We observe that the construc˙ data (with Gal(k) × -actions) (G, B0 , T0 ) and  tion of the triple (G, T, ϕ) in the definition of s involves passing from the given ˙ 0 ) ). It can be root datum  to a cocycle c in Z 1 (k, W (G, T0 )ϕ() ) = Z 1 (k, W ( checked that the passage from  to c is precisely the one occurring in the above definition of the map a . The proof of Theorem 1 proceeds to define (G, T, ϕ) in such a way that it depends only on c, and not directly on . Moreover, it follows from the proof that the equivalence class of (G, T, ϕ) depends only on the class of c in H 1 (k, Aut  (0 )). ˙ ˙ ˙ Therefore, we obtain a map b : H0 −→ T0 such that s factors through H0 according to the commutative diagram

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143 s

˙

R0

˙

T0

r

b

a ˙

H0 In particular, since a and s are bijections, b must also be a bijection.

References 1. J.D. Adler, J.M. Lansky, Lifting representations of finite reductive groups I: semisimple conjugacy classes. Canad. J. Math. 66, 1201–1224 (2014). https://doi.org/10.4153/CJM-2014-0136. Available at arXiv:1106.0786 2. J.D. Adler, J.M. Lansky, Lifting representations of finite reductive groups II: explicit conorm functions. Available at arxiv:1109.0794 3. M. Demazure, Automorphismes des groupes réductifs, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64), Inst. Hautes Études Sci., Paris, 1965, p. 87 (French). MR0228503 (37 #4083) 4. T.J. Haines, On Satake parameters for representations with parahoric fixed vectors. Int. Math. Res. Not. IMRN 20 (2015), 10367–10398. Available at arXiv:1402.3812. MR3455870 5. T.J. Haines, Dualities for root systems with automorphisms and applications to non-split groups. Represent. Theory 22, 1–26 (2018). Available at arXiv:1604.01468 6. R.E. Kottwitz and D. Shelstad, Foundations of twisted endoscopy. Astérisque255 (1999), vi+190 (English, with English and French summaries). MR1687096 (2000k:22024) 7. M.S. Raghunathan, Tori in quasi-split groups. J. Ramanujan Math. Soc. 19(4), 281–287 (2004). MR2125504 (2005m:20114) 8. T.A. Springer, Reductive groups, Automorphic forms, representations, and L-functions, ed. by A. Borel, W. Casselman. Part 1, Proceedings of Symposia in Pure Mathematics, XXXIII, American Mathematical Society, Providence, R.I., 99 (1979), pp. 3–27. MR546587 (80h:20062) 9. T.A. Springer, in Linear Algebraic Groups. Progress in Mathematics, vol. 9 (Birkhäuser Boston Inc., Boston, MA, 1998). MR1642713 (99h:20075) 10. T.A. Springer, in Linear Algebraic Groups. Algebraic geometry IV, Encyclopedia of Mathematical Sciences (Springer, 1994), pp. 1–121. MR1100484 (92g:20061) 11. R. Steinberg, in Endomorphisms of Linear Algebraic Groups. Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR0230728 (37 #6288)

The Character of a Simple Supercuspidal Representation of SL(2, F) Moshe Adrian

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Reduction Formula for the Character on the Split Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Character Formula on the Split Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Character Formula on Elliptic Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146 146 148 149 153 159

Abstract Let F be a non-Archimedean local field of characteristic zero with residual characteristic p. In this paper, we compute the character of a simple supercuspidal representation of SL(2, F), when p is arbitrary.

1 Introduction Let F be a non-Archimedean local field of characteristic zero, with residual characteristic p. Let G be a connected reductive group defined over F, and let G = G(F) be its F-points. If π is an irreducible admissible representation of G, we denote by θπ its distribution character, which is a linear functional on Cc∞ (G), the locally constant, compactly supported complex-valued functions on G. Harish-Chandra showed that θπ can be represented by a locally constant function on the regular semisimple set of G, which we will also denote by θπ . Suppose now that π is a supercuspidal representation. Much is known about θπ . The first supercuspidal characters were computed by Sally and Shalika in [7], where they investigated the supercuspidal representations of SL(2, F) when p = 2. Since then, there has been much work on understanding and computing supercuspidal characters, which has culminated in the work of Adler and Spice [1]. The computation M. Adrian (B) Department of Mathematics, Queens College, CUNY, 65-30 Kissena Blvd., Queens, NY 11367-1597, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_4

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of these characters has yielded important consequences, including applications to the study of stability of supercuspidal L-packets constructed by Debacker and Reeder [2], Reeder [5], and Kaletha [4]. In this paper, we compute the character values of a simple supercuspidal representation of SL(2, F) when p is arbitrary (see Theorem 5.3 and Theorem 6.6). Our computations give formulas that are geometric in nature: On the split torus case, we get a polynomial in q, whose powers are lengths of affine Weyl group elements that bound an explicit interval in the standard apartment of SL(2, F) (see Remark 5.4). In the elliptic torus case, a similar polynomial arises, together with a Gauss sum. We now briefly present an outline of the paper. In Sect. 2, we set up the notation that we will use throughout. In Sect. 3, we recall the definition of a simple supercuspidal representation as well as the Frobenius formula for the character of an induced representation. In Sect. 4, we present a reduction formula for the character of a simple supercuspidal representation of a split simply connected group on the split maximal torus. In Sect. 5, we compute the character of a simple supercuspidal representation of SL(2, F) on the split torus. In Sect. 6, we compute the character on an elliptic torus.

2 Notation Let F be a non-Archimedean local field of characteristic zero. Let o be the ring of integers of F and p its maximal ideal. Fix a uniformizer  in F. If p = 2, we fix a non-square unit  in o× . Let | · | denote the standard normalized absolute value on F, and val the associated valuation. We fix a Haar measure on F such that o has volume 1, and we use the abbreviation vol for volume. A character of o is said to be of level 1 if it is trivial on p, but nontrivial on o. If ψ is a character of F, we define the twist of ψ by a ∈ F by ψa (x) := ψ(ax).

3 Background In this section, we recall the definition of a simple supercuspidal representation following Gross and Reeder [3], as well as the Frobenius formula for the character of a supercuspidal representation following [6]. Let G be a split, simply connected, almost simple, connected reductive group, and T a maximal F-split torus in G. Associated to T we have the set of roots  of T in G, a set of affine roots , and an apartment A. Let Z be the center of G, and set G = G(F), T = T(F), and Z = Z(F). Let X ∗ (T ) denote the character lattice of T , let T0 be the maximal compact subgroup of T , and set T1 = t ∈ T0 : λ(t) ∈ 1 + p ∀λ ∈ X ∗ (T ).

The Character of a Simple Supercuspidal Representation …

147

Denoting the normalizer of T in G by N , we have the extended affine Weyl group W a := N /T0 . Now fix a Chevalley basis in the Lie algebra of G. To each ψ ∈ , we have an associated affine root group Uψ in G. Fix an alcove C in the apartment with corresponding simple affine and positive affine roots  ⊂  + . For w ∈ W a , we let (w) denote the length of w. Set I = T0 , Uψ : ψ ∈  + , I+ = T1 , Uψ : ψ ∈  + , I++ = T1 , Uψ : ψ ∈  + \ . Then I is the Iwahori subgroup corresponding to C and I+ is its pro-unipotent radical. We set H = ZI+ . The following results can be found in [3, §9]. Lemma 3.1 The subgroup I++ is normal in I+ , with quotient I+ /I++ ∼ =



Uψ /Uψ+1

ψ∈

as T0 -modules. Definition 3.2 A character χ : H → C∗ is called affine generic if (i) χ is trivial on I++ and (ii) χ is nontrivial on Uψ /Uψ+1 for every ψ ∈  (see Lemma 3.1). Theorem 3.3 Let χ : H → C∗ be an affine generic character. Then cIndHG χ is an irreducible supercuspidal representation, where cInd denotes compact induction. Gross and Reeder have named the representations from Theorem 3.3 simple supercuspidal representations. Now suppose that π is an irreducible smooth supercuspidal representation of G. Let K be a compact open subgroup of G, and suppose that σ is an irreducible representation of K such that π = cIndKG σ. Let θσ denote the character of σ. The following is the Frobenius formula for the induced character θπ . Theorem 3.4 [6, Theorem 1.9] Let g be a regular element of G. Then θπ (g) =





x∈K\G/K y∈K\KxK

where

θ˙σ (ygy−1 )

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θ˙σ (k) =



θσ (k) if k ∈ K, 0 if k ∈ G \ K.

4 Reduction Formula for the Character on the Split Torus In this section, we show that the formula in Theorem 3.4 simplifies if G is simply connected, π is a simple supercuspidal representation, T is the split torus in G, and g ∈ T . Let us first recall the following basic fact about double coset decompositions. Suppose that K is a compact open subgroup of G, and let us choose any set of representatives {tα } for the double cosets of K\G/K. Then Ktα K is the disjoint union of the cosets Ktα s1 , Ktα s2 , . . . , Ktα sm , where s1 , s2 , . . . , sm is a set of representatives of (K ∩ tα−1 Ktα ) \ K. We will use this fact repeatedly in this paper. Let [W a ] denote any set of representatives for the group W a . Proposition 4.1 Let π = cIndHG χ be a simple supercuspidal representation of G, as in Theorem 3.3. If g ∈ T and g is regular, then 



x∈[W a ]

y∈H \HxH

θπ (g) = |T0 /ZT1 | 

where χ(h) ˙ =

−1 χ(ygy ˙ ),

χ(h) if h ∈ H 0 if h ∈ G \ H

Proof Our starting point is the well-known affine Bruhat decomposition I \G/I ←→ W a . Since I /I+ ∼ = T0 /T1 , the affine Bruhat decomposition descends to a decomposition H \G/H ←→ N /ZT1 . Therefore, θπ (g) =





−1 χ(ygy ˙ ).

x∈N /ZT1 y∈H \HxH

We now analyze the inner sum. Write τ (x) :=



−1 χ(ygy ˙ ),

y∈H \HxH

for x ∈ N . We claim that τ (x) = τ (xt) ∀t ∈ T0 . To see this, write HxH =

m  i=1

Hxxi ,

(1)

The Character of a Simple Supercuspidal Representation …

149

where xi are representatives of H /(H ∩ x−1 Hx). Then the map Hxy → Hxty gives a bijection H \HxH → H \HxtH , for t ∈ T0 . Therefore, in particular, HxtH =

m 

Hxtxi .

i=1

We conclude that 

τ (xt) =

−1 χ(ygy ˙ )=

y∈H \HxtH

=

m 

m 

−1 −1 −1 χ(xtx ˙ i gxi t x )

i=1 −1

χ(xtx ˙ i t tgt

−1

txi−1 t −1 x−1 )

i=1

=

m 

−1 −1 −1 −1 χ(xtx ˙ i t gtxi t x ),

i=1

the last equality coming from the fact that g ∈ T . Since H /(H ∩ x−1 Hx) is a disjoint union of certain spaces of the form Uγ /Uγ+n , and since conjugation by an element t ∈ T0 preserves Uγ /Uγ+n , we get that m 

−1 −1 −1 −1 χ(xtx ˙ i t gtxi t x ) =

m 

i=1

χ(xx ˙ i gxi−1 x−1 ) = τ (x).

i=1

We have therefore concluded that τ (xt) = τ (x) ∀t ∈ T0 . Finally, note that we have the following short exact sequence 1 → T0 /ZT1 → N /ZT1 → W a = N /T0 → 1. Since we just showed that τ is constant along fibers of the above exact sequence, together with (2), we now have that θπ (g) = |T0 /ZT1 |





x∈[W a ]

y∈H \HxH

−1 χ(ygy ˙ ).

 

5 The Character Formula on the Split Torus In this section, we prove Theorem 5.3. In particular, we will compute the inner sum τ (x) in Proposition 4.1 for any set of representatives of W a by decomposing HxH into a union of left cosets. Afterwards, we sum over all x to obtain θπ . By definition of affine generic, we have that χ|I+ is given by

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χ|I+ : I+ → C∗ 

d11 d12 d21 d22

 → χ1 (d12 )χ2 (d21 )

where χ1 is some level 1 character of o and where χ2 (d21 ) = χ2 ( 1 d21 ), where χ2 is some level 1 character of o. Extend both χ1 and χ2 to functions on F by setting both equal to 0 on F \ o.   a 0 Fix an element g = ∈ T1 , set r := val(a − a−1 ), and again define 0 a−1 

τ (x) :=

−1 χ(ygy ˙ ),

y∈H \HxH

for x ∈ N .



Proposition 5.1 Let x =

 b 0 . 0 b−1

(a) If val(b) = n ≥ 0 and 2n < r, then τ (x) = q2n . (b) If val(b) = n < 0 and −2n < r, then τ (x) = q−2n . (c) Otherwise, τ (x) = 0. Proof It will be convenient for the proof to introduce the following notation. For r1 , r2 ∈ N, we write      x1 x2 1 + p pr1 r1 r2 := ∈ SL(2, F) : x . , x ∈ 1 + p, x ∈ p , x ∈ p 1 4 2 3 pr2 1 + p x3 x4 We prove (a). Let val(b) = n ≥ 0. We first write the double coset HxH as a finite union of single right cosets. Since H ∩ x−1 Hx = Z ·



 1+p o , p2n+1 1 + p

we obtain an explicit disjoint union decomposition H=



 Z·

z∈p/p2n+1

1+p o p2n+1 1 + p

   10 · . z1

Therefore, we have the disjoint union HxH =

z∈p/p2n+1

 Hx ·

 10 . z1

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Suppose that y ∈ HxH . We need to check when ygy−1 ∈ H since χ˙ vanishes outside H  . Using  our above double coset decomposition, write y = hxu, where u is of the form 10 for some z ∈ p/p2n+1 and for some h ∈ H . Then ygy−1 ∈ H ⇔ xugu−1 x−1 ∈ z1 H . Moreover,   a 0 . xugu−1 x−1 = b−2 z(a − a−1 ) a−1 We now write a − a−1 = r u for some unit u ∈ o× . Then b−2 z(a − a−1 ) =  r z  for some z  ∈ p/p2n+1 . Therefore,   −1 χ(ygy ˙ )= χ2 (−2n+r−1 z  ). −2n

y∈H \HxH

z  ∈p/p2n+1

Making a change of variables, we get 

χ2 (−2n+r−1 z  )



=

z  ∈p/p2n+1

χ2 (v)

r −1



= vol(p )

v∈p−2n+r /pr

χ2 (v)dv

p−2n+r ∩o

since χ˙ vanishes outside H . If p−2n+r ⊇ o, then this integral vanishes since the integral of a nontrivial character over a group vanishes. However, if p−2n+r  o, which is precisely the condition that 2n < r, then vol(pr )−1



χ2 (v)dv = vol(pr )−1

p−2n+r ∩o



dv = vol(pr )−1 vol(p−2n+r )

p−2n+r

since χ2 is trivial on p. By our choice of measure, we have that vol(pd ) = q−d for d > 0, thus proving part (a). We now prove (b). So suppose that val(b) = n < 0. Then similarly as in (a), we get   1 + p p−2n −1 , H ∩ x Hx = Z · p 1+p H=



 Z·

z∈o/p−2n

1 + p p−2n p 1+p

   1z · . 01

Therefore, we have a disjoint union HxH =

z∈o/p−2n

 Hx ·

 1z . 01

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Suppose that y ∈ HxH . By the above  double coset decomposition, we may write 1z y = hxu, where u is of the form for some z ∈ o/p−2n , and for some h ∈ H . 01   2 −1 a b z(a − a) −1 −1 A computation yields xugu x = . Therefore, 0 a−1 



−1 χ(ygy ˙ )=

y∈H \HxH

χ1 (b2 z(a−1 − a)).

z∈o/p−2n

Rewriting b2 z(a−1 − a) = 2n r z  where z  ∈ o/p−2n , then after a change of variables, we obtain   χ1 (b2 z(a−1 − a)) = χ1 (2n r z  ) z∈o/p−2n

z  ∈o/p−2n

=



χ1 (v) = vol(pr )−1

v∈p2n+r /pr

χ1 (v)dv.

p2n+r ∩o

If p2n+r ⊇ o, then again this integral vanishes. However, if p2n+r  o, which is precisely the condition that −2n < r then vol(pr )−1



χ1 (v)dv = vol(pr )−1

p2n+r ∩o



dv = vol(pr )−1 vol(p2n+r ),

p2n+r

since χ1 is trivial on p. The proofs of parts (a) and (b) immediately imply part (c).   0 c Proposition 5.2 Let x = . −c−1 0

 

(a) If val(c) = n ≥ 0 and 2n + 1 < r, then τ (x) = q2n+1 . (b) If val(c) = n < 0 and −2n − 1 < r, then τ (x) = q−2n−1 . (c) Otherwise, τ (x) = 0. Proof The proof is completely analogous to that of Proposition 5.1.

 

Theorem 5.3 Let π be a simple supercuspidal representation of SL(2, F).   a 0 ∈ T1 and set r = val(a − a−1 ). Let W a (r) := {w ∈ W a : (a) Let g = 0 a−1 (w) < r}. Then  q(w) , θπ (g) = cq w∈W a (r)

where cq :=

 q−1

if p = 2 2 q − 1 if p = 2

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(b) If g ∈ T \ ZT1 , then θπ (g) = 0. Proof We first prove (a). Let −1 + p denote the group generated by {−1 + z : z ∈ p}. Note that  q−1 if p = 2 × 2 |T0 /ZT1 | = |o /−1 + p| = q − 1 if p = 2 Therefore, by Propositions 5.1 and 5.2, and Proposition 4.1, we have θπ (g) = cq (1 + 2q + 2q2 + · · · + 2qr−1 ). 

b 0 0 b−1



and val(b) = k, It is a straight forward calculation to show that if x =   0 c then (x) = |2k|. Moreover, if x = and val(c) = k, then (x) = |2k + −c−1 0 1|. This concludes the proof of part (a). We now prove (b). In the proof of Proposition 5.1 (a), we showed that ygy

−1

 ∈ H ⇐⇒

a 0 b−2 z(a − a−1 ) a−1

 ∈ H.

This trivially implies that a ∈ −1 + p. Moreover, the condition a ∈ −1 + p is implied, for the same reason, when you compute the terms ygy−1 that appear in θπ (g) for any other representative x ∈ [W a ], as simple computations will show. Therefore,   θπ vanishes on T \ ZT1 . Remark 5.4 1. Up to central character, we have computed θπ on all of the split torus (since we assumed above that g ∈ T1 ). Since the central character is given by the data forming the simple supercuspidal representation, we have computed θπ on the entire split torus. 2. The set W a (r) has geometric meaning. If C denotes the fundamental alcove in the standard apartment for SL(2, F), then W a (r) · C is the symmetric interval about C containing 2r − 1 alcoves. 3. We note that the term a − a−1 in Theorem 5.3 is, up to a sign, a canonical square root of the Weyl denominator. In particular, if D denotes the Weyl denominator, then −D(g) = (a − a−1 )2 .

6 The Character Formula on Elliptic Tori In this section, we compute θπ on the elliptic tori of SL(2, F). We first recall some relevant notions.

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√ Let E/F be a quadratic √ extension. Then we may set E = F( ), with  a nonsquare in F, and we let δ = . Let E be a uniformizer in E and set  = Gal(E/F). If w ∈ E × , then we write w for the image of w under the non-trivial element of . We have the norm map NE/F : E × → F × given by NE/F (w) = ww. The elliptic tori of SL(2, F) are of the form E 1 := {w ∈ E × : NE/F (w) = 1}. Lemma 6.1 If w = a0 + a1 E ∈ E 1 , then a0 , a1 ∈ o. Proof Expanding the left-hand side of the equality NE/F (w) = 1 in a power series gives the result.   We now fix an embedding of the torus E 1 into SL(2, F). Specifically, if we fix the basis , 1 of F ⊕ F, then we have the embedding E 1 → SL(2, F)  a0 a1 . a1  a0

 a0 + a1 δ →

We now compute θπ (g), where g = a0 + a1 δ, which we identify with the above 2 × 2 matrix. Recall that −1 + p denotes the group generated by {−1 + z : z ∈ p}. We set r := val(a1 ) and we assume that a0 ∈ 1 + p. As we will see later, in order for θπ (g) = 0 to hold, we must have a0 ∈ −1 + p. To obtain the character formula in the a0 ∈ −1 + p case, we use the central character (see Remark 6.7). First, we would like to use some of the ideas from the proof of Proposition 4.1. Recall that   −1 χ(ygy ˙ ), (2) θπ (g) = x∈N /ZT1 y∈H \HxH

and that we have defined τ (x) :=



−1 χ(ygy ˙ ),

y∈H \HxH

for x ∈ N . It is not necessarily the case anymore that τ (x) = τ (xt) ∀t ∈ T0 . If x satisfies τ (xt) = τ (x) ∀t ∈ T0 , then we can and will collect the τ (xt) terms together so that in the character formula we pick up a factor of |T0 /ZT1 | = cq , as we did in Sect. 5. In the cases where x ∈ N does not satisfy τ (x) = τ (xt) for all t ∈ T0 , we need to sum over all of these elements xt, which are parameterized by T0 /ZT1 ∼ = Fq /−1. These summations appear in Theorem 6.6. We have   b 0 . Suppose first that E/F is ramified. Proposition 6.2 Let x = 0 b−1 (a) Let val(b) = n > 0. (i) If 2n < r, then τ (xt) = τ (x) = q2n ∀t ∈ T0 .

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155

(ii) If 2n = r, then τ (x) = q2n χ2 (b−2 a1 ). (b) Let val(b) = n < 0. (i) If −2n < r, then τ (xt) = τ (x) = q−2n ∀t ∈ T0 . (ii) If −2n = r, then τ (x) = q−2n χ1 (b2 a1 ). (c) If val(b) = 0, then τ (x) = χ1 (b2 a1 )χ2 (b−2 a1 ). (d) If x does not satisfy the conditions of (a)-(i), (a)-(ii), (b)-(i), (b)-(ii), or (c), then τ (x) = 0. Suppose now that E/F is unramified. (a’) Let val(b) = n ≥ 0. (i) If 2n + 1 < r, then τ (xt) = τ (x) = q2n ∀t ∈ T0 . (ii) If 2n + 1 = r, then τ (x) = q2n χ2 (b−2 a1 ). (b’) Let val(b) = n < 0. (i) If −2n < r, then τ (xt) = τ (x) = q−2n ∀t ∈ T0 . χ1 (b2 a1 (1 − z 2 )). (ii) If −2n = r, then τ (x) = z∈o/p−2n

(c’) If x does not satisfy the conditions of (a’)-(i), (a’)-(ii), (b’)-(i), or (b’)-(ii), then τ (x) = 0. Proof We first prove (a), (c), and (a’). Let val(b) = n ≥ 0. Recall from Proposition 5.1 the decomposition

HxH =

z∈p/p2n+1

 Hx ·

 10 . z1 

 10 Suppose that y ∈ HxH . Write y = hxu, where u is of the form for some z1 z ∈ p/p2n+1 and for some h ∈ H . Then ygy−1 ∈ H ⇔ xugu−1 x−1 ∈ H . Moreover, −1 −1

xugu x

 =

b2 a1 a0 − za1 −2 2 b a1 ( − z ) a0 + za1

 .

Suppose first that E/F is ramified, and recall that −1 + p was defined to be the group generated by {−1 + z : z ∈ p}. Then xugu−1 x−1 ∈ H if and only if a0 ∈ −1 + p and b−2 a1 ∈ o, in other words, if and only if a0 ∈ −1 + p and 2n ≤ r. Note that we used here that since z ∈ p, then the diagonal entries are indeed in −1 + p. Now, we have χ(xugu−1 x−1 ) = χ1 (b2 a1 )χ2 (b−2 a1 ) = χ2 (b−2 a1 ) since val(b) > 0, val(z) > 0, and val(a1 ) ≥ 0 (see Lemma 6.1). We see now that if 2n < r, then χ2 (b−2 a1 ) = 1 so that

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τ (x) =



−1 χ(ygy ˙ )=

y∈H \HxH

χ2 (b−2 a1 ) = vol(p/p2n+1 ) = q2n .

z∈p/p2n+1

If 2n = r, then τ (x) =



−1 χ(ygy ˙ )=

y∈H \HxH



χ2 (b−2 a1 ) = q2n χ2 (b−2 a1 ).

z∈p/p2n+1

Suppose now that E/F is unramified. Then xugu−1 x−1 ∈ H if and only if a0 ∈ −1 + p and val(b−2 a1 ) ≥ 1 (i.e., r ≥ 2n + 1) since val() = 0. Indeed, we are also again using that z ∈ p, so that the diagonal entries are still in −1 + p. Now, we have χ(xugu−1 x−1 ) = χ2 (b−2 a1 ). If r > 2n + 1, then χ2 (b−2 a1 ) = 1 so that τ (x) = q2n as above. Otherwise, if 2n + 1 = r, τ (x) =



−1 χ(ygy ˙ )=

y∈H \HxH



χ2 (b−2 a1 ) = q2n χ2 (b−2 a1 ).

z∈p/p2n+1

Let us now prove (b) and (b’). Suppose val(b) = n < 0. Then recall from Proposition 5.1, we have   1z HxH = Hx · . 01 −2n z∈o/p



 1z Suppose that y ∈ HxH . Then we may write y = hxu, where u is of the form 01 −2n −1 −1 for , and for  some z ∈ o/p  some h ∈ H . A computation yields xugu x = a0 + za1  b2 a1 (1 − z 2 ) . a0 − a1 z b−2 a1  Suppose first that E/F is ramified. A similar analysis as in case (a) shows that a0 ∈ −1 + p and val(b2 a1 ) ≥ 0, so that −2n ≤ r. Note that we have used here that  ∈ p, so that the diagonal entries are indeed in −1 + p. Then, similarly as in case (a), if −2n < r, we have that τ (x) = q−2n . If −2n = r, then τ (x) =



−1 χ(ygy ˙ )=

y∈H \HxH



χ1 (b2 a1 ) = q−2n χ1 (b2 a1 ).

z∈o/p−2n

Now suppose that E/F is unramified. Then we must have a0 ∈ −1 + p and val(b2 a1 (1 − z 2 )) ≥ 0 (which implies in particular that we must have val(a1 ) ≥ 1), so that in particular we have r ≥ −2n. Note here that  ∈ o and z ∈ o. But as just remarked, we must have val(a1 ) ≥ 1. Thus, the diagonal entries are indeed still in −1 + p. Then, similarly as in case (a), if r > −2n, we have that τ (x) = q−2n . If r = −2n, then   −1 χ(ygy ˙ )= χ1 (b2 a1 (1 − z 2 )). τ (x) = y∈H \HxH

z∈o/p−2n

The Character of a Simple Supercuspidal Representation …

157

  Remark 6.3 1. We note that the val(b) > 0 and val(b) = 0 cases needed to be separated for the E/F ramified case because of the b2 a1 term. Specifically, if val(a1 ) = 0, then we get a nontrivial contribution to τ (x) from b2 a1 . However, if val(a1 ) ≥ 1, then the two cases val(b) > 0 and val(b) = 0 can be combined to give τ (x) = q2n in both of these cases. If E/F is unramified, however, this phenomenon does not occur. Namely, the b2 a1 term does not contribute in the val(b) = 0 case. The reason is that the b−2 a1 ( − z 2 ) term forces val(a1 ) ≥ 1 if we assume that val(b) = 0. Therefore, we may combine the val(b) > 0 and val(b) = 0 cases into one case, as we have done. 2. In Proposition 6.2, it is a simple matter τ (xt) in the cases where  to compute  s 0 τ (xt) = τ (x). For example, let γ = with s ∈ o× so that γ ∈ T0 . In 0 s−1 (a), we computed τ (x) when 2n = r. Since xγ falls into the same 2n = r case, we conclude that τ (xγ) has also been computed. In particular, we have that τ (xγ) = q2n χ2 ((bs)−2 a1 ).   0 c . Suppose first that E/F is ramified. Proposition 6.4 Let x = −c−1 0 (a) Let val(c) = n ≥ 0. (i) if 2n + 1 < r, then τ (xt) = τ (x) = q2n+1 ∀t ∈ T0 . (ii) if 2n + 1 = r, then τ (x) = q2n+1 χ2 (−c−2 a1 ). (b) Let val(c) = n < 0. (i) if −2n − 1 < r, then τ (xt) = τ (x) = q−2n−1 ∀t ∈ T0 . (ii) if −2n − 1 = r, then τ (x) = q−2n−1 χ1 (−c2 a1 ). (c) If x does not satisfy the conditions of (a)-(i), (a)-(ii), (b)-(i), or (b)-(ii), then τ (x) = 0. Suppose now that E/F is unramified. Then (a’) Let val(c) = n ≥ 0. (i) if 2n + 1 < r, then τ (xt) = τ  (x) = q2n+1 ∀t ∈ T0 . (ii) if 2n + 1 = r, then τ (x) = χ2 (−c−2 a1 (1 − z 2 )). z∈o/p2n+1

(b’) Let val(c) = n < 0. (i) if −2n < r, then τ (xt) = τ (x) = q−2n−1 ∀t ∈ T0 . (ii) if −2n = r, then τ (x) = q−2n−1 χ1 (−c2 a1 ). (c’) If x does not satisfy the conditions of (a’)-(i), (a’)-(ii), (b’)-(i), or (b’)-(ii), then τ (x) = 0.

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Proof The proof is analogous to that of Proposition 6.2. Remark 6.5

1. In the case that E/F is unramified and val(c) ≥ 0, one can compute that val(a1 ) ≥ 1 is forced for the character to not vanish. Thus, there is no need to distinguish between val(c) > 0 and val(c) = 0 cases, as we did in Remark 6.3. 2. The same remark as in Remark 6.3 can be made about elements xγ where τ (xγ) = τ (x). We now present the character formula on elliptic tori. First recall the set W a (r) := {w ∈ W a : (w) < r} defined in Sect. 5. Finally, we note that if g = a0 + a1 δ and / −1 + p, then θπ (g) = 0 (this can be seen from the proofs of Propositions 6.2 a0 ∈ and 6.4). Theorem 6.6 Let π be a simple supercuspidal representation   of SL(2, F) defined a0 a1 by the affine generic character of Sect. 5. Let g = ∈ E 1 , set r = val(a1 ), a1  a0 and assume that a0 ∈ 1 + p. Let  cq :=

q−1 2

if p = 2 q − 1 if p = 2

(a) Suppose that E/F is ramified. (i) Suppose that r is even and r > 0. Then θπ (g) = cq





q(w) + qr

w∈W a (r)

(χ2 (−r s2 a1 ) + χ1 (−r s2 a1 )).

s∈F× q /−1

(ii) Suppose that r is odd. Then θπ (g) = cq





q(w) + qr

w∈W a (r)

(χ2 (−−r+1 s2 a1 ) + χ1 (−−r−1 s2 a1 )).

s∈F× q /−1

(iii) Suppose that r = 0. Then 

θπ (g) =

χ1 (s2 a1 )χ2 (s−2 a1 ).

s∈F× q /−1

(b) Suppose that E/F is unramified. (i) Suppose that r is even and r > 0. Then  θπ (g) = cq 1 + 2q + 2q2 + · · · + 2qr−2 + qr−1   χ1 (−r s2 a1 (1 − z 2 )) + qr−1 + z∈o/pr s∈F× q /−1

 s∈F× q /−1

χ1 (−−r s2 a1 ).

The Character of a Simple Supercuspidal Representation …

159

(ii) Suppose that r is odd. Then

 θπ (g) = cq 1 + 2q + 2q2 + · · · + 2qr−2 + qr−1   χ2 (−−r+1 s2 a1 (1 − z 2 )) + qr−1 + z∈o/pr s∈F× q /−1



χ2 (−r+1 s2 a1 ).

s∈F× q /−1

(iii) Suppose that r = 0. Then θπ (g) = 0. Remark 6.7 1. Up to central character, we have computed θπ on all of the elliptic torus (since we assumed above that a0 ∈ 1 + p). Since the central character is given by the data forming the simple supercuspidal representation, we have computed θπ on the entire elliptic torus. 2. The set W a (r) again arises naturally in the computation of the character on elliptic tori. The geometric meaning of this set was discussed⎛in Remark 5.4. ⎞ 

 3. The term 1 + 2q + 2q2 + · · · + 2qr−2 + qr−1 is precisely ⎝ q(w) ⎠ − w∈W a (r)

qr−1 . Acknowledgements It is a great pleasure to have been able to discuss this work with Paul Sally, Jr. This paper has benefited from conversations with Paul Sally, Jr., Gordan Savin, and Loren Spice.

References 1. J. Adler, L. Spice, Supercuspidal characters of reductive, p-adic groups. Amer. J. Math. 131(4), 1137–1210 (2009) 2. S. DeBacker, M. Reeder, Depth-zero supercuspidal L-packets and their stability. Ann. Math. 169(3), 795–901 (2009) 3. B. Gross, M. Reeder, Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 154, 431–508 (2010) 4. T. Kaletha, Epipelagic L-packets and rectifying characters. Inventiones Mathematicae 202(1), 1–89 (2015) 5. M. Reeder, Supercuspidal L-packets of positive depth and twisted coxeter elements. J. Reine Angew. Math. 620, 1–33 (2008) 6. P.J. Sally, Jr., Some remarks on discrete series characters for reductive p-adic groups, in Representations of Lie groups, Kyoto, Hiroshima, 1986, (91g:22026) (1988), pp. 337–348 7. P. Sally, J. Shalika, Characters of the discrete series of representations of SL(2) over a local field. Proc. Nat. Acad. Sci. U.S.A. 61, 1231–1237 (1968)

Classification of Strongly Positive Representations of Even General Unitary Groups Yeansu Kim and Ivan Mati´c

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Tadi´c’s Structure Formula: General Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Classification of Strongly Positive Representations of Even General Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Construction of the Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Classification of Strongly Positive Representations: D(ρ; σcusp ) . . . . . . . . . . . . . . 4.3 Classification of Strongly Positive Representations . . . . . . . . . . . . . . . . . . . . . . . . 5 Appendix: Even Unitary Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Notation for Even Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Tadi´c’s Structure Formula for Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Strongly Positive Representations for Even Unitary Groups . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162 162 162 163 166 167 167 169 171 172 173 173 174

Abstract We explicitly identify the structure of Jacquet modules of parabolically induced representations of even unitary groups and even general unitary groups over a p-adic field F of characteristic different than two. As an application, we obtain a classification of the strongly positive discrete series representations of those groups. Keywords Tadi´c’s structure formula · Jacquet modules · Strongly positive representations Mathematics Subject Classification (2000) 20C11 · 11F70 The first author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2017R1C1B2010081). The second author was supported by the Croatian Science Foundation under project 9364. Y. Kim Department of Mathematics Education, Chonnam National University, Yongbong-ro 77, Gwangju City, Korea e-mail: [email protected] I. Mati´c Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, Osijek, Croatia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_5

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1 Introduction The first purpose of this paper is to explicitly construct Tadi´c’s structure formula for even unitary groups and even general unitary groups. The structure formula explores the Jacquet modules of parabolically induced representations. In the case of general linear groups, the Jacquet modules of parabolically induced representations are studied in [2, 17]. The case of classical groups is very different due to the different Weyl groups and how they act on Levi subgroups. In [14], Tadi´c explicitly describes the structure of Jacquet modules in the cases of Sp2n , GSp2n , and SO2n+1 , later generalized to the cases of SO2n , metaplectic groups, and GSpin groups in [1, 3–6]. Using Tadi´c’s structure formula, one can determine all Jacquet modules of certain classes of representations [9, 10]. The Tadi´c’s structure formula also happens to be extremely useful for the study of reducibility and composition series of certain induced representations which happen to be important for understanding of the unitary dual, such as standard representations and generalized principal series [15]. As an application of Tadi´c’s structure formula, the second purpose of this paper is to obtain a classification of strongly positive representations of even unitary groups and even general unitary groups. We note that the strongly positive representations serve as basic building blocks in the classification of the discrete series of classical groups, including unitary ones, obtained in [11, 12], and in the classification of discrete series representations of odd GSpin groups, recently provided in [7]. This paper is organized as follows. In Sect. 2, we outline standard notation. In Sect. 3, we obtain Tadi´c’s structure formula for even general unitary groups, which describes the explicit structure of the Jacquet modules of parabolically induced representations of general unitary groups. In Sect. 4, we obtain a classification of strongly positive representations of even general unitary groups. In the appendix, we also discuss the even unitary group case.

2 Notation and Preliminaries 2.1 Notation Let F be a non-Archimedean local field of characteristic different than two and let E/F be a quadratic extension. Let  = Gal(E/F) and write x → x¯ for its non-trivial element. Choose an element β ∈ E such that E = F(β) and β¯ = −β. To define the unitary groups, we set   βIn . Jn = −βIn We let Hn = GU(n, n) be the quasi-split general unitary group in 2n variables defined with respect to E/F and Jn . Its F-points are

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Hn (F) = {g ∈ GL2n (E)|t gJ ¯ n g = λJn , λ ∈ E × } We fix λ throughout the paper. Let s and Ms be as in Remark 3.1. For a parabolic subgroup Ps = Ms Ns of Hn , n we denote the induced representation IndH Ps (ρ1 ⊗ · · · ⊗ ρk ⊗ τ ) by ρ1 × · · · × ρk  τ where each ρi (resp. τ ) is a representation of some GLni (E) (resp. Hn−n (F)). In parn ticular, IndH Ps is a functor from admissible representations of Ms (F) to admissible representations of Hn (F) that sends unitary representations to unitary representations. We also denote the normalized Jacquet module with respect to Ps by rs (τ ). In particular, rs is a functor from admissible representations of Hn (F) to admissible representations of Ms (F). The Grothendieck group of the category of all admissible representations of finite length of Hn (F), i.e. a free abelian group over the set of all irreducible representations of Hn (F) (resp. GLn (E)) is denoted by RGU (n) (resp. RGL (n)) and set RGU = ⊕ RGU (n), RGL = ⊕ RGL (n). n≥0

by

n≥0

n (ρ1 ⊗ · · · ⊗ ρk ) In the case of GL, we denote the induced representation IndGL P

ρ1 × · · · × ρk such that P = M N is the standard parabolic subgroup of GLn where M ∼ = GLn1 × GLn2 × · · · × GLnk and each ρi is a representation of GLni (E) for i = 1, . . . , k. We also follow the notation in [2]. Let ρ be an irreducible unitary cuspidal representation of some GLp (E). We define the segment, Δ := [ν a ρ, ν a+k ρ] = {ν a ρ, ν a+1 ρ, . . . ν a+k ρ} where a ∈ R and k ∈ Z≥0 . If a > 0, we call the segment Δ strongly positive.

3 The Tadi´c’s Structure Formula: General Unitary Groups We fix the F-Borel subgroup B of upper triangular matrices in Hn . Then, B = TU, where T is a maximal torus of diagonal elements in Hn and let A0 be the maximal F-split subtorus of T. Then, ⎧⎛ x1 ⎪ ⎪ ⎪⎜ . ⎪ ⎪ .. ⎜ ⎪ ⎪ ⎪ ⎨⎜ ⎜ xn T(F) = ⎜ ⎜ λ¯x1−1 ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ .. ⎪ ⎪ ⎝ . ⎪ ⎪ ⎩

⎞

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎟ ⎟ ⎟ ⎟ ⎟ |xi ∈ E × , λ ∈ F × ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭ −1

λ¯xn

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and

⎧⎛ x1 ⎪ ⎪ ⎪ ⎜ .. ⎪ ⎪ ⎜ ⎪ . ⎪ ⎪ ⎨⎜ ⎜ xn A0 (F) = ⎜ ⎜ λx1−1 ⎪ ⎜ ⎪ ⎪ ⎪⎜ .. ⎪ ⎪ ⎝ . ⎪ ⎪ ⎩

⎞

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎟ ⎟ ⎟ ⎟ ⎟ |xi , λ ∈ F × ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭ −1

λxn

For simplicity, we let a(x1 , . . . , xn ; λ) be an element of the form diag(x1 , . . ., xn , λx1−1 , . . . , λxn−1 ) in A0 (F). Let Φ(Hn (F), A0 (F)) be the restricted roots of Hn (F) with respect to A0 (F) and let Δ := {αi }ni=1 be the set of simple roots, where αi = ei − ei+1 , 1 ≤ i ≤ n − 1, αn = 2en − e0 . The Weyl group W (Hn (F)/A0 (F)) is isomorphic to Sn  {±1}n , where Sn is the permutation group of n letters. More precisely, for (ij) ∈ Sn , (ij) · a(x1 , . . . , xn ; λ) = a(x1 , . . . , xi−1 , xj , xi+1 , . . . , xj−1 , xi , xj+1 , . . . , xn ; λ) and for = ( 1 , . . . , n ) ∈ {±1}n , i · a(x1 , . . . , xn ; λ) = a(x1 , . . . , xi−1 , λxi i , xi+1 , . . . , xn ; λ). Remark 3.1 Let s = (n1 , n2 , . . . , nk ) be an ordered partition of some n such that n ≤ n and let Θ = Δ\{αn1 , αn1 +n2 , . . . , αn1 +...+nk }. Let As be the subtorus of A0 that corresponds to Θ and let Ms be the centralizer of As . Then, its F-points is of the form ⎧⎛ g1 ⎪ ⎪ ⎪ ⎜ . ⎪ ⎪ .. ⎜ ⎪ ⎪ ⎪⎜ ⎪ ⎜ ⎪ ⎪ gk ⎨⎜ ⎜ g Ms (F) = ⎜ ⎜ ⎪ ⎪ λt g¯ 1−1 ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪⎜ .. ⎪ ⎪ ⎝ . ⎪ ⎪ ⎩

⎞

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ |gi ∈ GLni (E), g ∈ Hn−n (F), λ ∈ F × . ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭ −1 λt g¯ n

Therefore, Ms (F) ∼ = GLn1 (E) × GLn2 (E) × · · · × GLnk (E) × Hn−n (F) and for simplicity, the element diag(g1 , g2 , . . . , gk , g, λt g¯1−1 , λt g¯2−1 , . . . , λt g¯k−1 ) in Ms (F) is denoted by (g1 , g2 , . . . , gk , g). Then, for an element (g1 , g2 , . . . , gk , g) ∈ Ms (F), the Weyl group W (Hn (F)/ As (F)) is a subgroup of Sk  {±1}k . In particular, for (ij) ∈ W (Hn (F)/As (F)) ⊂ Sk , (ij) · (g1 , g2 , . . . , gk , g) = (g1 , . . . , gi−1 , gj , gi+1 , . . . , gj−1 , gi , gj+1 , . . . , gk , g),

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and for = ( 1 , . . . , k ) ∈ {±1}k ⊂ W (Hn (F)/As (F)) with i = −1, k = 1 for k = i, (3.1) · (g1 , . . . , gi , . . . , gk , g) = (g1 , . . . , λt g¯i−1 , . . . , gk , g). Therefore, the Weyl group action on the maximal F-split torus A0 (F) and Levi subgroup Ms (F) of Hn is similar to that for general symplectic groups (Note that the main difference is the Weyl group action on the Levi subgroup (3.1)). In [14, Sect. 4], Tadi´c characterizes the representative element of the set [WΔ\α \W/WΔ\β ] and its explicit action on the simple roots for GSp2n . We also get the same results, i.e. from Lemmas 4.1 through 4.8 of [14] in the case of even general unitary groups, since those lemmas only depend on the simple roots, Weyl group and its action on the simple roots and we also know that simple roots for Hn is same as those for GSp2n . We now explain the Tadi´c’s structure formula for Hn and follow the notation in [14] for simplicity. Let i1 , i2 be integers which satisfy 1 ≤ i1 , i2 ≤ n. Take an integer d such that 0 ≤ d ≤ min{i1 , i2 }. Suppose that an integer k satisfies max{0, (i1 + i2 − n) − d } ≤ k ≤ min{i1 , i2 } − d . Let pn (d , k)i1 ,i2 ∈ Sn be defined by ⎧ j for 1 ≤ j ≤ k; ⎪ ⎪ ⎪ ⎪ for k + 1 ≤ j ≤ i2 − d ; ⎨ j + i1 − k pn (d , k)i1 ,i2 (j) = (i1 + i2 − d + 1) − j for i2 − d + 1 ≤ j ≤ i2 ; ⎪ ⎪ for i2 + 1 ≤ j ≤ i1 + i2 − d − k; j − i2 + k ⎪ ⎪ ⎩ j for i1 + i2 − d − k + 1 ≤ j ≤ n. Let qn (d , k)i1 ,i2 be (pn (d , k)i1 ,i2 , (1i2 −d , −1d , 1n−i2 )) where 1i = 1, . . . , 1 (1 appears i times). Let w = qn (d , k)i1 ,i2 . Then, for (g1 , g2 , g3 , g4 , h) ∈ GLk (E) × GLi2 −d −k (E) × GLd (E) × GLi1 −d −k (E) × Hn−i1 −i2 +d +k (F), we have w · (g1 , g2 , g3 , g4 , h) = (g1 , g4 , λ t g¯3−1 , g2 , h). Let πi be an irreducible smooth representation of GLni (E) for i = 1, 2, 3, 4 and let σ be an irreducible smooth representation of Hm . We have, w −1 · (π1 ⊗ π2 ⊗ π3 ⊗ π4 ⊗ σ) = π1 ⊗ π4 ⊗ πˇ 3 ⊗ π2 ⊗ ωπ3 σ. where π(g) ˇ := π(t g¯ −1 ). Set (π4 ⊗ σ) = πˇ 1 × π2 × π4 ⊗ π3  ωπ1 σ. (π1 ⊗ π2 ⊗ π3 )

(3.2)

(3.3)

Applying (3.2) and (3.3), we get Theorem 3.1 (Tadi´c’s structure formula for general unitary groups.) For π ∈ RGL (i) and σ ∈ RGU (n − i), the following structure formula holds  μ∗ (σ). μ∗ (π  σ) = M∗ (π) Lemma 3.2 Let ρ be an irreducible cuspidal representation of GLk (E) and a, b ∈ R be such that b − a ∈ Z≥0 . Let σ be an admissible representation of finite length

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of Hn (F). Write μ∗ (σ) =



π  ⊗ σ  . Then M∗ (δ([ν a ρ, ν b ρ])) =

π  ,σ 

b  b  i=a−1 j=i

ν i ρ]) ⊗ δ([ν j+1 ρ, ν b ρ]) ⊗ δ([ν i+1 ρ, ν j ρ]) and μ∗ (δ([ν a ρ, ν b ρ])  σ) = 

δ([ν a ρ,

b  b  i=a−1 j=i

δ([ν −i

t −1

ρ¯ , ν −a

t −1

ρ¯ ]) × δ([ν j+1 ρ, ν b ρ]) × π  ⊗ δ([ν i+1 ρ, ν j ρ])  ωσ  . We

π  ,σ 

omit δ([ν x ρ, ν y ρ]) if x > y. We recall the definition of the strongly positive representations of GSpin groups.

Definition 3.3 An irreducible representation σ of Hn (F) is called strongly positive if for every embedding σ → ν s1 ρ1 × ν s2 ρ2 × · · · × ν sk ρk  σcusp where ρi , i = 1, 2, . . . , k are irreducible unitary cuspidal representations of GL, σcusp is an irreducible cuspidal representation of Hn (F) and si ∈ R, i = 1, 2, . . . , k, then we have si > 0 for each i. The following lemma is also useful when we explicitly calculate Jacquet modules: Lemma 3.4 Let ρ be a cuspidal representation of GLk (E) and let σcusp be a cuspidal representation of Hn (F). Write ρ = ν e(ρ) ρu , where e(ρ) ∈ R and ρu is a unitary cuspidal representation. If ρ  σcusp has a strongly positive discrete series subrepresentation, then we have (i) ρu ∼ = ρ¯u , i.e. ρu is conjugate self-dual. (ii) ωρ σcusp ∼ = σcusp . Proof Let σ be a strongly positive subrepresentation of ν e(ρ) ρu  σcusp . Then, e(ρ) > 0 since σ is strongly positive. If (i) or (ii) does not hold, then due to Lemma 2.1 in [16], ν α ρu  σcusp is irreducible for every α. Then, we have the following embedding: ¯u  σcusp . σ → ν e(ρ) ρu  σcusp ∼ = ν −e(ρ) ρ Since −e(ρ) < 0, this contradicts the strong positivity of σ.

4 Classification of Strongly Positive Representations of Even General Unitary Groups In this section, we classify the strongly positive representations of even general unitary groups. We mostly follow the arguments in [8] and appendix to [5] and generalize those to our case.

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4.1 Construction of the Map In this section, we construct the map from the set of strongly positive representations into certain induced representations. We consider the induced representations of the following form (4.1) δ(Δ1 ) × δ(Δ2 ) × · · · × δ(Δk )  σcusp where Δ1 , Δ2 , . . . , Δk is a sequence of strongly positive segments (See Notation 2.1 for the definition of strongly positive segments) satisfying 0 < e(Δ1 ) ≤ e(Δ2 ) ≤ · · · ≤ e(Δk ) (we allow k = 0 here), σcusp an irreducible cuspidal representation of Hm (F). Then, we show that Theorem 4.1 (i) The induced representation δ(Δ1 ) × δ(Δ2 ) × · · · × δ(Δk )  σcusp of the form (4.1) has a unique irreducible subrepresentation which we denote by δ(Δ1 , . . . , Δk ; σcusp ). (ii) The strongly positive representation can be embedded into induced representation of the form (4.1). Proof (i) and (ii) are GU analogue of Theorem 3.3 and Theorem 3.4 in [8], respectively. Since the idea of their proofs depends on the behaviour of GL parts of Jacquet modules, we apply those in [8] to the case of even general unitary groups and we do not repeat here.

4.2 Classification of Strongly Positive Representations: D(ρ; σcusp ) Let ρ be a conjugate self-dual irreducible cuspidal representation of GLnρ (E) and σcusp be an irreducible cuspidal representation of Hm (F). Let D(ρ; σcusp ) be the set of strongly positive representations whose cuspidal supports are the representation σcusp and twists of the representation ρ by positive valued characters. Let a ≥ 0 be the unique non-negative real number such that ν a ρ  σcusp reduces [13]. Furthermore, we assume that this reducibility point a is in 21 Z (see (HI) of [12], page 771). Let kρ denote a, the smallest integer which is not smaller than a. In this section, we obtain the classification of strongly positive representations in D(ρ; σcusp ). In a previous section, Theorem 4.1 implies that every strongly positive representation can be viewed as the unique irreducible subrepresentation of induced representation of the form (4.1). Therefore, there exists an mapping from the set of strongly positive representations of Hn (F) into the set of induced representations of the form (4.1). Now, we further refine the image of this mapping when we restrict the mapping to D(ρ; σcusp ).

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Theorem 4.2 Let σsp be an irreducible strongly positive representation in D(ρ; σcusp ) and consider it as the unique irreducible subrepresentation of induced representation of the form (4.1). Write Δi = [ν ai ρ, ν bi ρ]. Then, ai = a − k + i, b1 < · · · < bk and k ≤ a. Proof We only consider the Theorem when a > 0. We use induction as in [8]. The cases k = 0 and k = 1 are exactly as in [8] and we skip the proof. We now consider the case when k = 2. Now, we have σsp → δ([ν a1 ρ, ν b1 ρ]) × δ([ν a2 ρ, ν b2 ρ])  σcusp As in the case k = 1, we easily show that a2 = a. Since σsp is the unique irreducible subrepresentation of δ([ν a1 ρ, ν b1 ρ]) × δ([ν a ρ, ν b2 ρ])  σcusp we also have σsp → δ([ν a1 ρ, ν b1 ρ])  δ([ν a ρ, ν b2 ρ]; σcusp ). This embedding gives us the following embedding σsp → δ([ν a1 +1 ρ, ν b1 ρ]) × ν a1 ρ  δ([ν a ρ, ν b2 ρ]; σcusp ) If ν a1 ρ  δ([ν a ρ, ν b2 ρ]; σcusp ) is irreducible, we have the embedding σsp → δ([ν a1 +1 ρ, ν b1 ρ]) × ν −a1 ρ  δ([ν a ρ, ν b2 ρ]; σcusp ) and this contradicts the strong positivity of σsp . Therefore, ν a1 ρ  δ([ν a ρ, ν b2 ρ]; σcusp ) is reducible. GU analogue of Proposition 4.3 [5] implies that a1 ∈ {a − 1, a, b2 + 1}. Let us first consider the case when a1 = a > 1/2. Similarly as in Proposition 3.1 in [14], we use the following calculation of Jacquet modules: ρ¯ × δ([ν a ρ, ν b2 ρ]) ⊗ σcusp rGL (ν a ρ  δ([ν a ρ, ν b2 ρ]; σcusp )) = ν −a + ν a ρ × δ([ν a ρ, ν b2 ρ]) ⊗ σcusp

(4.2)

μ∗ (ν a ρ  δ([ν a ρ, ν b2 ρ]; σcusp )) ≥ δ([ν a+1 ρ, ν b2 ρ]) ⊗ ν a ρ  δ(ν a ρ, σcusp ) (4.3) Due to GU analogue of Lemma 4.1 in [5], ν a ρ  δ(ν a ρ, σcusp ) is irreducible. Therefore, the irreducible subquotient of ν a ρ  δ([ν a ρ, ν b2 ρ]; σcusp ) that contains right-hand side of (4.3) in its Jacquet modules must also contain both terms in (4.2). This implies that ν a ρ  δ([ν a ρ, ν b2 ρ]; σcusp ) is irreducible, which is a contradiction. Now, we consider a1 = a = 1/2. In this case, GU analogue of Appendix of [5] (or Lemma 5.7 of [5]) implies that irreducible subrepresentation of δ([ν 1/2 ρ, ν b1 ρ]) × δ([ν 1/2 ρ, ν b2 ρ])  σcusp ) is not strongly positive, which is a contradiction. Similarly as in [8], we have a contradiction in the case a1 = b2 + 1. The remaining case is when a1 = a − 1, which is possible only if a > 1. In that case, we also have b1 < b2 . Completing argument of induction on k is also exactly as in [8] and we skip the proof.

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We also show that the mapping from D(ρ; σcusp ) to the set of induced representations of the form (4.1) is well defined in the following theorem: Theorem 4.3 Let σsp be an irreducible strongly positive representation in D(ρ; σcusp ). Then, there exist a unique set of strongly positive segments Δ1 , Δ2 , . . . , Δk , with 0 < e(Δ1 ) < e(Δ2 ) < · · · < e(Δk ), and a unique irreducible cuspidal representation σ  ∈ R such that σsp  δ(Δ1 , Δ2 , . . . , Δk ; σ  ). Proof The proof is similar to [8] and we, therefore, omit the proof in this case since we constructed all the tools that we need in Sect. 3. In Theorem 4.2 and Theorem 4.3, we construct an injective mapping from D(ρ; σcusp ) into the set of induced representations of the form (4.1) with refinement on the unitary exponents as in Theorem 4.2. More precisely, let Jord(ρ,a) stand for the set of all increasing sequences b1 , b2 , . . . , bkρ , where bi ∈ R, bi − a + kρ − i ∈ Z≥0 for i = 1, . . . , kρ and −1 < b1 < b2 < · · · < bkρ . So far, we construct the following injective mapping: D(ρ; σcusp ) → Jord(ρ,a) Now, it remains to show that this map is surjective. Let b1 , b2 , . . . , bkρ denote an increasing sequence appearing in Jord(ρ,a) . We showed in Sect. 4.1 that the induced representation δ([ν a−kρ +1 ρ, ν b1 ρ]) × δ([ν a−kρ +2 ρ, ν b2 ρ]) × · · · × δ([ν a ρ, ν bkρ ρ])  σcusp

(4.4)

has a unique irreducible subrepresentation, which we denote by σ(b1 ,...,bkρ ;a) . We apply the induction argument in [8] to show that the above subrepresentation is strongly positive and we do not repeat the argument here. Theorem 4.4 The representation σ(b1 ,...,bkρ ;a) is strongly positive.

4.3 Classification of Strongly Positive Representations Let ρi be a conjugate self-dual irreducible cuspidal representation of GLnρi (E) for i = 1, . . . , k and σcusp is an irreducible cuspidal representation of Hm (F). Let D(ρ1 , ρ2 , . . . , ρk ; σcusp ) be the set of strongly positive representations whose cuspidal supports are the representation σcusp and the twists of the representations ρi by positive-valued characters for i = 1, . . . , k. Let aρi ≥ 0 be the unique non-negative real number such that ν aρi ρi  σcusp reduces for each i = 1, . . . , k [13]. Furthermore, we assume that this reducibility point aρi is in 21 Z (see (HI) of [12], page 771). With Theorem 4.1, we use induction to prove the following two theorems as in [8]:

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Theorem 4.5 Let σsp be a strongly positive representation in D(ρ1 , ρ2 , . . . , ρk ; σcusp ). Then σsp can be considered the unique irreducible subrepresentation of the following induced representation: (

ki k  

(i)

δ([ν aρi −ki +j ρi , ν bj ρi ]))  σcusp ,

(4.5)

i=1 j=1 (i) where ki ∈ Z≥0 , ki ≤ aρi , b(i) j > 0 such that bj − aρi ∈ Z≥0 , for i = 1, . . . , k j = (i) 1, . . . , ki . Also, b(i) j < bj+1 for 1 ≤ j ≤ ki − 1.

Theorem 4.5 implies that we construct the mapping from D(ρ1 , ρ2 , . . . , ρk ; σcusp ) to the set of induced representations of the form (4.1). We now show that this mapping is well defined and injective. Theorem 4.6 Suppose that the representation σsp ∈ D(ρ1 , ρ2 , . . . , ρk ; σcusp ) can be embedded as the unique irreducible subrepresentations of both representations ki ki k  k    (i) a  −k  +j b(i) aρi −ki +j  j ( δ([ν ρi , ν ρi ]))  σcusp and ( δ([ν ρi i ρi , ν cj ρi ]))  σcusp i=1 j=1

i=1 j=1

 as in Theorem 4.5. Then we have k = k  , σcusp ∼ and { = σcusp

ki 

(i)

δ([ν aρi −ki +j ρi , ν bj ρi ])|

j=1 ki

i = 1, . . . , k} is a permutation of {



δ([ν

aρ −ki +j  i ρi ,

(i)

ν cj ρi ])|i = 1, . . . , k}.

j=1  ∼ Proof Since σsp ∈ D(ρ1 , ρ2 , . . . , ρk ; σcusp ), σcusp = σcusp and {ρi |i = 1, . . . , k  } ⊂ {ρi |i = 1, . . . , k}. Then, comparing the Jacquet modules, we easily see that k = k  ki ki   a  −k  +j b(i) aρi −ki +j j and { δ([ν ρi , ν ρi ])|i = 1, . . . , k} is a permutation of { δ([ν ρi i ρi , j=1 (i) ν cj ρi ])|i

j=1

= 1, . . . k}.

Now, we extend the above mapping to the set of all strongly positive representations of H(F). We first show the uniqueness of partial cuspidal support of strongly positive representation. Proposition 4.7 Let σsp denote a strongly positive representation of Hn (F). Then there is a unique, up to isomorphism, cuspidal representation σcusp of Hm (F) such that σsp is a subrepresentation of π  σcusp , for an irreducible representation π of GLn−m (E). Proof Suppose that there are non-isomorphic irreducible cuspidal representations σ1 of Hm1 (F) and σ2 of Hm2 (F), such that σsp → π1  σ1 and σsp → π2  σ2 for appropriate irreducible representations π1 and π2 .

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Thus, there are cuspidal representations ρ1 , ρ2 , . . . , ρk of general linear groups such that σsp → ν x1 ρ1 × ν x2 ρ2 × · · · × ν xk ρk  σ1 . Strong positivity of σsp implies xi > 0 for all i. Also, Frobenius reciprocity implies μ∗ (σsp ) ≥ π2 ⊗ σ2 , which implies that μ∗ (ν x1 ρ1 × ν x2 ρ2 × · · · × ν xk ρk  σ1 ) ≥ π2 ⊗ σ2 . Repeated application of Lemma 3.2 implies that π2 is an irreducible subquotient of ρ1 × ρ2 × · · · × ρk , where ρi ∈ {ν xi ρi , ν −xi t ρi −1 }, for i = 1, 2, . . . , k. Since σ1 is not isomorphic to σ2 , using Lemma 4.2 with obtain that there is an i ∈ {1, 2, . . . , k} such that ρi ∼ = ν −xi t ρi −1 . Since xi > 0, this contradicts strong positivity of σsp and the proposition is proved. Furthermore, by comparing Jacquet modules as in the proof of Theorem 4.6, we also show the uniqueness of GL cuspidal supports of strongly positive representation. Therefore, for any strongly positive representation σsp of H(F), there exists unique set of ρ1 , ρ2 , . . . , ρk and σcusp such that σsp can be considered to be the element in D(ρ1 , ρ2 , . . . , ρk ; σcusp ). Let SP be the set of all strongly positive representations of H(F). To see this mapping explicitly, let us collect the data from the induced representations of the ki k     {(ρi , b(i) form (4.5). Let LJ be the set of (Jord , σ ) where Jord = j )} and σ be i=1 j=1

an irreducible cuspidal representation in R such that (i) {ρ1 , ρ2 , . . . , ρk } is a (possibly empty) set of mutually non-isomorphic irreducible conjugate self-dual cuspidal unitary representations of GL such that  ν aρi ρi  σ  reduces for aρ i > 0 (this defines aρ i ), (ii) ki = aρ i , (i) (i) (iii) for each i = 1, 2, . . . , k, b(i) 1 , b2 , . . . , bki is a sequence of real numbers such (i) (i) (i) that aρ i − b(i) j ∈ Z, for j = 1, 2, . . . , ki , and −1 < b1 < b2 < · · · < bki . Now, the last step is to show that this mapping is surjective onto LJ . Following [8], we have Theorem 4.8 The maps described above give a bijective correspondence between the sets SP and LJ.

5 Appendix: Even Unitary Case We shortly derive the structural formula for unitary groups, which is also mentioned in Section 15 of [12], for the purpose of obtaining a classification of strongly positive

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representations of even unitary groups, following the same approach as in the previous sections. The strongly positive representations of even unitary groups have also been classified in Sections 7 and 15 of [12], using a different approach.

5.1 Notation for Even Unitary Groups We let Gn = U(n, n) be the quasi-split unitary group in 2n variables defined with respect to E/F and Jn and let RU (n) be the Grothendieck group of the category of all admissible representations of finite length of Gn (F) and set RU = ⊕ RU (n). As n≥0

in the even general unitary groups. Let also B = TU, A0 , T for Gn be defined as in Sect. 3. Then, ⎫ ⎧⎛ ⎞ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ .. ⎟ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ . ⎪ ⎪ ⎟ ⎪ ⎪ ⎬ ⎨⎜ ⎜ ⎟ x n × ⎟ |x T(F) = ⎜ ∈ E ⎜ ⎟ i x¯ 1−1 ⎪ ⎪ ⎟ ⎪ ⎪⎜ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪⎝ ⎠ . ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ −1 x¯ n and

⎧⎛ ⎪ ⎪ x1 ⎪ ⎜ ⎪ ⎪ ⎪⎜ . . . ⎪ ⎪ ⎨⎜ ⎜ xn A0 (F) = ⎜ ⎜ x1−1 ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ .. ⎪ ⎪ ⎝ . ⎪ ⎪ ⎩

⎞

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎟ ⎟ ⎟ ⎟ ⎟ |xi ∈ F × ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭ −1

xn

The F-points of Levi subgroups in Gn that corresponds to s = (n1 , n2 , . . . , nk ) is of the form ⎧⎛ g1 ⎪ ⎪ ⎪ ⎜ .. ⎪ ⎪ ⎜ ⎪ . ⎪ ⎪⎜ ⎪ ⎜ ⎪ gk ⎨⎜ g Ms (F) = ⎜ ⎜ ⎪ t −1 ⎜ ⎪ g¯1 ⎪ ⎪⎜ ⎪ ⎜ ⎪ .. ⎪ ⎪⎝ . ⎪ ⎪ ⎩

⎞

t

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ |gi ∈ GLn (E), g ∈ Gn−n (F) . i ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭ −1

g¯n

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5.2 Tadi´c’s Structure Formula for Unitary Groups Note that the Weyl group for unitary groups is isomorphic to general unitary groups. Therefore, we use the same notation for qn (d , k)i1 ,i2 (= ω) as in the general unitary group case. Then, for (g1 , g2 , g3 , g4 , h) ∈ GLk (E) × GLi2 −d −k (E) × GLd (E) × GLi1 −d −k (E) × Gn−i1 −i2 +d +k (F), we have w · (g1 , g2 , g3 , g4 , h) = (g1 , g4 , t g¯3−1 , g2 , h). Let πi be an irreducible smooth representation of GLni (E) for i = 1, 2, 3, 4 and let σ be an irreducible smooth representation of Gm . By our previous calculation, w −1 · (π1 ⊗ π2 ⊗ π3 ⊗ π4 ⊗ σ) = π1 ⊗ π4 ⊗ πˇ 3 ⊗ π2 ⊗ σ. Set



 (π4 ⊗ σ) = πˇ 1 × π2 × π4 ⊗ π3  σ. (π1 ⊗ π2 ⊗ π3 )

(5.1)

(5.2)

We follow argument in Sect. 3 by replacing (3.3) by (5.2), we have Theorem 5.1 (Tadi´c’s structure formula for unitary groups.) For π ∈ RGL (i) and σ ∈ RU (n − i), the following structure formula holds 

 μ∗ (σ). μ∗ (π  σ) = M∗ (π)

5.3 Strongly Positive Representations for Even Unitary Groups With Tadi´c’s structure formula for unitary groups (Sect. 5.2), we apply the arguments as in Sect. 4 to obtain the analogous results for even unitary groups. In this subsection, we only state the main result for even unitary groups and skip the proof since we already go through the similar arguments in Sect. 4. Let SP  be the set of all strongly positive representations of G(F) and LJ  be the set of (Jord , σ  ) where σ  be an irreducible cuspidal representation in RU and Jord be exactly as in the case of even general unitary groups. Then, one can repeat the same arguments as before to obtain the bijective correspondence between SP  and LJ  . Acknowledgements The first author would like to thank the organizers of the workshop on Representation theory of p-adic groups at IISER Pune, Professors Anne-Marie Aubert, Manish Mishra, Alan Roche, Steven Spallone for their invitation and hospitality. The authors would also like to thank the referee for his/her careful reading of the paper and for helpful comments and suggestions.

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References 1. D. Ban, Parabolic induction and Jacquet modules of representations of O(2n, F). Glas. Mat. Ser. III 34(54) 2, 147–185 (1999) 2. J. Bernstein, A.V. Zelevinsky, Induced representations of reductive p-adic groups I. Ann. Sci. ´ Ecole Norm. Sup. 10, 4411–472 (1977) 3. M. Hanzer, G. Mui´c, Parabolic induction and Jacquet functors for metaplectic groups. J. Algebra 323, 241–260 (2010) 4. C. Jantzen, Jacquet modules of induced representations for p-adic special orthogonal groups. J. Algebra 305, 802–819 (2006) 5. Y. Kim, Strongly positive representations of GSpin2 n + 1 and the Jacquet module method. with an appendix, “Strongly positive representations in an exceptional rank-one reducibility case” by Ivan Matic. Math. Z. 279, 271–296 (2015) 6. Y. Kim, Strongly positive representations of even GSpin groups. Pacific J. Math. 280, 69–88 (2016) 7. Y. Kim, I. Mati´c, Discrete series of odd GSpin groups. Preprint, available at arXiv:1706.01111 8. I. Mati´c, Strongly positive representations of metaplectic groups. J. Algebra 334, 255–274 (2011) 9. I. Mati´c, Jacquet modules of strongly positive representations of the metaplectic groups. Trans. Amer. Math. Soc. 365(5), 2755–2788 (2013) 10. I. Mati´c, M. Tadi´c, On Jacquet modules of representations segment type. Manuscripta Math. 147, 437–476 (2015) 11. C. Mœglin, Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc. 4, 143–200 (2002) 12. C. Mœglin, M. Tadi´c, Construction of discrete series for classical groups. J. Am. Math. Soc. 15, 715–786 (2002) 13. A. Silberger, Special representations of reductive p-adic groups are not integrable. Ann. of Math. 111, 571–587 (1980) 14. M. Tadi´c, Structure arising from induction and Jacquet modules of representations of classical p-adic groups. J. Algebra 177, 1–33 (1995) 15. M. Tadi´c, On reducibility of parabolic induction. Israel J. Math. 107, 29–91 (1998) 16. M. Tadi´c, On regular square integrable representations of p-adic groups. Amer. J. Math. 120, 159–210 (1998) 17. A.V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n)

On the Unicity of Types for Toral Supercuspidal Representations Peter Latham and Monica Nevins

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Construction of Toral Supercuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results on Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Mackey Decomposition and Strategies for Identifying Types . . . . . . . . . . . . . . . . . . . . . . . 7 Unicity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Inequivalence of Unicity and Strong Unicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract For tame arbitrary-length toral, also called positive regular, supercuspidal representations of a simply connected and semisimple p-adic group G, constructed as per Adler-Yu, we determine which components of their restriction to a maximal compact subgroup are types. We give conditions under which there is a unique such component and then present a class of examples for which there is not, disproving the strong version of the conjecture of unicity of types on maximal compact open subgroups. We restate the unicity conjecture and prove it holds for the groups and representations under consideration under a mild condition on depth.

1 Introduction Let G be a connected reductive p-adic group. In [2], J. Bernstein gives a decomposition of the category of smooth representations of G into indecomposable full subcategories, called blocks, that are indexed by the inertial support of the irreducible The second author’s research is supported by NSERC Canada Discovery Grant 06294. P. Latham Department of Mathematics, King’s College London, Strand, London, UK e-mail: [email protected] M. Nevins (B) Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_6

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representations they contain. Given an irreducible supercuspidal representation π of G, a type for π is a pair (K , λ) consisting an irreducible representation λ of a compact open subgroup K of G such that Ind GK λ is a projective generator of the block containing π. In this case, λ occurs as a subrepresentation of π| K , and we say π contains the type (K , λ). Types are now known to exist for many classes of supercuspidal representations. In particular, the work of Adler [1], generalized by Yu [24], shows that every essentially tame supercuspidal representation contains a type; the work of Kim [11] and Fintzen [7] assure us that under mild assumptions on the residual characteristic, then all supercuspidal representations of G are of this form. Given a supercuspidal representation π of G containing a type (K , λ), it is simple to produce additional types: Any G-conjugate of (K , λ) is a type, as is any pair (K  , τ ) where K  is a compact open subgroup of G containing K and τ is an irreducible representation of K  that contains λ upon restriction to K . A natural question to ask is whether π can contain any additional types, specifically on maximal compact open subgroups, that are not related to (K , λ) in this way. It is expected that this should never happen, and the conjecture that this is the case is known as the unicity of types. The name is due to Pa˘sk¯unas [21], whose thesis extended G. Henniart’s appendix “Sur l’unicité des types pour GL2 ” of the article [3]. The goal of this paper is to establish the unicity of types for a class of essentially tame supercuspidal representations which we call toral, defined below. The unicity conjecture is a theorem for G = GLn (F) [21], G = SL2 (F) [12], for essentially tame representations of G = SLn (F) [14], and for depth-zero supercuspidal representations of any connected reductive p-adic group G [13]. In each of these cases, it was seen that a stronger property holds, namely that if K ⊂ G is any maximal compact open subgroup, then there exists, up to G-conjugacy, at most one type defined on K for each supercuspidal representation π of G. We will refer to this as the strong unicity property. While strong unicity implies unicity, in Sect. 8 we provide counterexamples to prove that they are in fact inequivalent in general. From now on, let us specialize to the case that G is a simply connected semisimple p-adic group. Essentially tame supercuspidal representations are constructed from sequences of twisted Levi subgroups that split over a tamely ramified Galois extension, together with characters of these subgroups and a representation of the smallest twisted Levi subgroup. We restrict our attention here to those sequences for which the smallest twisted Levi subgroup is an anisotropic (also called elliptic) maximal torus of G. For the purposes of this paper, we call these toral supercuspidal representations, though we caution the reader that some authors reserve “toral” to mean the more restrictive case that the twisted Levi sequence has length d = 1. Relating to work of T. Kaletha [10], F. Murnaghan calls our representations “positive regular” (as justified in [18]). The strategy for proving the unicity conjecture for supercuspidal representations of G is as follows: Let π be an essentially tame supercuspidal representation and (K, κ) a type as arising from the above construction. Since G is semisimple and simply connected, we have both that c-IndKG κ is irreducible (hence equivalent to π) and that every maximal compact open subgroup of G is the stabilizer G y of a vertex

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y in the Bruhat-Tits building of G. Thus for any such G y containing K, it follows G directly that we have an induced type (G y , IndK y κ). The conjecture of unicity of types, therefore, amounts to the statement that, up to G-conjugacy, all types for π on a maximal compact open subgroup arise in this way. Strong unicity is the statement that furthermore any two types for π on G y are conjugate by an element of NG (G y ); this is equivalent to the statement that (K, κ) is not contained in two distinct but conjugate maximal compact open subgroups. The restriction of π to a maximal compact open subgroup G y decomposes as an infinite direct sum of irreducible representations of G y . Describing these branching rules is a difficult open problem of interest in its own right. Here, it suffices to note that with G as above, the Bernstein block corresponding to π is generated by π. Therefore, the types of π supported on G y are exactly those irreducible components of π|G y that do not occur in π  |G y for any other (inequivalent) irreducible representation π  of G. Proving this, in turn, is made possible by the major work of Hakim and Murnaghan [8] which establishes the equivalences among essentially tame supercuspidal representations entirely in terms of the Adler-Yu data used to construct them. To state our main result (Theorem 7.3), let T be an anisotropic maximal torus of G and let B T denote the fixed point set of T acting on the Bruhat-Tits building B(G) of G. This set contains in particular a point x which is the image of the building of T in the building of G. Let cT ≥ 0 be the simplicial radius of B T , relative to x, as defined in Sect. 5. Theorem Let T be a tame elliptic maximal torus of G and suppose π is a supercuspidal representation of G built from a datum containing T . If the character φ0 of T appearing in the datum has depth greater than 2cT , then π satisfies the unicity conjecture relative to any maximal compact open subgroup of G. Moreover, in this case, π satisfies the strong unicity property if and only if B T consists precisely of the closure of a single facet in B(G). We note that the theorem holds without any hypothesis on the depth of φ0 when T is unramified; see Corollary 7.4. An essential ingredient of the proof is the analysis of the fixed points of B(G) under the action of both the torus T and of the inducing subgroup K, using particularly Lemma 5.1. As we discuss in Sect. 8, while the hypothesis for strong unicity (e.g., that B T = {x}) holds in many cases (notably, for G = SLn , for unramified tori and for purely ramified Coxeter tori), it fails for many classes of anisotropic tori in general. In these cases, the number of inequivalent types of the form (G y , λ) can grow arbitrarily large as the rank of G increases; see Example 8.4. This paper is organized as follows: We establish our notation in Sect. 2, including particularly of the building B(G) and of Moy–Prasad filtration subgroups, and recall the definition of the essentially tame toral supercuspidal representations we study in Sect. 3. In Sect. 4, we prove a proposition about the inequivalence of toral supercuspidal representations under certain twists, based on [8, Cor 6.10] (recalled here as Lemma 4.1) and some results of F. Murnaghan in [17]. In Sect. 5, we establish a key

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result relating Moy-Prasad filtration subgroups to stabilizers of subsets of an apartment, generalizing a proposition in [20], and we define the notion of the simplicial radius of a bounded subset of B(G). In Sect. 6, we recall the Mackey decomposition of π|G y for a vertex y and prove two general results about components that can contain types. Our main results on unicity are proven in Sect. 7. We discuss the success and failure of strong unicity and provide examples where it fails, in Sect. 8. Several interesting problems remain open. The hypothesis on the depth of the character of T given in the statement of Theorem 7.3 arises as a result of our method of proof. There is no such restriction for the unicity theorems on G = GLn or on G = SLn , where a different argument (particular to type A) was employed by V. Pa˘sk¯unas to address the small-depth case. Finally, not all essentially tame supercuspidal representations arise from toral data; the consideration of general twisted Levi sequences is necessary to completely resolve the unicity conjecture in these cases.

2 Notation Let F be a non-Archimedean local field of residual characteristic p, with integer ring O and maximal ideal p. We normalize the valuation ν on F so that ν(F × ) = Z; if E is an extension field of F, then we also denote by ν the unique extension of this valuation to E. Fix an additive character  of F that is nontrivial on O but trivial on p. If H ⊂ G are groups and g ∈ G, let g H = {ghg −1 | h ∈ H } and for any representation τ of H let g τ denote the corresponding representation of g H . If σ is a representation of G, we write Res H σ for its restriction to H . Let G be a semisimple simply connected linear algebraic group defined over F and let G = G(F). Let B(G) = B(G, F) denote the reduced building of G over F; since G is semisimple, it coincides with the enlarged building. Since G is simply connected, the stabilizer of a point x in the building, G x , coincides with the parahoric subgroup G x,0 associated with the facet containing x. Both hypotheses together imply that the maximal compact open subgroups of G are exactly the maximal parahoric subgroups of G, that is, G y for each vertex y of the building. Let S be a maximal F-split torus in G defined over F. Fix a maximal unramified extension F un of F, and let S be a maximal F un -split torus of G defined over F and containing S. Let Z be the centralizer of S in G, which is a maximal torus of G defined over F. Denote by  = (G, S), the root system of G relative to S and by A = A(S, F) the apartment of B(G, F) corresponding to S. Let  be the set of affine roots, corresponding to our choice of valuation ν; these are functions on A. Denote the root subgroup of G corresponding to α ∈  by Gα . For α ∈ , we set G α = Gα (F); this group admits a filtration by compact open subgroups G ψ , as ψ runs over the affine roots with gradient ψ˙ = α. For more details, see [5, §4] or the careful exposition in [6, §2]. Let S = S(F), S  = S (F) and Z = Z(F). Let Z b be the maximal bounded subgroup of Z . Recall that any torus T = T(F) admits a natural filtration by subgroups

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Tr for r ≥ 0, and its Lie algebra t is filtered by lattices tr for r ∈ R. In particular, Z admits a natural filtration by subgroups Z r for r ≥ 0, with Z 0 ⊆ Z b . For any x ∈ A and r ≥ 0, Moy and Prasad [16] defined G α,x,r = G ψ | ψ(x) ≥ r, ψ˙ = α and thus filtration subgroups G x,r = Z r , G α,x,r | α ∈  of G x . We set G x,r + = ∪s>r G x,s . They similarly defined lattices gx,r in g = Lie(G)(F) and g∗x,r in g∗ , indexed by r ∈ R. Conjugation by G allows us to extend these definitions to any x ∈ B(G). We say that a group G (or its set of F-points G  = G (F)) is a twisted Levi subgroup of G if there is an extension E of F such that G is an E-Levi subgroup of G defined over F. We say G is tamely ramified if G (and thus G) splits over a tamely ramified extension. Suppose now that G is a (tamely ramified) twisted Levi subgroup of G and let T be a maximal torus of G ; let E be a tamely ramified splitting field of T over F. Let g denote the F-points of Lie(G ), and denote by z ∗ the F-points of the dual of the center of Lie(G ). Let r > 0. Following [8, Definition 3.7], an element X ∗ ∈ z ∗−r is called G-generic of depth -r if it satisfies the conditions GE1 and GE2 of [24, §8]. By [24, Lemma 8.1], if p is not a torsion prime for the root datum (in the sense of [22]) of (G, T), then these conditions reduce to the requirement that ν(X ∗ , Ha ) = −r , for each a ∈ (G, T) \ (G , T), where Ha ∈ Lie(G)(E) is the coroot associated with a. Fix also a Moy–Prasad isomorphism [8, §2.6] e : G x,r /G x,r + → gx,r /gx,r + . A character φ of G  of depth r is said to be realized by an element X ∗ ∈ z ∗−r on G x,r if for every Y ∈ gx,r , φ(e(Y + gx,r + )) = (X ∗ , Y ). By [8, Definition 3.9], φ is G-generic (relative to x) of depth r > 0 if φ is trivial on G x,r + , nontrivial on G x,r , and is realized on G x,r by an element X ∗ ∈ z ∗−r that is G-generic of depth −r . A particular consequence of the G-genericity of a character φ of G  is that for any g ∈ G, we have that g φ and φ coincide on G x,r if and only if g ∈ G  [24, Lemma 8.3].

3 The Construction of Toral Supercuspidal Representations Fix G semisimple and simply connected, and retain all the notations above. Following [24, §3] and [8, §3.1], we define a positive-depth generic toral supercuspidal datum where φ), of length d of G (hereafter: toral supercuspidal datum) to be a pair (G,

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= (G 0  G 1  · · ·  G d = G), for some d ≥ 1, is a sequence of tamely ram• G ified twisted Levi subgroups, such that in particular G 0 = T is an anisotropic maximal torus in G; • φ = (φ0 , φ1 , . . . , φd−1 ) where φi is a G i+1 -generic character of G i of depth ri , and these real numbers satisfy 0 < r0 < r1 < · · · < rd−1 . For convenience, we write si = ri /2 for each i = 0, . . . , d − 1. Given a tamely ramified twisted Levi subgroup G of G that is defined over F, one may embed B(G , F) into B(G, F) with canonical image. Since T is anisotropic, the image of B(T ) in B(G) consists of a single point x, and it is this point relative to which the characters φi are G i+1 -generic. Thus, a toral supercuspidal datum implies the datum d, r , s , x), and we will take this extra data for granted where there is no φ, (G, possibility of confusion. be a toral supercuspidal datum. Set K0 = G 0 = T . For each φ) Let  = (G, x 1 ≤ i ≤ d, define Ki := G 0x G 1x,s0 . . . G ix,si−1 ⊆ T G x,s0 ⊆ G x . The groups in these products have large pairwise intersections, so we next define subgroups Ji and Ji+ which will have the property that Ki Ji+1 = Ki+1 and Ki ∩ i Ji+1 = Ki ∩ Ji+1 + = G x,ri . Let E be a tamely ramified Galois extension of F over which T splits. For each 0 ≤ i ≤ d, let i = (Gi , T) be the corresponding root system. For 0 ≤ i < d, let J(E)i+1 be the group generated by T(E)ri together with the root subgroups Ga (E)x,ri for a ∈ i and Gb (E)x,si for b ∈ i+1 \ i . Similarly, let J(E)i+1 + = T(E)ri , Ga (E)x,ri , Gb (E)x,si + | a ∈ i , b ∈ i+1 \ i . These subgroups of Gi+1 (E) are invariant under Gal(E/F), and we set Ji+1 = i+1 i+1 . J(E)i+1 ∩ G i+1 , Ji+1 + = J(E)+ ∩ G Next for each 0 ≤ i < d, the subdatum ((G i , G i+1 ), φi ) is used to construct a representation φi of Ki+1 . The first step is to extend the restriction of φi to G ix,ri i+1 trivially to a character φˆ i of J+i+1 [24, §4]. Next, when Ji+1 , define a character + =J  i+1 φi of K by φi (k j) = φi (k)φˆ i ( j),

(3.1)

i+1 , then instead one for each k ∈ Ki , j ∈ Ji+1 . On the other hand, when Ji+1 + = J i i+1 ˆ uses φi to construct a Heisenberg–Weil representation ω of K  J , and then define the representation φi on k ∈ Ki , j ∈ Ji+1 by

φi (k j) = φi (k)ω(k, j).

(3.2)

Now set K = Kd . There is a well-defined inflation process extending each φi trivially across the remaining subgroups to give a representation of K, which we

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denote κi = inf KKi+1 φi [8, §3.4]. Note that this representation κi is independent of any other characters φ j , j = i, in the datum [8, Proposition 3.26]. Putting these together, we obtain a representation κ() = κ0 ⊗ · · · ⊗ κd−1 of K such that π() = c-IndKG κ() is an irreducible supercuspidal representation of G of depth r = rd−1 [24, Theorem 15.1]. It then follows that (K, κ()) is a type for π().

4 Results on Equivalence We begin by noting when two toral supercuspidal data give rise to equivalent representations, from [8, Cor 6.10]. ˙ be two toral ˙ φ) and  φ) ˙ = (G, Lemma 4.1 [Hakim-Murnaghan] Let  = (G, ∼ ˙ supercuspidal data. Then π() = π() if and only if there exists some g ∈ G such that G 0 = g G˙ 0 and ResG 0 (φ0 φ1 . . . φd−1 ) = ResG 0 g (φ˙ 0 φ˙ 1 . . . φ˙ d−1 ). In this case, G i = g G˙ i for all i ≥ 0. is a toral supercuspidal datum of length d with T = G 0 . φ) Now suppose  = (G, Let ξ be a character of T of depth strictly less than r0 . The G 1 -genericity of the character φ0 of G 0 of depth r0 depends only on ResG 0x,r φ. Since ξφ0 and φ0 coincide 0 with G 0x,r0 , we conclude that ξφ0 is also a G 1 -generic character of G 0 of depth r0 . Thus (ξφ0 , φ1 , . . . , φd−1 )) ξ = (G, is another toral supercuspidal datum. Lemma 4.2 The character ξ inflates to a character ξ of K such that κ(ξ ) = ξ κ(). Proof Since ξ is a character of T = G 0 = K0 of depth less than r0 , it is trivial on G 0x,r0 . Since K1 = K0 J1 and K0 ∩ J1 = G 0x,r0 , ξ may be uniquely inflated to a 1 character ξ  = inf KK0 ξ of K1 that is trivial on J1 . In particular, for any k ∈ K0 , j ∈ J1 ,  we have ξ (k j) = ξ(k). Since φ0 and ξφ0 coincide on G 0x,r0 , we may denote the extension to J1+ of their common restriction by φˆ 0 and, if applicable, the corresponding Heisenberg–Weil representation of K0  J1 by ω. Using now (3.1) and (3.2), as applicable, to construct (ξφ0 ) and φ0 , we deduce that in both cases (ξφ0 ) = ξ  φ0 . Set ξ = inf KK1 ξ  ; then inf KK1 ξ  φ0 = ξκ0 . Since for i > 0, κi depends only on the character φi , we conclude that ξ factors out of the tensor product to yield κ(ξ ) = ξκ(), as desired. 

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be a toral supercuspidal datum and suppose ξ is a φ) Proposition 4.3 Let  = (G, (ξφ0 , φ1 , . . . , φd−1 )), character of T = G 0 of depth less than that of φ0 . If ξ = (G, then π(ξ ) is an irreducible supercuspidal representation of G and it is equivalent to π() if and only if ξ is trivial. Proof Suppose that π() ∼ = π(ξ ). In the setting of Lemma 4.1, this means there is some g ∈ NG (T ) such that ResT (φ0 φ1 . . . φd−1 ) = ResT g (ξφ0 φ1 . . . φd−1 ). Then g (ξφ0 )g φ1 . . . g φd−1 is a refactorization of φ0 . . . φd−1 , in [17, Definition the sense of 5.3], and so by [17, Proposition 5.4], for each i > 0, dj=i φ j and dj=i g φ j coincide with G ix,ri−1 + . The genericity of the characters φ j recursively implies, as in the proof of [17, Proposition 5.6], that g ∈ G 1 , and so g φi = φi for all i ≥ 1. Therefore, our equality reduces to ResTr0 φ0 = ResTr0 g (ξφ0 ) = ResTr0 g φ0 , whence g ∈ T by genericity. Returning to the first equality, we conclude that ξ is the trivial character of T . 

5 Stabilizers Let S, Z , A = A(S),  be as in Sect. 2. Given a subset  ⊆ A, the subgroup of G that fixes  pointwise is generated by Z b (the maximal bounded subgroup of the maximal torus Z ) and those G ψ satisfying ψ(z) ≥ 0 for each z ∈  [4, §6.4]. For x, y ∈ A, we write [x, y] for the geodesic from x to y; then G [x,y] = G x ∩ G y . The following result is a generalization of [20, Prop 3.3], and it allows us to relate Moy–Prasad filtration subgroups at x to stabilizers of its neighborhoods in any apartment A containing x. Note that A is the affine space under X ∗ (S) ⊗Z R. Given two points x, z ∈ A, we identify z − x with a vector in X ∗ (S) ⊗Z R, and then for each α ∈ , α(z − x) is a well-defined real number. Lemma 5.1 Let A = A(S, F) be an apartment in B(G, F) containing x and let  = (G, S) be the corresponding root system. Let s ≥ 0 and define A (x, s) = {z ∈ A | ∀α ∈ , α(z − x) ≤ s}. Then G x,s ⊆ G A (x,s) = G A (x,s) , where A (x, s) denotes the simplicial closure of A (x, s) in A. Proof That G A (x,s) = G A (x,s) follows from [4, 2.4.13], since G is semisimple and simply connected. Let z ∈ A (x, s). For each affine root ψ such that ψ(x) ≥ s, let α denote its gradient. Then ψ(x) − ψ(z) = α(x − z) ≤ s so ψ(z) ≥ ψ(x) − s ≥ 0. Thus for all affine roots ψ, if G ψ ⊂ G x,s then G ψ ⊂ G A (x,s) . Since also Z s ⊆ Z b , we conclude G x,s ⊆ G A (x,s) . More generally, we may define (x, s) =

 Ax

A (x, s)

On the Unicity of Types for Toral Supercuspidal Representations

183

which is the union over all apartments of B(G) containing x. By local compactness, this is reduced to a finite union. Let (x, s) denote its simplicial closure. Definition 5.2 Let  ⊂ B(G) be a bounded convex set and suppose x ∈ . For each apartment A = A(S, F) containing x define r (, A, x) = sup{α(z − x) | z ∈  ∩ A, α ∈ (G, S)}. Then, the simplicial radius of  with respect to x is defined to be c(, x) = sup{r (, A, x) | x ∈ A}. As one motivating example, note that the simplicial radius of (x, s) with respect to x is c((x, s), x) = s. For another, letting {x} denote the simplicial closure of {x} in B(G), we have c({x}, x) < 1 since {x} is constrained between adjacent affine root hyperplanes in any apartment containing x. Note that in each of these examples, we have the equality c(, x) = r (, A, x) for each apartment A containing x. One can be slightly more precise about c({x}, x) when x arises as the point identified with B(T ) from a datum . In this case, x is an optimal point of B(G), in the sense of [16, §6.1], whence the family of such values cx = c({x}, x) could be computed for any G. For example, if G = SLn (F), then the optimal points are among the barycentres of the facets, whence if x lies in a k-dimensional facet F then cx = 1 − k1 .

6 Mackey Decomposition and Strategies for Identifying Types Let y be a vertex of B(G) and G y the corresponding maximal compact open subgroup be a toral supercuspidal datum with T = G 0 and let π = π(). φ) of G. Let  = (G, We are interested in the irreducible representations of G y occurring in ResG y π. Mackey theory gives a decomposition ResG y π ∼ =



G

Ind G yy ∩g K g κ,

g∈G y \G/K G

where each Mackey component τ (g) := Ind G yy ∩g K g κ is a finite-dimensional representation of G y . Note that since G y = NG (G y ), the Mackey components are parametrized by a subset of the G-orbit of the vertex y in B(G). We emphasize that these Mackey components are not, in general, irreducible; a first strategy for identifying those that contain types is the following.

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Proposition 6.1 If K ⊆ G g−1 y then (G y , τ (g)) is a type for π. Proof Consider instead the twisted Mackey component g −1

G g−1 y −1 G τ (g) = g (Ind G yy ∩g K g κ) ∼ = Ind G −1 ∩K κ g

y

G g−1 y −1 which is a representation of G g−1 y . If K ⊆ G g−1 y , then g τ (g) ∼ = IndK κ. By the  transitivity of compact induction, c-Ind G G g−1 y τ = π, whence by Frobenius reciprocity −1

(G g−1 y , g τ (g)) is a type for π, and the result follows. On the other hand, the key strategy to discern Mackey components that cannot contain a type of π is the following. Theorem 6.2 Given y, , g as above, suppose that there exists a nontrivial character ξ of T of depth less than r0 such that its inflation ξ to K is trivial on K ∩ G g−1 y . Then no irreducible subrepresentation of the Mackey component G

Ind G yy ∩g K g κ() is a type. −1

Proof The twisted Mackey component g τ (g) depends only ResG g−1 y ∩K κ(). By Lemma 4.2, κ(ξ ) = ξκ(); by hypothesis ResG g−1 y ∩K ξ is trivial. Thus, G

Ind G yy ∩g K g κ() is a common component of both π() and π(ξ ). By Proposition 4.3, π()  π(ξ ). Since these are inequivalent irreducible supercuspidal representations, they have distinct inertial support, and thus, no irreducible representation of G G y occurring in Ind G yy ∩g K g κ() can be a type.

7 Unicity Results We first identify the obvious types occurring in ResG y π. Lemma 7.1 Suppose y is a vertex of the facet containing x. Then (G y , τ (1)) is a type. Proof Let F be the facet containing x. We have K ⊆ G x . Since y ∈ F by hypothesis, G x ⊆ G y . This implies that K ⊆ G y , whence the first statement by Proposition 6.1. Now, continuing with the notation of the previous section, we identify g ∈ G for which the associated Mackey components do not contain types. Proposition 7.2 Let g ∈ G. Suppose that there exists z ∈ [x, g −1 y] ∩ (x, s0 ) such that T  G z . Then no irreducible subrepresentation of τ (g) is a type.

On the Unicity of Types for Toral Supercuspidal Representations

185

Proof Let A be an apartment of B(G) containing x and g −1 y; then A contains the geodesic [x, g −1 y]. Let z ∈ [x, g −1 y] ∩ A (x, s0 ) as in the proposition. Since z is on the line [x, g −1 y], whose pointwise stabilizer is G x ∩ G g−1 y , we have that G x ∩ G g−1 y ⊆ G z . At the same time, Lemma 5.1 implies that G x,s0 ⊆ G A (x,s0 ) ⊆ G z . Thus K ∩ G g−1 y ⊆ T G x,s0 ∩ G x ∩ G g−1 y ⊆ T G x,s0 ∩ G z = (T ∩ G z )G x,s0 . Noting that Ts0 ⊆ T ∩ G z , we deduce that any character of T that is trivial on T ∩ G z has depth strictly less than s0 . By hypothesis T ∩ G z  T ; so let ξ be a nontrivial character of T that is trivial on T ∩ G z . Since its depth is less than s0 , its inflation ξ is trivial on (T ∩ G z )G x,s0 . We deduce that ξ satisfies the hypotheses of Theorem 6.2, whence the proposition. Let B T = {z ∈ B(G) | T ⊆ G z } be the set of fixed points of T acting on B(G). This is a convex subset of B(G) containing x, and it is compact since T is an anisotropic maximal torus. Let cT denote the simplicial radius of B T with respect to x, as per Definition 5.2. Now let A be an apartment containing x. The hypothesis of Proposition 7.2 will be satisfied for all g −1 y ∈ A if AT = A ∩ B T is contained in the interior of the simplicial closure A (x, s0 ) of A (x, s0 ). In particular this holds if s0 > cT ≥ r (B T , A, x). Putting these geometric ideas together yields our main theorem. Theorem 7.3 Let T be a tamely ramified anisotropic maximal torus of G and let be any toral supercuspidal datum such that G 0 = T and φ) s0 > cT . Let  = (G, such that the depth of φ0 is at least 2s0 . Then π() satisfies the conjecture of unicity of types relative to any maximal compact open subgroup of G. If moreover B T is the closure of a single facet of B(G) then π() has the property of strong unicity. Proof Let g ∈ G y \G/K. Suppose first that g −1 y ∈ B T , so that T ⊆ G g−1 y . Since s0 > cT , we have g −1 y ∈ (x, s0 ) so G x,s0 ⊆ G g−1 y as well. Thus K ⊆ T G x,s0 ⊆ G g−1 y and by the same argument as Lemma 7.1, we conclude that (G y , τ (g)) is a type for π. / B T , so that T ⊂ G g−1 y . Since s0 > cT , (x, s0 ) conNow suppose that g −1 y ∈ tains an open neighborhood of B T , so the line [x, g −1 y] must meet (x, s0 ) \ B T in at least one point z. Using now Proposition 7.2, we conclude that the corresponding Mackey component contains no types for π(). Finally, we note that if B T = F for a facet F ⊂ B(G), then since G is semisimple simply connected, each orbit of a vertex y in B(G) meets B T at most once. By the above arguments, this implies strong unicity. By [23, 3.6.1], if T is an anisotropic maximal torus which splits over an unramified extension, then B T consists of a single vertex, namely {x}, whence cT = 0. Since all our toral supercuspidal representations have positive depth, we have the following immediate corollary.

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Corollary 7.4 Let T be an unramified anisotropic maximal torus of G. Let  = be any toral supercuspidal datum such that G 0 = T . Then π() satisfies φ) (G, strong unicity of types relative to any maximal compact open subgroup of G.

8 The Inequivalence of Unicity and Strong Unicity Recall that strong unicity of types is the statement that Res K π should contain at most one type, for each choice of maximal compact open subgroup K . We have the following converse to the strong unicity statement in Theorem 7.3. Note that this result is without a condition on depth. Lemma 8.1 Let T be a tamely ramified anisotropic maximal torus. If B T contains two distinct but G-conjugate vertices y and y  of alcoves whose closure contain x, such that G 0 = T , the strong φ) then for any toral supercuspidal datum  = (G, unicity property fails to hold for π(), that is, there exist at least two nonisomorphic types in ResG y π(). Proof Let x ∈ B(T ) ⊆ B(G) and set π = π(). Let y = y  be G-conjugate vertices of two distinct alcoves C, C  of B(G) such that x ∈ C ∩ C  . Then for any s0 > 0, we have that y, y  ∈ (x, s0 ). Let g ∈ G be such that y  = g −1 y. If both y and g −1 y lie in B T , then by the proof of Theorem 7.3, we conclude that both (G y , τ (1)) and (G y , τ (g)) are types for π occurring in ResG y π. Note that although these types are induced from the G-conjugate types (K, κ) and (g K, g κ), they are not themselves G-conjugate since g ∈ / NG (G y ) = G y . Neither are these types isomorphic, since G τ (1) implies π∼ c-Ind = Gy C∼ = Hom G y (τ (1), ResG y π) = Hom G (π, π) ∼ by Frobenius reciprocity. We next prove the existence of pairs (G, T ) satisfying the hypotheses of Lemma 8.1 with an example such that G has rank 2. We thank Jeff Adler for providing this instructive example of a torus that stabilizes more than the closure of a single facet.  0 I Example 8.2 Consider G = Sp4 (F), given in matrix form relative to J = −I 0 ∈ M4 (F) as the set G = {g ∈ GL4 (F) | t g J g = J } where t g denotes the transpose. Note that G contains a generalized Levi subgroup G  isomorphic to SL2 (F) × SL2 (F), defined by the long roots. Let be a uniformizer of F. For i = 1, 2, we choose an anisotropic torus Ti of SL2 (F) that isomorphic to the norm-one elements of some quadratic ramified √ extension E i = F[ γi ] with ν(γi ) = 0. Let T = T1 × T2 be the corresponding anisotropic torus of G  ; explicitly we embed T into G as the subgroup

On the Unicity of Types for Toral Supercuspidal Representations

⎧⎡ a 0 ⎪ ⎪ ⎨⎢ 0 c T = ⎢ ⎣bγ1 0 ⎪ ⎪ ⎩ 0 dγ2

b 0 a 0

187

⎤   0  a, b, c, d ∈ O, ⎥ d ⎥  2 a − b2 γ1 = 1, 0 ⎦  2 2  c − d γ2 = 1 c

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

For each i, the point in the building B(SL2 , F) corresponding to Ti is the midpoint of the fundamental alcove. Thus, the point x representing B(T ) ⊂ B(G) is the midpoint of the diagonal facet F in the closure of the fundamental alcove C in the standard apartment A of B(G), as illustrated in Fig. 1. Evidently F ⊆ AT . Let us now prove that AT = F. Let y denote the (non-special) vertex of C opposite F, and w ∈ G a reflection in the wall of A containing F (viewed as a representative of the corresponding element of the affine Weyl group). Adopting the convention that a matrix ring stands for its intersection with Sp4 (F), we compute directly that ⎡

O O O O

⎢p O GC = ⎢ ⎣p p p p

⎤

⎡

O O⎥

O O

O

p

O O

⎤

⎥ ⎢ ⎥ and G wC = ⎢O O O O⎥ , ⎣ p p O O⎦ p⎦ O p p pO

(8.3)

each of which contains T as a subgroup. It, therefore, follows that {y, wy} ⊂ C ∪ wC ⊆ AT , even though neither y nor wy lie in F. Applying now Lemma 8.1, we deduce that strong unicity fails for any supercuspidal representation π() constructed   from G 0 = T . Remark 8.3 In the setting of the preceding example, we can conclude slightly more. First note that if z ∈ A and T ⊆ G z , then for each positive long root α, we must have

wy wC x

F C y

Fig. 1 A portion of the standard apartment A of Sp4 (F), identifying an alcove C , the point x that is the image of B(T ) in B(G), the facet F containing x and the vertex y of C not in F . The images of y and of C under a reflection w in the Weyl group are also indicated. The subset AT of A fixed 1 by T is the closed region shaded in gray. For sake of example, the set A (x, s0 ), with s0 = 10 , is T indicated with a dotted line; its closure is A

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0 ≤ α(z) ≤ 1. It follows that AT = C ∪ wC, as indicated in Fig. 1. We claim that in fact B T = AT . Since G x ∩ G y = G C , the orbit of y is parametrized by the set G x /G C  GL2 (O)/I, where I denotes its Iwahori subgroup. A set of representatives is ⎧ ⎡ 1a ⎪ ⎪ ⎨ ⎢0 1 a=⎢ ⎣ 1 ⎪ ⎪ ⎩ −a

⎤

⎡

0 1 ⎥ ⎢−1 0 ⎥,w = ⎢ ⎣ 0⎦ 0 1 −1

⎫ ⎤   ⎪ ⎪  ⎬ ⎥  ⎥  a ∈ O/p .  1⎦  ⎪ ⎪ ⎭ 0 

One can verify directly that for each a ∈ O× , a T  G y . Thus since the fixed point set B T is closed and convex but contains no vertices of chambers adjacent to x outside of A, we conclude that B T = C ∪ wC = AT , which is contained entirely in the standard apartment. Therefore, we can compute the simplicial radius of B T with respect to x, yielding cT = 21 . We had seen that τ (1) and τ (w) gave types. For a ∈ O× , since the choice z = −1 a y satisfies the hypotheses of Proposition 7.2 for any s0 > 0, we deduce that the corresponding Mackey components τ (a) do not contain types. For the remaining Mackey components, g −1 y is not in the closure of an alcove adjacent to x, and Theorem 7.3 implies that if s0 > cT = 21 , any corresponding toral supercuspidal representation contains exactly the two types on G y identified above. Example 8.2 generalizes immediately, to give families of supercuspidal representations for which the number of distinct types supported on a given non-special maximal compact subgroup grows exponentially with the rank of G. Example 8.4 Let n ≥ 2 and consider the subgroup G  ∼ = SL2 (F)n of G = Sp2n (F) generated by the root subgroups corresponding to long roots. To allow us to be explicit, let the roots of Sp2n with respect to the diagonal torus S be the set  = {εi ± ε j , ±2εi | 1 ≤ i = j ≤ n}, with simple system  = {εi − εi+1 , 2εn | 1 ≤ i < n}. With respect to the basis {e1 , . . . , en } of X ∗ (S) ⊗Z R (whose affine space is A), dual to {ε1 , . . . , εn }, the vertices of the fundamental alcove C are vi = ij=1 21 e j , for 0 ≤ i ≤ n. Let W = WG denote the Weyl group of G relative to S. In SL2 (F), there are two conjugacy classes of unramified anisotropic tori, attached to the distinct conjugacy classes of vertices in B(SL2 , F). There are between 2 and 4 conjugacy classes of ramified anisotropic tori, attached to the midpoint of facets; see [19], for example. With respect to the coordinates above, the roots of each SL2 (F) subgroup are ±2εi , and thus up to conjugacy we can arrange that the vertices have ei -coordinates in {0, 21 }, whereas the midpoints have ei -coordinate 14 . Let each of T1 , . . . , Tn represent an anisotropic torus of SL2 (F), ordered so that: for 1 ≤ i ≤ m, Ti is unramified and attached to 21 ; for m + 1 ≤ i ≤ m + , Ti is ramified and attached to 14 ; and for i > m + , Ti is again unramified, but attached to 0. Then T = T1 × · · · × Tn embeds as an anisotropic maximal torus of G, and {x} = B(T ) ⊆ B(G) has coordinates

On the Unicity of Types for Toral Supercuspidal Representations

x=

m  1 j=1

2

ej +

189

m+ 

1 ej. 4 j=m+1

This is the midpoint ofthe 1-dimensional facet F whose closure is the geodesic  1 [vm , vm+ ]. Let  = { mj=1 21 e j + m+ j=m+1 a j e j | 0 ≤ a j ≤ 2 }; then F ⊆  and T one can verify directly that  = {z ∈ A | T ⊆ G z } = A using the argument of Remark 8.3. Note that since every maximal facet of  has x in its closure,  ⊆ A (x, s) for any s > 0. Note that F =  if and only if  > 1. In this case, let W  = {w ∈ W | w ⊆ } and let W = {w ∈ W | ∀z ∈ , wz = z}. The elements of W (T ) := W  /W permute the vertices of . Let G = Sp2m × SL2 × Sp2n−2m−2 ⊆ G; then we can identify W (T ) with WG /WG . The vertices of  ∩ C are vm , . . . , vm+ . The vertices of W (T ), but the orbit of y = vm+t under vm and vm+ are fixed by each element  W (T ), for 0 < t < , contains t distinct vertices of . Let  be any toral supercuspidal datum such that G 0 = T , and π = π(). By the argument of the  proof of Lemma 8.1, for each vertex y = vm+t with 0 < t < ,   ResG y π contains t > 1 inequivalent types. Remark 8.5 The quadratic tori of Examples 8.2 and 8.4 are the smallest of a broad class of tori to which the preceding arguments apply. For example, using L. Morris’s classification of anisotropic maximal principal tori of Sp2n (F) in [15] one can explicitly construct products of tori of arbitrary rank, obtaining analogous results. Similar constructive arguments may be made for groups of type Bn and Dn . The question of determining the fixed points of an anisotropic maximal torus T acting on B(G) was partially addressed by F. Hurst in his thesis [9]. Under the hypotheses that G is simple, connected and split over F (but not of type E 7 or E 8 ), that T splits over a purely tamely ramified cyclic Galois extension, and some mild assumptions on F, Hurst proves that B T0+ = {x} if and only if T is a Coxeter torus (in which case x lies in an alcove) [9, Satz 13.14]. For other T , he shows that x lies on the wall of an alcove. Then it follows that B T0+ contains the closure of the G x -orbit of this alcove in B(G), since T0+ ⊆ G x,0+ ⊆ G C for all alcoves C adjacent to x. (This is deduced via more explicit arguments in [9, Satz 13.15].) F. Hurst computes a range of examples in [9, §13], including of particular interest one in type F4 (labeled 6 p 6 p ) where T0+ fixes pointwise an alcove that is not adjacent to x. Hence, this example may yield a torus T with a more interesting set of fixed points B T and consequently may be an unusual example to explore. Acknowledgements The second author warmly thanks Anne-Marie Aubert, Manish Mishra, Alan Roche and Steven Spallone for the invitation to the excellent conference Representation theory of p-adic groups at IISER Pune, India. The stimulating environment of the workshop contributed significantly to this article; in particular, fellow participant Jeff Adler provided invaluable insight into tori and buildings, and he offered up the torus of Example 8.2.

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References 1. J.D. Adler, Refined anisotropic K -types and supercuspidal representations. Pacific J. Math. 185(1), 1–32 (1998) 2. J. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a local field, ed. by P. Deligne (Travaux en Cours, Hermann, Paris, 1984), pp. 1–32 3. C. Breuil, A. Mézard, Multiplicités modulaires et représentations de GL2 (Z p ) et de Gal(Q p /Q p ) en l = p, Duke Math. J. 115(2), 205–310 (2002). With an appendix by Guy Henniart 4. F. Bruhat, J. Tits, Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972) 5. F. Bruhat, J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math. 60, 197–376 (1984) 6. J. Fintzen, On the Moy-Prasad filtration. Preprint arXiv:1511.00726v3 [math.RT] (2017) 7. J. Fintzen, Types for tame p-adic groups. Preprint arXiv:1810.04198 [math.TR] (2018) 8. J. Hakim, F. Murnaghan, Distinguished tame supercuspidal representations. Int. Math. Res. Pap. IMRP 2, Art. ID rpn005, 166 (2008) 9. F. Hurst, Primitive Tori in einfachen spaltenden Gruppen und ihre Fixpunkte im Bruhat-Tits Gebäude, Inaugural-Dissertation zur Erlangung der Doktorwürde der NaturwissenschaftlichMathematischen Gesamtfakultät der Ruprecht-Karls-Universität Heidelberg (2005), 164pp 10. T. Kaletha, Regular supercuspidal representations, Preprint arXiv:1602.03144v2 (2016) 11. J.-L. Kim, Supercuspidal representations: an exhaustion theorem. J. Amer. Math. Soc. 20(2), 273–320 (electronic) (2007) 12. P. Latham, Unicity of types for supercuspidal representations of p-adic SL2 . J. Number Theor. 162, 376–390 (2016) 13. P. Latham, The unicity of types for depth-zero supercuspidal representations. Represent. Theor. 21, 590–610 (2017) 14. P. Latham, On the unicity of types in special linear groups. Manuscripta Math. 157(3–4), 445–465 (2018) 15. L. Morris, Some tamely ramified supercuspidal representations of symplectic groups. Proc. London Math. Soc. 63(3), 519–551 (1991) 16. A. Moy, G. Prasad, Unrefined minimal K -types for p-adic groups. Invent. Math. 116(1–3), 393–408 (1994) 17. F. Murnaghan, Parametrization of tame supercuspidal representations, in On certain Lfunctions. Clay Mathematical Proceedings 13, American Mathematical Society, Providence, RI, 2011, pp. 439–469 18. F. Murnaghan, Distinguished positive regular representations. Bull. Iranian Math. Soc. 43(4), 291–311 (2017) 19. M. Nevins, Branching rules for supercuspidal representations of S L 2 (k), for k a p-adic field. J. Algebra 377, 204–231 (2013) 20. M. Nevins, On branching rules of depth-zero representations. J. Algebra 408, 1–27 (2014) 21. V. Paskunas, Unicity of types for supercuspidal representations of GL N . Proc. Lond. Math. Soc. 91(3), 623–654 (2005) 22. T.A. Springer, Reductive groups, automorphic forms, representations and L-functions, in (Proceedings Symposium Pure Mathematical, Oregon State University, Corvallis, Ore, 1977), Part 1 (XXXIII, American Mathematical Society, Providence, R.I, Proceeding Symposium Pure Mathematical, 1979), pp. 3–27 23. J. Tits, Reductive groups over local fields, automorphic forms, representations and L-functions (Oregon State University, Corvallis, Ore, 1977), Part 1, Proceedings Symposium Pure Mathematical, XXXIII, American Mathematical Society, Providence, R.I., 1979, pp. 29–69 24. J.-K. Yu, Construction of tame supercuspidal representations. J. Amer. Math. Soc. 14(3), 579– 622 (2001) (electronic)

Local Gamma Factors, Converse Theorems and Related Problems Chufeng Nien

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Gamma Factors in the p-Adic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Local Converse Problem and Jacquet’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Local Converse Problem Over p-Adic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Local Converse Problem Over Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Sharpness on the Bound [ 2n ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Explicit Computation of Gamma Factors Over Finite Fields . . . . . . . . . . . . . . . . . . . . . . . 3.1 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 n × 1 Gamma Factor and Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Some n × 1 Local Converse Theorem for GLn (Fp ) . . . . . . . . . . . . . . . . . . . . . . . . 4 Gamma Factors for Level Zero Cuspidals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Converse Theorems and Distinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 p-Adic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Finite Field Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192 192 193 194 194 195 196 197 197 198 199 201 202 202 204 204

Abstract This paper reviews the development of Local Converse Theorems and related results on distinction for representations of general linear groups over finite and p-adic fields. Keywords Generic representation · Whittaker models · Gamma factors · Local Converse Theorem · Distinguished representation · Gauss sum 2000 Mathematics Subject Classification. Primary 20C33 · Secondary 11L05

C. Nien (B) Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_7

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1 Introduction We will give a survey on local gamma factors, Local Converse Theorems and related problems. For a more complete picture on these subjects, please see some other articles. For example, the references [9, 10] contain nice and substantial introductions.

1.1 Notation Let Fq be the finite field of q elements and F be a p-adic field. We will write K for such a field, finite or p-adic, depending on context. Denote by Bn the standard Borel subgroup of upper-triangular matrices in GLn . Let Un be its unipotent radical and write An for the Levi subgroup of Bn consisting of diagonal matrices. Denote by Zn the center of GLn . Let ψ be a fixed nontrivial additive character of K. The standard non-degenerate character ψn is given by ψn (u) = ψ(

n−1 

ui,i+1 ), for u = (ui,j ) ∈ Un .

i=1

A character ψ  of Un is called non-degenerate if ψ  (u) = ψ(

n−1 

ai ui,i+1 ), for u = (ui,j ) ∈ Un ,

i=1

with ai ∈ K × for all i. We call an irreducible representation π of GLn ψ  -generic if  n dim HomGLn (π, IndGL Un ψ )  = 0

for ψ  non-degenerate. By transitivity of the conjugation action of An on the set of generic characters, if π is ψ  -generic, then it is also ψ  -generic for any other non-degenerate character ψ  of Un . Hence, we often say “generic” rather than “ψ  generic.” n For π irreducible, the image of π under an embedding in IndGL Un ψn is unique. Theorem 1.1 [Uniqueness of Whittaker models] n dim HomUn (π |Un , ψn ) = dim HomGLn (π, IndGL Un ψn ) ≤ 1,

for any irreducible representation π of GLn (K). When π is generic, the above Hom-space has dimension one. Moreover, any irreducible cuspidal representation of GLn (K) is generic.

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Let ψn ∈ HomUn (π |Un , ψn ) be nonzero. Such an element is called a Whittaker functional of π . Let Wv (g) = ψn (π(g)v), for v ∈ Vπ and g ∈ GLn (K). The function n Wv ∈ IndGL Un ψn is called the Whittaker function attached to the vector v. By Theorem 1.1, the subspace generated by all Whittaker functions Wv is unique. We denote it by W(π, ψ). This space is called the Whittaker model of π . For Wv ∈ W(π, ψ), v the function on GLn given by denote by W v (g) = Wv (wn (t g −1 )), g ∈ GLn (K) W where wn is the element of GLn (K) with 1’s on the anti-diagonal and zeros elsewhere (the longest Weyl element). Let π˜ be the contragredient of π . Then, v ∈ W(π˜ , ψ −1 ). W

1.2 Gamma Factors in the p-Adic Case Let n > t ≥ 1 be integers. Let π be an irreducible generic representation of GLn (F) with central character ωπ and let τ be an irreducible generic representation of GLt (F) with central character ωτ . Let Wπ ∈ W(π, ψn ) and Wτ ∈ W(τ, ψt−1 ) be Whittaker functions of π and τ . For g ∈ GLn (F), we denote by Rg the  right translation  action by It 0 . of g on C-valued functions on GLn (F) to C. Let wn,t = 0 wn−t For n − t − 1 ≥ j ≥ 0, a local zeta integral for Wπ ∈ W(π, ψ), Wτ ∈ W(τ, ψ −1 ) is defined by   Z(Wπ , Wτ , s; j) :=

g

⎛

⎞ g 0 0 n−t Wπ ⎝x In−t−1−j 0 ⎠ Wτ (g)| det g|s− 2 dxd g, x 0 0 Ij+1

where integration in the variable g is over Ut (F)\GLt (F) and integration in the variable x is over Mat (n−t−j−1)×t (F). In the celebrated paper [18], Jacquet, Piatetski-Shapiro, and Shalika established the following functional equations. Theorem 1.2 [18, Section 2.7] (1) Each integral Z(Wπ , Wτ , s; j) is absolutely convergent for (s) sufficiently large and is a rational function of q−s . More precisely, for fixed j, the integrals Z(Wπ , Wτ , s; j) span a fractional ideal (independent of j) C[qs , q−s ]L(s, π × τ ) of the ring C[qs , q−s ], where the local L-factor L(s, π × τ ) has the form P(qs )−1 , with P ∈ C[x] and P(0) = 1.

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(2) For n − t − 1 ≥ j ≥ 0, there is a factor (s, π × τ, ψ) independent of j, such that τ , 1 − s; n − t − j − 1) π , W Z(Rwn,r W L(1 − s, π˜ × τ˜ ) = ωτ (−1)n−1 (s, π × τ, ψ)

Z(Wπ , Wτ , s; j) . L(s, π × τ )

The local Gamma factor1 attached to a pair of representations π and τ is defined by (s, π × τ, ψ) = (s, π × τ, ψ)

L(1 − s, π˜ × τ˜ ) . L(s, π × τ )

(1.1)

The functional equation in Theorem 1.2 (2) can be rewritten as τ , 1 − s; n − t − j − 1) π , W Z(Rwn,t W

(1.2)

= ωτ (−1)n−1 (s, π × τ, ψ)Z(Wπ , Wτ , s; j).

2 Local Converse Problem and Jacquet’s Conjecture 2.1 Local Converse Problem Over p-Adic Fields It is natural to ask if an irreducible generic representation is determined by sufficiently many of its twisted Gamma factors. More precisely, we ask the following. Question 2.1 [ n × m Local Converse Problem for GLn (F)] Let π1 and π2 be irreducible generic representations of GLn (F). If the (local) Gamma factors (s, π1 × τ, ψ) and (s, π2 × τ, ψ) agree for any irreducible generic representation τ of GLt (F), with t = 1, 2, . . . , m, can we deduce that π1 and π2 are isomorphic? In 1993, in [14], Henniart proved n × (n − 1) Local Converse Theorem for GLn (F) and Jiang-Ping (Jeff) Chen proved n × (n − 2) Local Converse Theorem [8] in 1996. However, a conjecture credited to Jacquet predicts (essentially) that n × [ 2n ] twisted Gamma factors should be enough. Conjecture 2.2 [Jacquet’s conjecture] Let π1 and π2 be irreducible smooth generic representations of GLn (F) with the same central character. Assume that the local Gamma factors (s, π1 × τ, ψ) and (s, π2 × τ, ψ) are equal for all irreducible generic representations τ of GLt (F), 1 ≤ t ≤ [ 2n ]. Then, π1 and π2 are isomorphic. For a while, no further progress was made toward the conjecture and people turned to consider finite field analogue for inspiration. 1 In this paper, we use Gamma or -factors to indicate the gamma factors in the p-adic case and use gamma or γ -factors in the case of finite fields.

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2.2 Local Converse Problem Over Finite Fields Roditty in her thesis considered the finite field analogue of zeta integrals and gamma factors. Theorem 2.3 [26] Let π be an irreducible cuspidal representation of GLn (Fq ) and τ an irreducible generic representation of GLt (Fq ) with n > t. Then, there exists a complex number γ (π × τ, ψ) such that ⎛ ⎞ m 0 0 Wπ ⎝ x In−t−j−1 0 ⎠ Wτ (m) γ (π × τ, ψ)qtj 0 0 Ij+1 m∈Ut \GLt (Fq ) x∈Mn−t−j−1,t ⎛ ⎞ 0 In−t−j 0   = Wπ (⎝ 0 0 Ij ⎠)Wτ (m), m 0 y m∈Uj \GLt (Fq ) y∈Mt,j 



for all 0 ≤ j ≤ n − t − 1, Wπ ∈ W(π, ψ) and Wτ ∈ W(τ, ψ −1 ). Due to lack of uniqueness of some auxiliary models, the above functional equations can only be built when n > t and the irreducible representation π of GLn (Fq ) is cuspidal. As in the p-adic case, we can consider the n × m Local Converse Problem. Question 2.4 [n × m Local Converse Problem for GLn (Fq )] Let π1 and π2 be irreducible cuspidal representations of GLn (Fq ) with the same central character.2 If the (local) gamma factors γ (π1 × τ, ψ) and γ (π2 × τ, ψ) agree for any irreducible generic representation τ of GLt (Fq ), with t = 1, 2, . . . , m, can we deduce that π1 and π2 are isomorphic? In [26], Roditty verified n × (n − 1) and n × (n − 2) Local Converse Theorem for GLn (Fq ). In 2014, the author proved the following finite field analogue of Jacquet’s conjecture. Theorem 2.5 [n × [ 2n ] Local Converse Theorem, [19]] Let π1 and π2 be irreducible cuspidal representations of GLn (Fq ) with the same central character. If γ (π1 × τ, ψ) = γ (π2 × τ, ψ), for all irreducible generic representation of GLr (Fq ), 1 ≤ r ≤ [ 2n ], then π1 ∼ = π2 . Denote by Pn the mirabolic subgroupof GLn ,consisting of matrices in GLn with 0 In−1 last row equal to (0, . . . , 0, 1). Let α = . 1 0 The proof of the Theorem 2.5 is based on the following geometric decomposition of GLn . condition can be removed if we assume that γ (π1 × τ, ψ) = γ (π2 × τ, ψ) for any τ ∈ F× q. Refer to [22].

2 This

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Proposition 2.6 [19] An W (GLn ) ⊂ ∪0≤r≤[ 2n ],n−[ 2n ]≤k≤n α r Pn α k Zn . Or equivalently, GLn = Un (∪0≤r≤[ 2n ],n−[ 2n ]≤k≤n α r Pn α k )Zn Un . 2.3 In [17], a project with Dihua Jiang and Shaun Stevens, we removed the criterion on the agreement of central characters in Conjecture 2.2 by appealing to the stability of Gamma factors. Corollary 2.7 [17] Let π1 and π2 be irreducible generic representations of GLn (F). If their local Gamma factors (s, π1 × χ , ψF ) and (s, π2 × χ , ψF ) are equal (as functions in the complex variable s) for any character χ of F × , then they possess the same central character. Applying the multiplicativity of Gamma factors, we reduce Conjecture 2.2 to the following, which allows us checking the agreement of twisted Gamma factors against irreducible cuspidals only instead of all irreducible generic representations. Conjecture 2.8 Let π1 and π2 be irreducible smooth generic representations of GLn (F). Assume that the local Gamma factors (s, π1 × τ, ψ) and (s, π2 × τ, ψ) are equal (as functions in the complex variable s) for all irreducible cuspidal representations τ of GLt (F), 1 ≤ t ≤ [ 2n ]. Then, π1 and π2 are isomorphic. We further set up a scheme for verifying Conjecture 2.8 and used this to confirm many cases. Based on our formulation, Adrian et al. [1] verified Conjecture 2.2 for GLp (F) for p prime. Finally, Jacquet and Liu [16], using an analytic approach, and Chai [7], using p-adic Bessel functions, independently settled this longstanding conjecture.

2.3 Sharpness on the Bound [ n2 ] After confirmation of Jacquet’s conjecture, it was natural to consider sharpness of the bound [ 2n ]. The paper [3] showed that [ 2n ] is the sharp bound for necessary twisting in the Local Converse Theorem for a pair of cuspidal representations of GLn (F) for n prime. However, for special families of cuspidal representations, you may wonder if it is possible to lower the upper bound for necessary twisting? For instance, for simple cuspidal representations(i.e., cuspidal representations of minimal positive depth), the upper bound may be lowered to 1: See [2, 27] when the residue characteristic is coprime to n and [5] for general case. Question 2.9 In the case of depth zero cuspidal representations, can we lower the upper bound for necessary twisting to 1? Since depth zero cuspidal representations of GLn (F) are constructed from cuspidal representations of GLn (Fq ), we start by studying the finite field analogue.

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3 Explicit Computation of Gamma Factors Over Finite Fields 3.1 Bessel Functions Bessel functions were introduced by Gelfand in [11]. They play an important role in explicit computations of gamma factors and in Converse Theorems over finite fields. Proposition 3.1 Let π be an irreducible generic representation of GLn (Fq ), and let χπ be its character. The genericity of π guarantees the existence of a function B ∈ W(π, ψ) satisfying B(u1 gu2 ) = ψn (u1 )ψn (u2 ), for all u1 , u2 , ∈ Un , g ∈ GLn (Fq ).

(3.1)

By Theorem 1.1, this function is unique up to a scalar. Proposition 3.2 [[11, Proposition 4.5] or [26, Lemma 6.1.1]] Let π be an irreducible generic representation of GLn (Fq ), and let χπ be its character. Define B(g) = |Un (Fq )|−1



ψn (u−1 )χπ (gu), for g ∈ GLn (Fq ).

u∈Un

Then, B satisfies (3.1). Moreover, B(In ) = 1. We call B the Bessel function of π in terms of ψ and denote it by Bπ,ψ to indicate the dependence on π and ψ. We then have the following nice expression for twisted gamma factors in terms of the associated Bessel functions. Proposition 3.3 [26] Let π be an irreducible cuspidal representation of GLn (Fq ) and τ an irreducible generic representation of GLr (Fq ) with r < n. Then, γ (π × τ, ψ) =

 Ur \GLr

 Bπ,ψn

 0 In−r Bτ,ψr−1 (m). m 0

(3.2)

Bessel functions have nice symmetry properties. These together with the geometric decomposition, Lemma 2.6, are the key to the proof of Theorem 2.5. Furthermore, this symmetry also contributes to the proof of the converse theorem on distinction, which is to be discussed in later sections. Proposition 3.4 [Symmetry of Bessel functions, [26]] Let π be an irreducible generic representation of GLn (Fq ). Then, Bπ,ψ (g −1 ) = Bπ,ψ (g), g ∈ GLn (Fq ).

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n × 1 Gamma Factor and Gauss Sums

Let Fq denote the algebraic closure of Fq and Fqd be the unique field extension of Fq with index d such that Fq ⊂ Fqd ⊂ Fq . In [12], Green showed that the character of an irreducible representation of GLn (Fq ) can be computed in terms of a suitable associated character of × × F× qn1 × Fqn2 × · · · × Fqnk

where n1 + · · · + nk = n. Let η be a character of F× qn satisfying the following properties: ηq

m

−1

= 1, for all positive m < n.

(3.3)

Such a character η is called regular. Theorem 3.5 (Green’s construction, [12]) A regular character η of F× qn correspond to irreducible cuspidal representation πη of GLn (Fq ). Moreover, any irreducible cuspidal representation corresponds to some regular character. Regular characters η1 and η2 correspond to the same cuspidal π if and only if qt

η1 = η2 for some t ≥ 1.

(3.4)

× Let tr n : Fqn → Fq and Nr n : F× qn → Fq be the trace map and norm map (resp.) for the extension Fqn / Fq . Using Green’s formula, we can relate n × 1 gamma factors over finite fields and abelian Gauss sums.

Theorem 3.6 [20, Theorem 1.1] Let π be an irreducible cuspidal representation of GLn (Fq ), n ≥ 2 and χ ∈ F× q . Then, γ (π × χ , ψ) = (−q−1 χ (−1))n−1



ψn (tr n σ −1 )ηπ (σ )χ (Nr n (σ )),

σ ∈F× qn

where ηπ is the regular character of F× qn corresponding to π in Green’s construction. The above theorem and [6, Theorem 9.3] imply that the gamma factors for GLn (Fq ) defined by Roditty coincide with the ones constructed through cohomology by Braverman and Kazhdan in [6]. The next result is an analogue of a well known property of abelian Gauss sums. Corollary 3.7 (absolute values of gamma factors, [21, Corollary 4.1]) Let π be an irreducible cuspidal representation of GLn (Fq ) and τ an irreducible generic representation of GLt (Fq ) with n > t. Then, |γ (π × τ, ψ)| = q

−t(n−t−1) 2

.

Local Gamma Factors, Converse Theorems and Related Problems

Proof Use

199

γ (π × τ, ψ)γ (π˜ × τ˜ , ψ −1 ) = q−t(n−t−1)

and Bπ,ψ ˜ −1 = Bπ,ψ . Theorem 3.6 and Corollary 3.7 suggest the following question. Question 3.8 Let π be an irreducible cuspidal representation of GLn (Fq ) and τ an irreducible, cuspidal representation of GLn (Fq ). Can we express the gamma factor γ (π × τ, ψ) in terms of the characters ηπ and ητ ? Moreover, if this is possible, can we view the resulting expression as some sort of generalization of a Gauss sum?

3.3 Some n × 1 Local Converse Theorem for GLn (Fp ) × Let Nr n:d denote the norm map from F× qn to Fqd where d | n. Guided by Langlands functoriality, there should exist a tensor lifting taking an irreducible representation π ⊗ τ of GLn (Fq ) × GLm (Fq ) to an irreducible representation  of GLmn (Fq ) such that  corresponds to the character ηπ ◦ Nr mn:n · ητ ◦ Nr mn:m of F× qmn . Note that the character ηπ ◦ Nr mn:n · ητ ◦ Nr mn:m is possibly not regular so that the corresponding representation  need not be cuspidal. We expect that  corresponds to the unique generic subrepresentation of mn IndGL GLn ×···×GLn π1 ⊗ · · · ⊗ πk , 1

k

where n1 + · · · + nk = mn and each πi is cuspidal. If we extend Roditty’s gamma factors to generic representations through multiplicativity,3 then γ ( × 1, ψ) :=

k i=1 γ (πi × 1, ψ) and we expect that γ ( × 1, ψ) = q

2−m2 −m 2

γ (π × τ, ψ).

Based on the known case m = 1 (Theorem 3.6), we propose the following conjectural formula for n × m gamma factors in terms of the corresponding characters of F× qmn . Conjecture 3.9 For n > m, let π and τ be irreducible cuspidal representations of GLn (Fq ) and GLm (Fq ) respectively, and χ and η be the corresponding regular × characters of F× qn and of Fqm . Then, γ (π × τ, ψ) = c · χ (−1)m−1 η(−1)n−1 G(χ ◦ Nr mn:n · η ◦ Nr mn:m , ψ),

(3.5)

where 3 Multiplicativity

note.

of gamma factors of GLn (Fq ) has been established by Soudry in an unpublished

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G(β −1 , ψ) :=



β −1 (a)ψ(tr N a) =

a∈F× qN



β(a)ψ(tr N a−1 )

a∈F× qN

is the classical Gauss sum for a character β −1 of F× , and qN c = (−1)m(n−1) q−mn+

m2 +m 2

.

The above conjecture interprets representation theoretic matters in terms of number theoretic objects. For the case m = 1, this conjectural formula has been verified in Theorem 3.6. When m divides n, Conjecture 3.9 and the multiplicativity of gamma factors imply the Hasse-Davenport relation for Gauss sums. The Gauss sums involved in n × 1 gamma factors over finite fields are algebraic integers, and the results developed for Gauss sums over Fp enable us to extract the invariants for characterizing our target representations. Using Theorem 3.6 and known results about Gauss sums developed in number theory, the author and Lei Zhang translated the n × 1 Local Converse Problem for GLn (Fp ) into a combinatorial statement and obtained the following results. Theorem 3.10 [22] Let π1 and π2 be irreducible cuspidal representations of GLn (Fp ) where n ≤ 5 and p is prime.4 If γ (π1 × χ , ψ) = γ (π2 × χ , ψ) ∼ for all χ ∈ F× p , then π1 = π2 . Recall that Mersenne primes are primes of the form 2n − 1. In this case, n must be a prime and all non-trivial characters of F× 2n are regular. Proposition 3.11 [22] If 2n − 1 is a prime (Mersenne prime), then the n × 1 Local Converse Theorem holds for GLn (F2 ). The 50th Mersenne prime, M50 = 277232917 − 1, was found on December 26, 2017, and it became the largest prime number known to mankind. Thus, the 77232917 × 1 Local Converse Theorem holds for GL77232917 (F2 ). Moreover, in the appendix to [22], Zhiwei Yun gives a series of examples confirming the n × 1 Local Converse Theorem by transferring the questions on Gauss sums to ones concerning Kloosterman sheaves. Theorem 3.12 (Appendix of [22] by Zhiwei, Yun) Let n ≥ 1 be an integer satisfying × √ + 1. Let χ1 , χ2 : F× n < 2q−1 qn → C be characters. Suppose that q G(χ1 · η, ψ) = G(χ2 · η, ψ), 4 Here,

sums.

the cardinality of Fq is required to be a prime to make use of certain properties of Gauss

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× for any character η : F× q → C . Then, χ1 and χ2 are in the same Frobenius orbit;  qj i.e., there exists j ∈ Z such that χ1 = χ2 , where G(χ , ψ) = x∈F×n χ (x)ψ(tr n x). q

However, we also found counterexamples to the 6 × 1 Local Converse Problem. Theorem 3.13 [22] There exist non-isomorphic irreducible cuspidal representations π1 and π2 of GL6 (F3 ) such that γ (π1 × χ , ψ) = γ (π2 × χ , ψ) for all χ ∈ F× 3. To interpret these different outcomes to n × 1 Local Converse Problems, we introduced a notion of primitive form for cuspidal representations and made the conjecture below, drawing inspiration from Arthur’s work [4]. Let r be a prime divisor of n. For each regular character χ of F× qn , denote by πχ the cuspidal representation of GLr (Fqn/r ) corresponding to χ which is called a primitive representation associated to χ . Conjecture 3.14 [22] For i = 1, 2, let χi be a regular character of F× qn and r be a prime divisor of n. If γ (πχ1 × η, ψ) = γ (πχ2 × η, ψ) for all characters η of F× qn/r . ∼ Then πχ1 = πχ2 .

4 Gamma Factors for Level Zero Cuspidals Let o be the ring of integers in F and p the prime ideal in o. Denote by k the residue field of F. Let cInd denote the functor of compact induction and Zn (F) the center of GLn (F). Let Kn = GLn (o), the unique maximal compact subgroup of GLn (F) (up to conjugacy), and set Pn := In + Mat n (p). Let ψF be an additive complex character of F such that ψF is nontrivial on o and trivial on p. Let ψ be the character of o/p induced by ψF . Let τ be an irreducible cuspidal representation of GLn (k). We write τ for the representation of Kn obtained by composing τ with the natural map ¯ : Kn → GLn (k) given by reduction modulo p. GLn (F) ωτ¯ , where ω is a character of Zn (F) such that ω|Kn ∩Zn (F) = Let π ∼ = cIndZn (F)K n (F) τ¯ |Kn ∩Zn (F) . Theorem 4.1 If τ is an irreducible cuspidal representation of GLn (k), then π is an irreducible cuspidal representation of GLn (F). The representations π are the level zero cuspidal (or depth zero cuspidal) representations of GLn (F). By using (generalized) Bessel functions of cuspidal representations of GLn (F) introduced by V. Pašk¯unas and Stevens in [25], the author and Lei Zhang were able to establish the following relation between Gamma factors for a pair of level

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zero cuspidal representations of GLn (F) and GLt (F) and gamma factors for the corresponding cuspidal representations of GLn (Fq ) and GLt (Fq ). Theorem 4.2 [22] For n ≥ 2, let π1 and π2 be irreducible level zero cuspidal representations of GLn (F) and GLt (F)) (resp.) with t < n such that π ∼ = GLt (F) n (F)  ∼ ω τ and π ω τ . Then, cIndGL cInd = 1 1 2 2 Zn (F)Kn (F) Zt (F)Kt (F) ωτ2 (−1)n−1 (s, π × π  , ψF ) = q

t(n−t−1) 2

γ (τ1 × τ2 , ψ).

Given two level zero cuspidal representations of GLn (F), GLn (F) ωτi , πi ∼ = cIndZn (F)K n (F)

for i = 1, 2, where τ1 (resp. τ2 ) is an irreducible cuspidal representation of GLn (Fq ) (resp. GLt (Fq )), and q is the cardinality of the residue field of F. Assume that F ×. (s, π1 × χ , ψF ) = (s, π2 × χ , ψF ), for all χ ∈ Then, π1 and π2 possess the same central character ω, by Corollary 2.7. Therefore, n × 1 Local Converse Theorem for level zero cuspidals for n ≤ 5, q prime or n < q−1 √ + 1 follows the results of Theorem 3.10, 3.12 and 4.2. 2 q Theorem 4.3 ([22] and the appendix by Zhiwei Yu) Let π be an irreducible, level zero cuspidal, unitary representations of GLn (F), where the cardinality of the √ + 1. Then, π is uniquely residue field q of F is either a prime, and n ≤ 5 or n < 2q−1 q F × }. determined by the set of twisted Gamma factors {(s, π × χ , ψF ) | χ ∈ Theorem 4.4 [22] Let F2 be the 2-adic field with residue field of two elements. Assume that 2n − 1 is a Mersenne prime. Let π1 and π2 be a pair of irreducible, unitarizable, level zero supercuspidal representations of GLn (F2 ). Suppose that × , (s, π1 × χ , ψF2 ) = (s, π2 × χ , ψF2 ), for all χ ∈ F 2 then π1 and π2 are isomorphic.

5 Converse Theorems and Distinction 5.1 p-Adic Results Definition 5.1 Let H be a subgroup of G. A representation (π, V ) of G is called H -distinguished if Hom G (π, IndHG 1) ∼ = HomH (π |H , 1) = 0.

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Let E be a quadratic extension of a p-adic field F such that the characteristic of the residue field of F is odd. Let ψE be a non-trivial character of E whose restriction to F is trivial. Let G n = GLn (E) and Hn = GLn (F). Hakim studied distinction for generic representations of G 2 using special values of Gamma factors. Theorem 5.2 [13] Let π be an irreducible unitary generic representation of G 2 with central character whose restriction to F × is trivial. If 1 ( , π × τ, ψE ) = 1 2 for all F × -distinguished character τ of E × , then π is H2 -distinguished. In 1997, under the supervision of Jacquet, Youngbin Ok in his thesis proved the n × (n − 1) Local Converse Theorem on distinction. Theorem 5.3 [24] Let π be an irreducible unitary cuspidal representation of GLn (E) with central character whose restriction to F × is trivial. Then, the following are equivalent: (1) π is Hn -distinguished. (1) ( 21 , π × τ, ψE ) = 1 for any Hr -distinguished unitary generic representation τ of G r , for r = 1, . . . , n − 1. In 2011, Omer Offen gave a criterion for representation of G n to be Hn distinguished in terms of special values of twisted  and  factors. Theorem 5.4 [23] Let π (resp. τ ) be an irreducible generic and Hn -distinguished (resp. Ht -distinguished) representation of G n (resp. G t ). Then, 1 1 ( , π × τ ) = ( , π × τ, ψE ) = 1. 2 2 In 2015, Hakim and Offen established n × (n − 2) Local Converse Theorem on distinction over p-adic fields. Theorem 5.5 [15] Let π be a cuspidal representation of G n , n ≥ 3. Then, π is Hn -distinguished if and only if 1 ( , π × τ, ψE ) = 1 2 for all irreducible generic Hr -distinguished representations τ of G r , for r = 1, . . . , n − 2. Imitating Jacquet’s conjecture on the Local Converse Problem, we proposed the following conjecture on distinction. Conjecture 5.6 [n × [ 2n ] Local Converse Problem on distinction for GLn (F)] The n × [ 2n ] Local Converse Theorem for distinction hold for GLn (F). As a first step, we worked on the finite field analogue to seek supporting evidence for a positive answer.

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5.2 Finite Field Results Definition 5.7 Let π be an irreducible representation of GLn (Fq ) and τ irreducible representation of GLr (Fq ), 1 ≤ r ≤ n. We call τ exceptional to π or (π, τ ) an exceptional pair if there exists an irreducible representation ρ of GLn−r (Fq ) such that n dim Hom(π, IndGL P τ × ρ)  = 0, where P is the standard parabolic subgroup of GLn , corresponding to partition (r, n − r). Here, we allow r = n and the only irreducible representation of GLn (Fq ) which is exceptional to a cuspidal π is π itself. Definition 5.8 Let π be an irreducible generic representation of GLn (Fq ) and τ be an irreducible generic representation of GLr (Fq ), r < n. Then, we define γ (π × τ, ψ) =

 Ur \GLr

 Bπ,ψn

 0 In−r Bτ,ψr−1 (m). m 0

(5.1)

When π is cuspidal, by Eq. (3.2), γ (π × τ, ψ) in this definition coincides with the one defined in Theorem 2.3. However, the functional equations in Theorem 2.3 may fail for generic non-cuspidal π . Let q be a power of an odd prime and ψ0 be a non-trivial additive character of Fq2 , whose restriction to Fq is trivial. Theorem 5.9 [20] Let π be an irreducible generic representation of GL2 (Fq2 ) with central character ωπ whose restriction to F× q is trivial. Then, π is GL2 (Fq )distinguished if and only if γ (π × χ −1 , ψ0 ) = 1 for all F× q -distinguished χ ∈  GL1 ((Fq2 ) which are non-exceptional to π . Theorem 5.10 [[21], n × [ 2n ] Local Converse Theorem on distinction over finite fields] For n ≥ 3, let π be an irreducible cuspidal representation of GLn (Fq2 ) with central character ωπ whose restriction to F× q is trivial. If γ (π × ρ, ψ0 ) = q−r(n−r−1) for all generic GLr (Fq )-distinguished representations ρ of GLr (Fq2 ) for r = 1, . . . , [ 2n ], then π is GLn (Fq )-distinguished. Acknowledgements This work was supported by the Ministry of Science and Technology, Taiwan 105-2115-M-006-010-MY2.

References 1. M. Adrian, B. Liu, S. Stevens, P. Xu, On the Jacquet conjecture on the local converse problem for p-adic GLN . Represent. Theor. 20, 1–13 (2016) 2. M. Adrian, B. Liu, Some results on simple supercuspidal representations of GLn (F). J. Number Theor. 160, 117–147 (2016)

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3. M. Adrian, B. Liu, S. Stevens, K.-F. Tam, On the sharpness of the bound for the Local Converse Theorem of p-adic GL(prime). To appear in Proc. Amer. Math. Soc. (2017) 4. J. Arthur, Functoriality and the Trace Formula. preprint 5. C.J. Bushnell, G. Henniart, Langlands parameters for epipelagic representations of GLn . Math. Ann. 358(1–2), 433–463 (2014) 6. A. Braverman, D. Kazhdan, γ -functions of representations and lifting. With an appendix by V. Vologodsky. GAFA 2000 (Tel Aviv, 1999). Geometric Functional Analysis, Special Volume, Part I, pp. 237–278 (2000) 7. J. Chai, Bessel functions and local converse conjecture of Jacquet. To appear in J. Eur. Math. Soc. (JEMS) 8. J.-P.J. Chen, The n × (n − 2) Local Converse Theorem for GL(n) over a p-adic field. J. Number Theor. 120(2), 193–205 (2006) 9. J. Cogdell, I. Piatetski-Shapiro, Converse theorems for GLn . Publications Mathématiques de l’IHÉS (79), 157–214 (1994) 10. J. Cogdell, I. Piatetski-Shapiro, Converse theorems for GLn . II. J. Reine Angew. Math. 507, 165–188 (1999) 11. S.I. Gelfand, Representation of the full linear group over a finite field. Math. USSR Sbornik 12(1), 13–39 (1970) 12. J.A. Green, The characters of the finite general linear groups. Trans. Amer. Math. Soc. 80, 402–447 (1955) 13. J. Hakim, Distinguished p-adic representations. Duke Math. J. 62(1), 1–22 (1991) 14. G. Henniart, Characterization of the local Langlands correspondence by -factors of pairs. Invent. Math. 113(2), 339–350 (1993) 15. J. Hakim, O. Offen, Distinguished representations of GL(n) and Local Converse Theorems. Manuscripta Math. 148(1–2), 1–27 (2015) 16. H. Jacquet, B. Liu, On the Local Converse Theorem for p-adic GLn . To appear in Amer. J. Math 17. D. Jiang, C. Nien, S. Stevens, Towards the Jacquet conjecture on the Local Converse Problem for p-adic GLn . J. Eur. Math. Soc. (JEMS) 17(4), 991–1007 (2015) 18. H. Jacquet, I. Piatetski-Shapiro, J. Shalika, Rankin-Selberg convolutions. Amer. J. Math. 105, 367–464 (1983) 19. C. Nien, A proof of finite field analogue of Jacquet’s conjecture. Amer J. Math. 136(3), 653–674 (2014) 20. C. Nien, n × 1 Local Gamma factors and Gauss Sum. Finite Fields Appl. 46, 255–270 (2017) 21. C. Nien, Gamma factors and quadratic extension over finite fields. manuscripta math. 158(1–2), 31–54 (2019) 22. C. Nien, L. Zhang, Converse Theorem meets Gauss sums, with an appendix by Zhiwei Yun. submitted 23. O. Offen, On local root numbers and distinction. J. Reine Angew. Math. 652, 165–205 (2011) 24. Y. Ok, Distinction and Gamma factors at 21 : supercuspidal case. Ph.D. Thesis, Columbia University (1997) 25. V. Pašk¯unas, S. Stevens, On the realization of maximal simple types and epsilon factors of pairs. Amer. J. Math. 130, 1211–1261 (2008) 26. E.-A. Roditty, On gamma factors and Bessel functions for representations of general linear groups over finite field. M.Sc. Thesis, Tel-Aviv University (2010) 27. P. Xu, A remark on the simple cuspidal representations of GL(n, F). Preprint, available at arxiv.org/pdf/1310.3519v2.pdf

On Completions of Hecke Algebras Maarten Solleveld

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Affine Hecke Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Schwartz and C ∗ -completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Space of Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Reductive p-adic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Plancherel Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Space of Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Morita Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Conditions and First Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preservation of Temperedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Comparison of Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Hecke Algebras from Bushnell–Kutzko Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Hecke Algebras from Bernstein’s Progenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208 212 217 224 228 231 236 238 238 243 249 253 258 260

Abstract Let G be a reductive p-adic group and let H(G)s be a Bernstein block of the Hecke algebra of G. We consider two important topological completions of H(G)s : a direct summand S(G)s of the Harish-Chandra–Schwartz algebra of G and a two-sided ideal Cr∗ (G)s of the reduced C ∗ -algebra of G. These are useful for the study of all tempered smooth G-representations. We suppose that H(G)s is Morita equivalent to an affine Hecke algebra H(R, q) – as is known in many cases. The latter algebra also has a Schwartz completion S(R, q) and a C ∗ -completion Cr∗ (R, q), both defined in terms of the underlying root datum R and the parameters q. We prove that, under some mild conditions, a Morita equivalence H(G)s ∼M H(R, q) extends to Morita equivalences S(G)s ∼M S(R, q) and Cr∗ (G)s ∼M Cr∗ (R, q). We also check that our conditions are fulfilled in all known cases of such Morita equivalences between Hecke algebras. This is applied to compute the topological K-theory of the reduced C ∗ -algebra of a classical p-adic group. The author was supported by an NWO Vidi grant “A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528). M. Solleveld (B) IMAPP, Radboud Universiteit, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_8

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2010 Mathematics Subject Classification 20C08 · 22E50 · 22E35

1 Introduction Let G be a connected reductive group over a non-Archimedean local field. Let Rep(G) be the category of smooth G-representations on complex vector spaces. To study such representations, it is often useful to consider various group algebras of G. Most fundamentally, there is the Hecke algebra H(G), the convolution algebra of locally constant, compactly supported functions f : G → C. The category Rep(G) is equivalent to the category Mod(H(G)) of nondegenerate H(G)-modules. For the purposes of harmonic analysis, and in particular for the study of tempered smooth G-representations, the Harish-Chandra–Schwartz algebra S(G) [14] can be very convenient. This is a topological completion of H(G), consisting of locally constant functions f : G → C that decay rapidly in a specified sense. By [43, §III.7], an admissible smooth G-representation is tempered if and only if it is naturally an S(G)-module. For larger representations, it is best to define the category of tempered smooth G-representations as the category Mod(S(G)) of nondegenerate S(G)-modules [32, Appendix]. Further, from the point of view of operator algebras or noncommutative geometry, the reduced C ∗ -algebra Cr∗ (G) may be the most suitable. The modules of Cr∗ (G) are Banach spaces, so they are usually not smooth as G-representations. But S(G) ⊂ Cr∗ (G) and the smooth vectors in any Cr∗ (G)-module do form a nondegenerate S(G)module and hence a smooth G-representation. Moreover, this operation provides a bijection between the irreducible representations of Cr∗ (G) and those of S(G). This feature distinguishes Cr∗ (G) from other Banach group algebras like L1 (G) or the maximal C ∗ -algebra of G. Let Rep(G)s be a Bernstein block of Rep(G) [5]. It is well known that in many cases (see Sects. 5 and 6) Rep(G)s is equivalent to the category of modules of an affine Hecke algebra H(R, q). Here R is a root datum and q is a parameter function for R. In such cases, it would be useful if one could detect, in terms of H(R, q), whether a G-representation in Rep(G)s (i) is tempered; (ii) is unitary; (iii) admits a continuous extension to a Cr∗ (G)-module. The structure needed to make sense of this is available for (extended) affine Hecke algebras with positive parameters. They have a natural ∗-operation, so unitarity is defined. Temperedness of finite-dimensional H(R, q)-modules can be defined conveniently either in terms of growth of matrix coefficients or by means of weights for a large commutative subalgebra of H(R, q) [25, §2.7]. Furthermore, there exists a Schwartz completion S(R, q) of H(R, q) [25, §6.2] that parallels the completion S(G) of H(G) [12]. By [25, Corollary 6.7], a finite

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dimensional H(R, q)-module is tempered if and only if it extends continuously to an S(R, q)-module. As in the group case, we define the category of tempered H(R, q)-modules to be the module category of S(R, q). The algebra H(R, q) is a Hilbert algebra, so it acts by multiplication on its own Hilbert space completion. Then one can define the reduced C ∗ -completion Cr∗ (R, q) as the closure of H(R, q) in the algebra of bounded linear operators on that Hilbert space. It is reasonable to expect that this algebra plays a role analogous to Cr∗ (G). Let H(G)s (resp. S(G)s and Cr∗ (G)s ) be the direct summand of H(G) (resp. S(G) and Cr∗ (G)) corresponding to Rep(G)s via the Bernstein decomposition Mod(H(G)) ∼ = Rep(G) =

 s∈B(G)

Rep(G)s .

In view of the above, it is natural to ask whether an equivalence of categories ∼ Rep(G)s ∼ = Mod(H(G)s ) −→ Mod(H(R, q))

(1)

extends to Morita equivalences S(G)s ∼M S(R, q) and Cr∗ (G)s ∼M Cr∗ (R, q).

(2)

That would solve the issues (i) and (iii) completely, and would provide a partial answer to (ii). Namely, since irreducible tempered representations are unitary, it would imply that (1) matches the unitary tempered representations on both sides. (It is not clear what it could say about unitary non-tempered representations.) Furthermore, (2) would make Mod(S(G)s ) and Mod(Cr∗ (G)s ) amenable to much more explicit calculations, in terms of the generators and relations from H(R, q). That is important for topological K-theory, where one deals with finitely generated projective Cr∗ (G)modules. While (2) looks fairly plausible, it is not automatic. To prove it, we impose three conditions on the Morita equivalence H(G)s ∼M H(R, q), Conditions 4.1 and 4.2 (described in Sect. 4.1) and Condition 2.1 (described in Sect. 2). • Condition 4.1 is about compatibility with parabolic induction and restriction. • Condition 4.2 says roughly that under this Morita equivalence every (suitable) parabolic subgroup of G should give rise to a parabolic subalgebra of H(R, q). In all such instances, the relation between a parabolic subgroup and a parabolic subalgebra should respect the appropriate notions of positivity. • Sometimes we obtain, instead of H(R, q), an extended affine Hecke algebra H(R, q)  , where  is a finite group. Then we require Condition 2.1, which says that  respects all the relevant structure. Here Condition 4.1 has little to do with affine Hecke algebras. If it holds, then it does so based on general functoriality principles. Condition 4.2 is needed to bring affine Hecke algebras into play. If it does not hold, then our results simply cannot be formulated in such terms. The positivity part is innocent and can usually be achieved

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by a good choice of a standard minimal parabolic subgroup of G. Condition 2.1 is more technical. It serves to rule out some phenomena that could happen for arbitrary  but not for reductive groups. Notice that the above conditions do not say anything about ∗-homomorphisms between H(R, q) and H(G)s . Instead, our conditions are chosen so that they will hold true for affine Hecke algebras arising from reductive p-adic groups via the two main methods: Bushnell–Kutzko types and Bernstein’s progenerators. Theorem 1 (see Theorem 4.12) Suppose that there is a Morita equivalence between H(G)s and an extended affine Hecke algebra H(R, q)   with positive parameters, such that Conditions 4.1, 4.2 and 2.1 hold. Then it induces Morita equivalences S(G)s ∼M S(R, q) and Cr∗ (G)s ∼M Cr∗ (G). Hitherto this was proved only for the Schwartz completions in the case of Iwahori– spherical representations of split groups [12, Theorem 10.2]. In all cases where a Morita equivalence on the Hecke algebra level is known (to the author), we check that the conditions of Theorem 1 are fulfilled. This includes principal series representations of F-split groups (F is any non-Archimedean local field), level-zero representations, inner forms of GLn (F), inner forms of SLn (F), symplectic groups (not necessarily split), and special orthogonal groups (also possibly non-split). In all these cases, we obtain a pretty good picture of Cr∗ (G)s , up to Morita equivalence. This can, for instance, be used to compute the topological K-theory of these algebras. Indeed, in [41] the author determined the K-theory of Cr∗ (R, q) for many root data R (it does not depend on q). These calculations, together with Theorem 1 for classical groups, lead to a result which is useful in relation to the Baum–Connes conjecture. Theorem 2 (see Theorem 6.3) Let G be a special orthogonal or a symplectic group over F (not necessarily split), or an inner form of GLn (F). Then K∗ (Cr∗ (G)) is a free abelian group. For every Bernstein block Rep(G)s , the rank of K∗ (Cr (G)s ) is finite and can be computed explicitly. Let us also discuss other approaches to the topics (i), (ii), and (iii) from page 208. In most cases where a Morita equivalence H(G)s ∼M H(R, q)   is known, these issues are not discussed in the literature. When the Morita equivalence comes from a type (K, λ) in the sense of Bushnell–Kutzko [9], several relevant techniques are available. Let eλ ∈ H(G) and H(G, J , λ) = End G (indJG λ) be the idempotent and the algebra associated with the type (K, λ) in [9, §2]. In this setting, the Morita equivalence can be implemented by injective algebra homomorphisms ϒλ

H(R, q)   −−→ H(G, J , λ) → eλ H(G)eλ → H(G)s ,

(3)

where we assume that ϒλ is an isomorphism. It follows quickly from the definition of types that the last two maps in (3) induce Morita equivalences [9, (2.12) and Theorem

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4.3]. Both algebras H(G, J , λ) and eλ H(G)eλ are endowed with a natural trace and ∗-operation, which are preserved by the injections H(G, J , λ) → eλ H(G)eλ → H(G)s .

(4)

Considerations with Hilbert algebras show that the maps (4) induce equivalences between the associated categories of finite length tempered representations [8]. Moreover, after the first version of this paper appeared, Ciubotaru showed that (4) also induces equivalences between the respective subcategories of unitary modules [11]. Hence whenever ϒλ is a ∗-isomorphism, the categories of unitary modules of all the algebras in (3) are equivalent. Earlier, this had been proved under certain additional hypotheses [3, 4]. Notice that in (3), all algebras are endowed with extra structure, which on the left-hand side comes from affine Hecke algebras and for the other terms from the embedding in H(G). In particular, both H(G) and H(R, q)   are endowed with a canonical trace, which stems from evaluation of functions at the unit element of G. Usually, (3) will transfer the trace on H(G)s to a positive scalar multiple of the trace on H(R, q)  . If that is the case and ϒλ is a ∗-isomorphism, then by [12, Theorem 10.1], (3) induces an equivalence between the category of finite length tempered Grepresentations in Rep(G)s and the category of finite dimensional tempered modules of H(R, q)  . This relies on properties of the Plancherel measures of G and of H(R, q)  , and it uses that ϒλ preserves these Plancherel measures, up to a scalar multiple. When the Morita equivalence H(G)s ∼M H(R, q)   does not arise from a type, fewer techniques for (i), (ii), and (iii) were known. Heiermann established such Morita equivalences for symplectic and special orthogonal groups and for inner forms of GLn (F) [15], by using Bernstein’s progenerators of Rep(G)s . In [16], he showed that these equivalences preserve temperedness of finite length modules. Here, it is unknown whether the ∗-operation and the trace are preserved by the Morita equivalence. Since maps like (3) are lacking, it is even unclear how such a statement could be formulated in this setting. Summarizing, in the literature, already several results about the behaviour of finite length modules under a Morita equivalence between a Bernstein block Rep(G)s and the module category of an (extended) affine Hecke algebra can be found, but there is so far almost nothing about the Schwartz completions and the C ∗ -completions. Let us briefly describe the contents of the paper. In the first section, we recall the definitions of affine Hecke algebras and their topological completions. We formulate the Plancherel isomorphism for these completions, from [12], and we establish suitable versions for affine Hecke algebras extended by finite groups. In addition, we analyse the space of irreducible representations and the subspace of irreducible tempered representations, mainly relying on [39]. This is formulated in terms of the Langlands classification and induction from discrete series representations of parabolic subalgebras.

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After that, we look at the aforementioned group algebras for a reductive p-adic group G. We recall the Plancherel isomorphism for the Schwartz algebra of G [14, 43] and for the reduced C ∗ -algebra of G [26]. In parallel to our discussion of affine Hecke algebras, we analyse the space of irreducible smooth G-representation in terms of the Langlands classification and parabolic induction of square-integrable representations, following [37]. This forms the setup for the proof of Theorem 1, our main result, which occupies Sect. 4. The crucial idea behind our argument is that in the Plancherel isomorphisms for S(G)s and S(R, q), very similar algebras appear. In both settings, one encounters a bundle of matrix algebras over a compact torus, one takes C ∞ -sections of those, and then invariants with respect to a finite group acting via intertwining operators. We compare the resulting algebras on both sides, analyzing the data used to describe the Plancherel isomorphisms. First, we prove that a Morita equivalence between H(G)s and H(R, q), plus the mild extra conditions listed on page 209, imply that the two necessary sets of data, for S(G)s and for S(R, q), become equivalent after some manipulations. The most problematic part is to prove that the Morita equivalences preserve temperedness and match square-integrable G-representations with discrete series H(R, q)-modules. For that, we use very specific information about the spaces of irreducible representations and their subspaces of tempered representations. When we have compared all the data needed for the Plancherel isomorphisms on both sides, we establish the desired Morita equivalences between topological algebras. In Sect. 5, we check that the conditions from Sect. 4 are fulfilled in (most) known cases of Morita equivalences coming from types. In the Sect. 6, we do the same for Heiermann’s Morita equivalences constructed with the use of projective generators, and we derive Theorem 2.

2 Affine Hecke Algebras Let a be a finite dimensional real vector space and let a∗ be its dual. Let Y ⊂ a be a lattice and X = HomZ (Y , Z) ⊂ a∗ the dual lattice. Let R = (X , R, Y , R∨ , ).

(5)

be a based root datum. Thus R is a reduced root system in X , R∨ ⊂ Y is the dual root system,  is a basis of R and the set of positive roots is denoted R+ . Furthermore, a bijection R → R∨ , α → α∨ is given, such that α , α∨  = 2 and such that the corresponding reflections sα : X → X (resp. sα∨ : Y → Y ) stabilize R (resp. R∨ ). We do not assume that R spans a∗ . The reflections sα generate the Weyl group W = W (R) of R, and S := {sα | α ∈ } is the collection of simple reflections. We have the affine Weyl group W aff = ZR  W and the extended (affine) Weyl group W e = X  W . Both can be considered the groups of affine transformations of a∗ . We denote the translation corresponding to x ∈ X by tx . As is well known,

On Completions of Hecke Algebras

213

W aff is a Coxeter group, and the basis  of R gives rise to a set S aff of simple (affine) reflections. More explicitly, let ∨M be the set of maximal elements of R∨ , with respect to the dominance ordering coming from . Then S aff = S ∪ {tα sα | α∨ ∈ ∨M }. The length function  of the Coxeter system (W aff , S aff ) extends naturally to W e . The elements of length zero form a subgroup  ⊂ W e and W e = W aff  . A complex parameter function for R is a map q : S aff → C× such that q(s) = q(s ) if s and s are conjugate in W e . This extends naturally to a map q : W e → C× which is 1 on  and satisfies q(ww ) = q(w)q(w ) if (ww ) = (w) + (w ). Equivalently (see [20, §3.1]), one can define q as a W -invariant function q : R ∪ {2α : α ∈ R, α∨ ∈ 2Y } → C× .

(6)

We speak of equal parameters if q(s) = q(s ) ∀s, s ∈ S aff and of positive parameters if q(s) ∈ R>0 ∀s ∈ S aff . We fix a square root q1/2 : S aff → C× . The affine Hecke algebra H = H(R, q) is the unique associative complex algebra with basis {Nw | w ∈ W e } and multiplication rules N if (ww ) = (w) + (w ) , ww  w Nw = N   1/2 −1/2 Ns − q(s) Ns + q(s) = 0 if s ∈ S aff .

(7)

In the literature, one also finds this algebra defined in terms of the elements q(s)1/2 Ns , in which case the multiplication can be described without square roots. This explains why q1/2 does not appear in the notation H(R, q). For q = 1 (7) just reflects the defining relations of W e , so H(R, 1) = C[W e ]. The set of dominant elements in X is X + = {x ∈ X : x , α∨  ≥ 0 ∀α ∈ }. The subset {Ntx : x ∈ X + } ⊂ H(R, q) is closed under multiplication and isomorphic to X + as a semigroup. For any x ∈ X we put , where x1 , x2 ∈ X + and x = x1 − x2 . θx = Ntx1 Nt−1 x 2

This does not depend on the choice of x1 and x2 , so θx ∈ H(R, q)× is well defined. We write T = HomZ (X , C× ), so that O(T ) ∼ = C[X ]. The Bernstein presentation of H(R, q) [20, §3] says that: • {θx : x ∈ X } forms a C-basis of a subalgebra of H(R, q) isomorphic to C[X ], which we identify with O(T ).

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• H(W, q) := C{Nw : w ∈ W } is a finite dimensional subalgebra of H(R, q) (known as the Iwahori–Hecke algebra of W ). • The multiplication map O(T ) ⊗ H(W, q) → H(R, q) is a C-linear bijection. • There are explicit cross relations between H(W, q) and O(T ), deformations of the standard action of W on O(T ). To define parabolic subalgebras of affine Hecke algebras, we associate some objects with any set of simple roots Q ⊂ . Let RQ be the root system they generate, R∨Q the root system generated by Q∨ , and WQ their Weyl group. We also define   XQ = X X ∩ (Q∨ )⊥ YQ = Y ∩ QQ∨ TQ = HomZ (XQ , C× ) aQ = YQ ⊗Z R RQ = (XQ , RQ , YQ , R∨Q , Q) HQ = H(RQ , qQ )

X Q = X /(X ∩ QQ), Y Q = Y ∩ Q⊥ , T Q = HomZ (X Q , C× ), aQ = Y Q ⊗Z R, RQ = (X , RQ , Y , R∨Q , Q), HQ = H(RQ , qQ ).

Here, qQ and qQ are derived from q via (6). Both HQ and HQ are called parabolic subalgebras of H. The quotient map X → XQ yields a natural projection HQ → HQ : θx Nw → θxQ Nw .

(8)

In this way, one can regard HQ as a “semisimple” quotient of HQ . The algebra HQ is embedded in H via the Bernstein presentation, as the image of O(T ) ⊗ H(WQ , q) → H. Any t ∈ T Q and any u ∈ T Q ∩ TQ give rise to algebra automorphisms ψu : HQ → HQ , θxQ Nw → u(xQ )θxQ Nw , ψt : HQ → HQ , θx Nw → t(x)θx Nw .

(9)

Let  be a finite group acting on R, i.e. it acts Z-linearly on X and preserves R and . We also assume that  acts on T by affine transformations, whose linear part comes from the action on X . Thus  acts on O(T ) ∼ = C[X ] by γ(θx ) = zγ (x)θγx ,

(10)

for some zγ ∈ T . We suppose throughout that q1/2 is -invariant, and that  acts on H(R, q) by the algebra automorphisms Ad(γ) :

 w∈W,x∈X

cx,w θx Nw →



cx,w zγ (x)θγ(x) Nγwγ −1 .

(11)

w∈W,x∈X

This being a group action, the multiplication relations in H(R, q) imply that we must have zγ ∈ T W . We build the crossed product algebra H(R, q)  .

(12)

On Completions of Hecke Algebras

215

In [39], we considered a slightly less general action of  on H(R, q), where the elements zγ ∈ T W from (10) were all equal to 1. But the relevant results from [39] do not rely on  fixing the unit element of T , so they are also valid for the actions as in (11). In this paper, we will tacitly use some results from [39] in the generality of (11). We note that nontrivial zγ ∈ T W are sometimes needed to describe Hecke algebras coming from p-adic groups, for example in [30, §4]. Since H(R, q) is of finite rank as a module over its commutative subalgebra O(T ), all irreducible H(R, q)-modules have finite dimension. The set of O(T )-weights of a H(R, q)-module V will be denoted by Wt(V ). We regard t = a ⊕ ia as the polar decomposition of t, with associated real part map  : t → a. The vector space t can be interpreted as the Lie algebra of the complex torus T = HomZ (X , C× ). The latter has a polar decomposition T = Trs × Tun , where Trs = HomZ (X , R>0 ) and Tun = HomZ (X , S 1 ). The polar decomposition of an element t ∈ T is written as t = |t| (t |t|−1 ). The exponential map exp : t → T becomes bijective when restricted to a → Trs . We denote its inverse by log : Trs → a. We write a+ = {μ ∈ a : α , μ ≥ 0 ∀α ∈ }, a∗+ = {x ∈ a∗ : x , α∨  ≥ 0 ∀α ∈ }, a− = {λ ∈ a : x , λ ≤ 0 ∀x ∈ a∗+ } = The interior a−− of a− equals otherwise. We write



α∈ λα α

∨

 α∈

 λα α ∨ : λα ≤ 0 .



: λα < 0 if  spans a∗ , and is empty

T − = exp(a− ) ⊂ Trs and T −− = exp(a−− ) ⊂ Trs . We say that a module V for H(R, q) (or for H(R, q)  ) is tempered if |Wt(V )| ⊂ T − , and that it is discrete series if |Wt(V )| ⊂ T −− . The latter is only possible if R spans a, for otherwise a−− and T −− are empty. We alleviate these notions by calling a H  -module essentially discrete series if its restriction to H is discrete series. Equivalently, essentially discrete series means that Wt(V ) ⊂ T −− Tun T  . Such a representation is tempered if and only if Wt(V ) ⊂ T −− Tun . We denote the set of (equivalence classes of) irreducible tempered essentially discrete series representations by Irr L2 (H(R, q)  ). It follows from the Bernstein presentation [20, §3] that the centre of H(R, q)   contains O(T )W  = O(T /W ),

(13)

with equality if W  acts faithfully on T . By Schur’s Lemma, O(T )W  acts on every irreducible H  -representation π by a character. Such a character can be identified with a W -orbit W t ⊂ T . We will just call W t the central character of π. Then W |t| ⊂ Trs and cc(π) := W  log |t| is a single W -orbit in a. We fix a

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W -invariant inner product on a and we define cc(π) = log |t| .

(14)

In some cases that we will encounter, the appropriate parabolic subalgebras of H(R, q)   are not H(RQ , qQ ), but H(RQ , qQ )  Q for some subgroup Q ⊂ . To make this work well, we need some assumptions on the groups Q for Q ⊂ . Condition 2.1 (1) Q ⊂ Q if Q ⊂ Q ; (2) the action of Q on T stabilizes TQ and T Q ; (3) Q acts on T Q by multiplication with elements of KQ . Notice that Q is a subgroup of (Q, Q) = {γ ∈  : γ(Q) = Q}, but that we do not require these two groups to be equal. We also note that Conditions 2.1 entail that ∅ acts trivially on O(T ) = H∅ , so Irr(H∅  ∅ ) ∼ = T × Irr(∅ ).

(15)

Remark 2.2 Often there is a larger root system R˜ ⊃ R in X , such that WQ Q is ˜ associated with R˜ ∩ QQ. Then parts (2) contained in the parabolic subgroup of W (R) and (3) of Condition 2.1 are automatically satisfied (and part (1) is usually obvious). Under Conditions 2.1 Q commutes with KQ . The conditions also entail that the projection HQ → HQ and the isomorphisms φt : HQ → HQ (t ∈ T Q ) are Q equivariant, so they extend to algebra homomorphisms HQ  Q → HQ  Q and φt : HQ  Q → HQ  Q (t ∈ T Q ).

(16)

Via the first map of (16), we can inflate any representation of HQ  Q to HQ  Q , which we often do tacitly. For any representation π of HQ  Q and any t ∈ T Q we write π ⊗ t = π ◦ φt ∈ Mod(HQ  Q ). Lemma 2.3 (a) Every irreducible HQ  Q -representation is of the form πQ ⊗ t Q for some πQ ∈ Irr(HQ  Q ) and t Q ∈ T Q . Q (b) πQ ⊗ t Q is tempered if and only if π is tempered and t Q ∈ Tun . Q (c) πQ ⊗ t is essentially discrete series if and only if πQ is discrete series. Proof. (a) First we consider the situation without Q . Let π ∈ Irr(HQ ) with central character WQ t ∈ T /WQ . The group WQ = W (RQ ) acts trivially on T Q , so WQ t = t Q WQ tQ for some t Q ∈ T Q , tQ ∈ TQ . Then π ⊗ (t Q )−1 factors through HQ → HQ (say as πQ ), and π = πQ ⊗ t Q .

On Completions of Hecke Algebras

217

To include Q , we use Clifford theory [27, Theorem A.6]. It says that every irreducible HQ  Q -representation is of the form HQ 

π  ρ := indHQ QQ,π (π ⊗ ρ). Here Q,π is the stabilizer of π ∈ Irr(HQ ) in Q and (ρ, Vρ ) is an irreducible representation of a twisted group algebra of Q,π . If O(T ) acts by t Q t1 on a vector subspace V1 ⊂ Vπ , then for γ ∈ Q it acts by the character γ −1 (t Q t1 ) on Nγ (V1 ⊗ Vρ ). By Condition 2.1(3) γ −1 (t Q t1 ) ∈ t Q KQ γ −1 (t1 ). Hence (π  ρ) ⊗ (t Q )−1 factors through HQ  Q → HQ  Q as πQ  ρ, and π  ρ = (πQ  ρ) ⊗ t Q . Q

(b) As T = T Q TQ with T Q ∩ TQ ⊂ Tun , there is a factorization Trs = Trs × TQ,rs and − . Also |Wt(π ⊗ t Q )| = |t Q | |Wt(π)|. These (with respect to RQ ) Trs− = {1} × TQ,rs observations imply the result. (c) This is obvious from Wt(π ⊗ t Q ) = t Q Wt(π).

2.1 The Schwartz and C ∗ -completions To get nice completions of H(R, q), we assume from now on that q is a positive parameter function for R. As a topological vector space, the Schwartz completion of H(R, q) will consist of rapidly decreasing functions on W e , with respect to a suitable length function N . For example, we can take a W -invariant norm on X ⊗Z R and put N (wtx ) = x for w ∈ W and x ∈ X . Then we can define, for n ∈ N, the following norm on H:   hw Nw = supw∈W e |hw |(N (w) + 1)n . pn e w∈W

The completion of H with respect to these norms is the Schwartz algebra S = S(R, q). It is known from [25, Sect. 6.2] that it is a Fréchet algebra. The -action on H extends continuously to S, so the crossed product algebra S(R, q)   is well defined. By [25, Lemma 2.20], a finite dimensional H  -representation is tempered if and only if it extends continuously to an S  -representation. We define a *-operation and a trace on H(R, q) by  w∈W

τ



c N e w w

w∈W e

∗ 

=

 w∈W e

cw Nw−1 ,

cw Nw = ce .

Since q(sα ) > 0, * preserves the relations (7) and defines an anti-involution of H(R, q). The set {Nw : w ∈ W e } is an orthonormal basis of H(R, q) for the inner product

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h1 , h2  = τ (h∗1 h2 ). This gives H(R, q) the structure of a Hilbert algebra. The Hilbert space completion L2 (R) of H(R, q) is a module over H(R, q), via left multiplication. Moreover, every h ∈ H(R, q) acts as a bounded linear operator [25, Lemma 2.3]. The reduced C ∗ algebra of H(R, q) [25, §2.4], denoted Cr∗ (R, q), is defined as the closure of H(R, q) in the algebra of bounded linear operators on L2 (R). By [25, Theorem 6.1] H(R, q) ⊂ S(R, q) ⊂ Cr∗ (R, q). As in (12), we can extend this to a C ∗ -algebra Cr∗ (R, q)  , provided that q is -invariant. Let us recall some background about Cr∗ (R, q)  , mainly from [25, 39]. It follows from [12, Corollary 5.7] that it is a finite type I C ∗ -algebra and that Irr(Cr∗ (R, q)) is precisely the tempered part of Irr(H(R, q)). According to [25, Theorem 4.23] all irreducible S(R, q)  -representations extend continuously to Cr∗ (R, q)  . Hence we can regard the representation theory of Cr∗ (R, q)   as the tempered unitary representation theory of H(R, q)  ). The structure of Cr∗ (R, q)   is described in terms of parabolically induced representations. As induction data, we use triples (Q, δ, t) where Q ⊂ , δ ∈ Irr L2 (HQ ) and t ∈ T Q . We regard two triples (Q, δ, t) and (Q , δ , t ) as equivalent if Q = Q , t = t , and δ ∼ = δ . Notice that HQ comes from a semisimple root datum, so it can have discrete series representations. We inflate such a representation to HQ via the projection (8). To a triple (Q, δ, t), we associate the H  -representation (δ ◦ ψt ). π  (Q, δ, t) = indH HQ

(17)

(When  = 1, we often suppress it from these and similar notations.) For t ∈ Q Tun = T Q ∩ Tun , these representations extend continuously to the respective C ∗ completions of the involved algebras. Let un be the set of triples (Q, δ, t) as above, such that, moreover, t ∈ Tun . Considering Q and δ as discrete variables, we regard un as a disjoint union of finitely many compact real tori (of different dimensions). Let V be the vector bundle over un , whose fibre at ξ = (Q, δ, t) is the vector space underlying π  (Q, δ, t). That vector space is independent of t, so the vector  bundle is trivial. Let End(V ) be the algebra bundle with fibres EndC π  (Q, δ, t) . These data give rise to a canonical map   H(R, q)   → O ; End(V )  h → ξ → π  (ξ)(h)

(18)

which we refer to as the Fourier transform. By [25, Lemma 2.22], every discrete series representation is unitary, so Vδ carries an HQ -invariant inner product and EndC (Vδ ) has a natural *-operation. For any t ∈ T Q , this becomes an HQ -invariant nondegenerate pairing between δ ◦ φt and δ ◦ φt|t|−2 . By [25, Proposition 4.19], this extends canonically to an inner product on the vector space

On Completions of Hecke Algebras

219

π  (Q, δ, t) =   H(W, q) ⊗H(WQ ,q) Vδ .

(19)

That yields an anti-involution on EndC (π  (Q, δ, t)) and a nondegenerate H   −2 invariant pairing between π  (Q,  δ, t) and π (Q, δ, t |t| ).   The algebra O ; End(V ) is endowed with the anti-involution (f ∗ )(Q, δ, t) = f (Q, δ, t |t|−2 )∗ .

(20)

With respect to this anti-involution, (18) is a *-homomorphism. To administer the upcoming intertwining operators, we use a finite groupoid G which acts on End(V ). It is made from the elements of W   and of KQ := TQ ∩ T Q . More precisely, its base space is the power set of , and for Q, Q ⊆  the collection of arrows from Q to Q is GQQ = {(g, u) : g ∈   W, u ∈ KQ , g(Q) = Q }.

(21)

Whenever it is defined, the multiplication in G is (g , u ) · (g, u) = (g g, g −1 (u )u). In particular, writing W (Q, Q) = {w ∈ W  : w(Q) = Q}, we have the group GQQ = W (Q, Q)  KQ .

(22)

Usually, we will write elements of G simply as gu. There is an analogous groupoid Q G Q for HQ  Q , which under Conditions 2.1 satisfies GQQ = Q × KQ . For γ ∈ W with γ(Q) = Q ⊂ , there are algebra isomorphisms ψγ : HQ → HQ , θxQ Nw → θγ(xQ ) Nγwγ −1 ,

ψγ : HQ → HQ , θx Nw → θγx Nγwγ −1 .

(23)

The groupoid G acts from the left on un by (g, u) · (Q, δ, t) := (g(Q), δ ◦ ψu−1 ◦ ψg−1 , g(ut)),

(24)

the action being defined if and only if g(Q) ⊂ . Suppose that g(Q) = Q ⊂  and δ ∼ = δ ◦ ψu−1 ◦ ψg−1 . By [25, Theorem 4.33] and [39, Theorem 3.1.5], there exists an intertwining operator   π  (gu, Q, δ, t) ∈ HomH(R,q) π  (Q, δ, t), π  (Q , δ , g(ut)) , Q

(25)

which depends algebraically on t ∈ Tun . This implies that, for all ξ ∈  and g ∈ G such that gξ is defined, π  (ξ) and π  (gξ) have the same irreducible constituents, counted with multiplicity [39, Lemma 3.1.7].

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M. Solleveld

The action of G on the continuous sections C(un ; End(V )) is given by (g · f )(gξ) = π  (g, ξ)f (ξ)π  (g, ξ)−1

g ∈ GQQ , ξ = (Q, δ, t).

(26)

The next result is the Plancherel isomorphism for affine Hecke algebras, proven in [12, Theorem 5.3 and Corollary 5.7] and [39, Theorem 3.2.2]. Theorem 2.4 The Fourier transform (18) induces *-homomorphisms G  H(R, q)   → O ; End(V ) ,  G S(R, q)   → C ∞ un ; End(V ) ,  G Cr∗ (R, q)   → C un ; End(V ) . The first is injective, the second is an isomorphism of Fréchet algebras, and the third is an isomorphism of C ∗ -algebras. For q = 1, these simplify to the well-known isomorphisms  W  H(R, 1)   = O(T )  W  → O T ; EndC (C[W ]) ,  W  ∞ ∞ , S(R, 1)   = C (Tun )  W  → C Tun ; EndC (C[W ])  W  . Cr∗ (R, 1)   = C(Tun )  W  → C Tun ; EndC (C[W ])

(27)

Unfortunately, the bookkeeping in Theorem 2.4 is not entirely suitable for our purposes, because sometimes the parabolic subalgebras need to be extended by diagram automorphisms. In those cases, we should rather use induction data based on Irr L2 (HQ  Q ) than based on Irr L2 (HQ ) or Irr L2 (HQ ). We fix a system of subgroups Q ⊂  (Q ⊂ ), satisfying Condition 2.1. With Lemma 2.3 in mind, we define new induction data. They are triples (Q, σ, t) with Q ⊂ , t ∈ T Q , and σ ∈ Irr L2 (HQ  Q ). We regard another such triple (Q , σ , t ) as equivalent if and only if Q = Q, t = t, and σ ∼ = σ. We keep the same groupoid G as before, and it also acts on the new triples via (24). To such a triple, we associate the representation (σ ⊗ t). (28) π(Q, σ, t) = indH HQ Q The vector space underlying π(Q, σ, t) does not depend on t, we denote it by VQ,σ . There is a natural homomorphism Q H(R, q)   → O(T ) ⊗ EndC (VQ,σ )  h → t → π(Q, σ, t)(h) .

(29)

We refer to the system of these maps, for all Q and σ, as the Fourier transform for H(R, q)  . The recipe for the intertwining operators from [25, §4] and [39, Theorem 3.1.5] remains valid, so we get

On Completions of Hecke Algebras

221

  π(gu, Q, σ, t) ∈ HomH π(Q, σ, t), π(g(Q), σ , g(ut))

(30)

with the same properties as in (25). In particular, π(Q, σ, t) and π(g(Q), σ , g(ut)) have the same irreducible constituents, counted with multiplicity. With these notions, we can vary on the Plancherel isomorphism (Theorem 2.4). To do so, we first consider essentially discrete series representations of HQ  HQ  Q Q . Pick δ1 ∈ Irr L2 (HQ ) and t1 ∈ Tun . We note that indHQ Q (δ ⊗ t1 ) is unitary Q and essentially discrete series, because Q stabilizes Q. Write GQQ δ1 = {δi }i . The   G Q associated with (Q, δ1 ) is summand of C ∞ Q,un ; End(VQQ )

 i

Q  Q  GQQ C ∞ Tun ; EndC (π Q (Q, δi , ti )) .

(31) H Q

Let {σj }j be the members of Irr L2 (HQ  Q ) contained in indHQQ

(δ1 ◦ ψu )

Q GQQ

= Q × KQ . The summand of for some u ∈ KQ . This set is stable under

G Q   Q ∞ Tun ; EndC (Vσ ) corresponding to the σj is σC

 j

Q  Q  GQQ C ∞ Tun ; EndC (Vσj ) .

(32)

Lemma 2.5 The algebras (31) and (32) are naturally isomorphic. Proof. For σ ∈ Irr L2 (HQ  Q ) we write (σ, t) > (δ1 , t1 ) if   HQ  HomHQ Q σ ⊗ t, indHQ Q (δ ⊗ t1 ) ∼ = HomHQ (σ ⊗ t, δ ⊗ t1 ) is nonzero. Since Q is finite, the set of such (σ, t) is finite. Hence the map 



σ ◦ ψt : H Q  Q →

(σ,t)>(δ1 ,t1 )

EndC (Vσ )

(σ,t)>(δ1 ,t1 ) Q

is surjective. The specialization of (31) at GQQ (Q, δ1 , t1 ) is also EndC (Vσ ), for that specialization is just HQ Q

indHQ

(σ,t)>(δ1 ,t1 )

(δ ◦ ψt1 )(S(RQ , qQ )  Q ).

Similarly,

specializing the algebra (32) at all (σ, t) > (δ1 , t1 ) gives a surjection from (32) to (σ,t)>(δ1 ,t1 ) EndC (Vσ ). Now we can explicitly compare (32) with (31). Both are algebras of smooth sections of (trivial) algebra bundles, and specialization at the points associated with Q (δ1 , t1 ) yields the same algebra in both cases. This holds for any t1 ∈ Tun and that accounts for all base points of these algebra bundles, so (31) and (32) are isomorphic.

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M. Solleveld

Moreover, the isomorphism is canonical: It is the composition of the inverse of the  map in Theorem 2.4 and the map induced by (29) (both for HQ  Q ). Next we formulate our variation on the Plancherel isomorphism. Proposition 2.6 The Fourier transform from (29) induces isomorphisms of Fréchet *-algebras

 G  Q ∞ S(R, q)   → , Tun ; EndC (VQ,σ ) Q,σ C

G   Q Cr∗ (R, q)   → . Q,σ C Tun ; End C (VQ,σ ) Proof. We will analyse the right-hand side of Theorem 2.4 (for the Schwartz algebras) and compare it with the current setting. For every Q ⊂ , Lemma 2.5 yields a canonical isomorphism

 σ∈Irr L2 (HQ Q )

Q  Q  GQQ ∼ C ∞ Tun ; EndC (Vσ ) =

 δ∈Irr L2 (HQ )

Q  GQQ  Q C ∞ Tun ; EndC (C[Q ] ⊗ Vδ ) . (33)

To obtain the right-hand side of Theorem 2.4 from (33), we must apply indH to HQ Q

H Q (δ ⊗ t) and then take invariants with respect to the larger C[Q ] ⊗ Vδ ∼ = indHQ groupoid G ⊃ G Q . The formula [39, (3.12)] is the same for G Q and for G, so the intertwiners associated with elements of G Q need not be adjusted in this process. With exactly the same procedure, we can turn (33) into the right-hand side of the current proposition. The intertwining operators associated with elements of G agree , because in both under the isomorphisms obtained from (33) by applying indH HQ Q settings they were constructed with [39, (3.12) and Theorem 3.1.5]. Consequently Q

 Q,σ

 G G  Q  ∼ C ∞ Tun ; EndC (VQ,σ ) = C ∞ un ; End(V ) ,

proving the proposition for the Schwartz algebras. For Cr∗ (R, q)  , one can use  the same argument, with everywhere C ∞ replaced by continuous functions. Choose representatives Q for P() modulo W -association. For every such Q, we choose representatives σ for the action of GQQ = W (Q, Q) × KQ on Irr L2 (HQ  Q Q ). By Lemma 2.3, these σ also form representatives for the action of GQQ  Tun on Q Irr L2 (H  Q ). We denote the resulting set of representatives of pairs by (Q, σ)/G. Q Let GQ,σ be the setwise stabilizer of (Q, σ, Tun ) in the group GQQ . Proposition 2.6 can be rephrased as isomorphisms G  Q

S(R, q)   → (Q,σ)/G C ∞ Tun ; EndC (VQ,σ ) Q,σ ,  Q GQ,σ

. Cr∗ (R, q)   → (Q,σ)/G C Tun ; End C (VQ,σ )

(34)

On Completions of Hecke Algebras

223

Sometimes we have to consider the opposite algebra (H(R, q)  )op and its completions. It is, morally, clear that all the previous results can also be developed for right H  -modules, that is, for (H  )op -modules. However, none of that has been written down, so we prefer more steady ground. For every H  -representation (π, Vπ ), the full linear dual Vπ∗ becomes a (H  op ) -representation π ∗ by π ∗ (hop )λ = λ ◦ π(h). This sets up a bijection between finite dimensional left and right modules of H  . In view of the canonical inner products from on the spaces (19), this bijection commutes with induction from parabolic subalgebras. For infinite dimensional representations, there is often some choice for which dual space of Vπ we use here. In particular, when Vπ is a Hilbert space we can use Vπ also as dual space. With this convention, one checks easily that π is unitary if and only if π ∗ is unitary. The O(T )-weights of π ∗ are the same as for π, so π ∗ is tempered or (essentially) discrete series if and only if π is so. Thus the pairs (Q, σ) with σ ∈ Irr L2 (HQ  Q ) are in natural bijection with the pairs (Q, σ ∗ ) in  Q⊂

  Irr L2 (H(RQ , qQ )  Q )op .

(35)

The bijection is G-equivariant for the G-action on (35) as in (24). Hence GQ,σ = GQ,σ∗ , and we can take (Q, σ ∗ )/G to be the image of (Q, σ)/G. Lemma 2.7 The Fourier transform for right H(R, q)  -modules induces isomorphisms of Fréchet *-algebras  Q G ∗

(S(R, q)  )op → (Q,σ∗ )/G C ∞ Tun ; EndC (VQ,σ∗ ) Q,σ ,  

GQ,σ∗ Q . (Cr∗ (R, q)  )op → (Q,σ ∗ )/G C Tun ; End C (VQ,σ ∗ ) Proof. The opposite algebra of   EndC (VQ,σ ) = EndC indH V HQ Q δ  ∗  , which by [25, Proposition 4.19] is canonically is naturally isomorphic to EndC VQ,σ isomorphic with   (Vσ∗ ) = EndC (VQ,σ∗ ). EndC indH HQ Q For g ∈ GQ,σ , we take π(g, Q, σ ∗ , t|t|−2 ) to be the transpose inverse of π(g, Q, σ, t).  Q  Thus an element of C Tun ; EndC (VQ,σ ) is GQ,σ -invariant if and only if its transpose   Q in C Tun ; EndC (VQ,σ∗ ) is GQ,σ∗ -invariant for the action (g · f )(g(Q, σ ∗ , t)) = π(g, Q, σ ∗ , t)f (Q, σ ∗ , t)π(g, Q, σ ∗ , t)−1 .

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Now we take the opposite algebras in Theorem 2.4 and we find the desired isomorphisms. The implementing algebra homomorphisms are given by transpose, the Fourier transform from (29) and again transpose, which works out to the Fourier transform for (H(R, q)  )op -modules. Since the correspondence between left and right H(R, q)  -modules preserves unitarity, the latter Fourier transform is still a *homomorphism. 

2.2 The Space of Irreducible Representations We compare the irreducible representations of H   to its representations induced from proper parabolic subalgebras (i.e., the algebras HQ  Q with Q  ). Let Gr(H  ) be the Grothendieck group of the category of finite length H  representations and write Gr Q (H  ) = Q ⊗Z Gr(H  ). Then (parabolic) induction induces a Q-linear map Gr Q (HQ  Q ) → Gr Q (H  ). Theorem 2.8 (a) The collection of irreducible H  -representations whose image in Gr Q (H  ) is not a Q-linear combination of representations induced from proper parabolic subalgebras is a nonempty union of T W  -orbits in Irr(H  ).  (b) Suppose that, for every w ∈ W  \ Q W (RQ )Q , the set Tw is finite. Then the set in part (a) is a finite union of T W  -orbits. Proof. (a) Recall from (13) that O(T )W (RQ )Q ⊂ Z(HQ  Q ). Hence all irreducible representations of HQ  Q come in families parametrized by T W (RQ )Q . Since T W  ⊂ T W (RQ )Q for all Q ⊂ , the set of irreducible representations under consideration is a union of T W  -orbits. By [40, Lemma 2.3], Irr(H  ) can be parametrized by the extended quotient T //W  =

 w∈W 

{w} × T w



W ,

where W  acts on the union by w · (w, t) = (w ww −1 , w (t)). This parametrization respects central characters, up to a twists which are constant on connected components of T //W  [40, Theorem 2.6]. In this parametrization of Irr(H  ), almost all elements of a piece {w} × T w with w ∈ W (RQ )Q come from representations induced from HQ  Q , and such w account for all representations induced from proper parabolic subalgebras. Hence the set considered in the statement can be parametrized by 

 (36) {w} × T w W  w∈W :w ∈W / (RQ )Q ∀Q

This set is nonempty because every Coxeter element of W = W (R) contributes at least (w, 1) to it. (b) This is obvious from (36). 

On Completions of Hecke Algebras

225

The induction data from  give rise to a partition Irr(H  ) into finite packets. Theorem 2.9 [39, Theorem 3.3.2(b)] For every π ∈ Irr(H  ) there exists a unique G-association class G(Q, δ, t) ∈ /G such that π is a constituent of π  (Q, δ, t) and the invariant cc(δ) from (14) is maximal for this property. With the new induction data (Q, σ, t) from (28), we can vary on Theorem 2.9. (Now σ ∈ Irr L2 (HQ  Q ), whereas the above δ was a representation of HQ .) Theorem 2.10 (a) For every π ∈ Irr(H  ) there exists a triple (Q, σ, t) as above, such that π is a constituent of π(Q, σ, t) and cc(σ) is maximal for this property. In this situation we say that π is a Langlands constituent of π(Q, σ, t). (b) In the setting of part (a), the restriction of σ ⊗ t to HQ is a direct sum of irreducible representations in one Q -orbit, say Q (δ ⊗ t) ⊂ Irr L2 (HQ ). Then (Q, δ, t) is uniquely determined by π, up to the action of G. (c) Let (Q, σ, t) be any induction datum as in (28). Every constituent of π(Q, σ, t) is either a Langlands constituent or a constituent of π(Q , σ , t ) for some induction  

  datum with cc(σ ) > cc(σ). (d) π is tempered if and only if t ∈ Tun , where t ∈ T Q comes from part (a). Proof. (a) Let (Q, δ, t) be as in Theorem 2.9. Thus π is a constituent of  HQ Q  π  (Q, δ, t) = indH indHQ (δ ⊗ t) , HQ Q and the norm of the central character of δ is maximal for this property. Let T −−Q be the HQ  subset T −− of T , but computed with respect to Q. Every A-weight of indHQ Q (δ ⊗ t) lies in one of the Q -orbits of weights of δ ⊗ t, which are entirely contained in HQ 

T −−Q Tun because Q stabilizes Q. In other words, indHQ Q (δ ⊗ t) is a direct sum of finitely many irreducible essentially discrete series representations of HQ  , all with central characters in the same WQ  Q -orbit. By Lemma 2.3, all these summands are of the form σi ⊗ t with σi ∈ Irr L2 (HQ  Q ). Hence π  (Q, δ, t) =

 i

π(Q, σi , t)

(37)

and π is a constituent of (at least) one π(Q, σi , t). The central characters of the σi and of δ have the same norm, and that of δ was maximal given π. Hence the norm of the central character of σi is also maximal, given π. (b) Suppose that (Q, σ, t) satisfies the requirements of part (a), that is, π is a Langlands constituent of π(Q, σ, t). In the proof of Lemma 2.3, we observed that the restriction of σ ⊗ t to HQ is a direct sum of representations of the form γ((δ ⊗ k −1 ) ⊗ kt) with γ ∈ Q and k ∈ KQ . For all γ ∈ Q , π is a constituent of π  (Q, δ, t) = π  (Q, γ(δ ⊗ k −1 ), γ(kt)).

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The norm of the central character of γ(δ ⊗ k −1 ) is maximal for this property, by the assumption on σ. By Theorem 2.9, all (Q, γ(δ ⊗ k −1 ), γ(kt)) lie in a unique G-association class in  determined by π. (c) In view of (37), it suffices to consider the constituents of π  (Q, δ, t). Then the statement is implicit in [39], we make it more explicit here. By [39, Lemma 3.1.7 and Theorem 3.3.2], we may, furthermore, assume that ξ = (Q, δ, t) is in positive position, that is, |t| ∈ T Q+ = exp(aQ ∩ a+ ). Write P(ξ) = {α ∈  : |α(t)| = 1} and HP(ξ) (P(ξ),P(ξ)) consider the representation indHQ (δ ⊗ t). By [39, Proposition 3.1.4] that representation is completely reducible, and all itsirreducible summands are of  the form δ ⊗ t where δ ∈ Irr HP(ξ)  (P(ξ), P(ξ)) is tempered and t ∈ T P(ξ) ∩ tTP(ξ) with   |t | ∈ T P(ξ)++ = exp {μ ∈ aP(ξ) : α , μ > 0 ∀α ∈  \ P(ξ)} . Thus every constituent π(Q, σ, t) is a constituent of (δ ⊗ t ) indH HP(ξ) (P(ξ),P(ξ))

(38)

for such a δ ⊗ t . By [39, Theorem 3.3.2], the Langlands constituents of π(Q, σ, t) (or π  (Q, δ, t)) are precisely those constitutents which are quotients of a representation (38). The Langlands classification for extended affine Hecke algebras [39, Corollary 2.2.5] says that the latter representation has a unique irreducible quotient (called the Langlands quotient, hence our terminology Langlands constituents). Moreover, by [39, Lemma 2.2.6(b)], all other constituents of (38) are Langlands quotients for data where the norm of the central character is bigger than cc(σ). Then [39, Proposition 3.1.4 and Theorem 3.3.2] entail that those otherirreducible representations occur as  Langlands constituents of a π  (Q , δ , t ) with cc(δ ) > cc(σ) = cc(δ). (d) As observed in the proof of part (c), in the construction for part (a), and for [39, Theorem 3.3.2], we may assume that (Q, σ, t) is positive and that π is the unique Langlands quotient of (38), for one of the above δ ⊗ t . By the uniqueness in the Langlands classification for extended affine Hecke algebras [39, Corollary 2.2.5], π  . is tempered if and only if P(ξ) =  and t ∈ Tun

  says that |t| = 1. ConHere P(ξ) =  implies |t| = |t | ∈ T , and then t ∈ Tun

  versely, t ∈ Tun implies P(ξ) =  and t = t ∈ T . With Theorem 2.10, one can express the structure of Irr(H  ) in terms of its subset of irreducible tempered representations. In essence, the former is the complexification of the latter (which is something like a real algebraic variety with multiplicities). We do not need the full strength of this. For our purposes, it suffices to consider half-lines in the parameter space, such that Q, σ and the unitary part of t are fixed and the absolute value of t can be scaled by a positive factor. Proposition 2.11 Let (Q, σ, t) be an induction datum with σ ∈ Irr L2 (HQ  Q ). (a) For r ∈ R>−1 , the number of inequivalent Langlands constituents of π(Q, σ, t |t|r ) does not depend on r.

On Completions of Hecke Algebras

227

(b) For all but finitely many r ∈ R>−1 , π(Q, σ, t |t|r ) is completely reducible. Then all its irreducible subquotients are Langlands constituents. Proof. (a) Let ξ = (Q, δ, t) ∈  be as in Theorem 2.10. Notice that the intertwining operators π(gu, Q, σ, t) depend algebraically on t ∈ T Q . This implies that, for every gu ∈ G, π(Q, σ, t) and π(gu(Q, σ, t)) have the same irreducible subquotients (counted with multiplicity), see [39, Lemma 3.1.7] or [38, Lemma 3.4]. Since every G-orbit in  contains an element in positive position, we may assume that (Q, δ, t) is positive. Then (Q, δ, t |t|r ) is positive for r ≥ −1 and for r > −1 its stabilizer in G does not depend on r. Now the statement for π  (Q, δ, t) is an instance of [39, Proposition 3.4.1]. Together with (37) and Theorem 2.10(c), this implies the statement for π(Q, σ, t). (b) By [39, Proposition 3.1.4(a)], the representation HP(ξ) (P(ξ),P(ξ))

indHQ

(δ ⊗ t |t|r ) =



HP(ξ) (P(ξ),P(ξ))

i

indHQ Q

(σi ⊗ t |t|r )

(39)

is completely reducible. Therefore, it suffices to consider all irreducible direct summands of (39) separately. This brings us to representations of the form (δ ⊗ t |t|r ). indH HP(ξ) (P(ξ),P(ξ))

(40)

Then (P(ξ), δ , t |t|r ) is a datum for the Langlands classification for extended affine Hecke algebras [39, Corollary 2.2.5]. This result says that such a representation has a unique irreducible quotient, so (40) is irreducible as soon as it is completely reducible. In that case, its Langlands constituent obviously is the whole of (40). That implies the claim about the constituents of π(Q, σ, t |t|r ) when that representation is completely reducible. Next we show that (40) is irreducible for almost all r ∈ R>−1 . From [39, Lemma 2.2.6(a)] we know that the space of H  -endomorphisms of (40) is just CId. For all z ∈ C, the isotropy group G(P(ξ),δ ,t ) also fixes (P(ξ), δ , t |t|z ). Since GP(ξ),δ acts on T Q by group automorphisms and translations, G(P(ξ),δ ,t |t|z ) equals G(P(ξ),δ ,t ) for almost all z ∈ C. In particular, this happens for some z ∈ −1 + iR. Then t |t|z ∈ P(ξ) Tun and by [39, Corollary 3.1.3] the representation π  (P(ξ), δ , t |t|z ) is unitary (and in particular completely reducible). By [39, Theorem 3.3.1(b)]   (δ ⊗ t |t|z ) EndH indH HP(ξ) (P(ξ),P(ξ))

(41)

is spanned by intertwining operators coming from GP(ξ),δ ,t , just like for (40). Therefore, (41) consists only of CId, which implies that the representation is irreducible for that z ∈ −1 + iR. In the above algebraic family of finite dimensional representations parametrized by z ∈ C, irreducibility is an open condition: Slightly varying z cannot destroy irreducibility. Hence the locus of z’s where the representation is reducibile is a Zariski-closed subset of C, that is, it is finite. In particular, (40) is irreducible for  all but finitely many r ∈ R>−1 .

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3 Reductive p-adic Groups Let F be a non-Archimedean local field and let G be the group of F-points of a connected reductive algebraic group defined over F. We endow G with the topology coming from the metric on F and we fix a Haar measure. Let H(G) be the Hecke algebra of G, the vector space of locally constant compactly supported functions f : G → C endowed with the convolution product (with respect to the Haar measure). Let S(G) be the Harish-Chandra–Schwartz algebra of G, as defined in [14] and [43, §III.6]. By definition, a smooth G-representation (by default on a complex vector space) is tempered if and only if it extends continuously to a module for S(G). Let Cr∗ (G) be the reduced C ∗ -algebra of G, the completion of H(G) in the algebra of bounded linear operators on the Hilbert space L2 (G). By [42, Theorem 29], there are dense inclusions H(G) ⊂ S(G) ⊂ Cr∗ (G). The *-operation and the trace on these algebras are f ∗ (g) = f (g −1 ) and τ (f ) = f (1G ). Fix a minimal parabolic F-subgroup P0 = M0 U0 and let W (G, M0 ) be the Weyl group of G with respect to the maximal F-split torus A0 in the centre of M0 . Write a0 = X ∗ (AM0 ) ⊗Z R and endow this vector space with a W (G, M0 )-invariant inner product. Every Levi F-subgroup of G is conjugate to a standard Levi subgroup, that is, one that contains M0 . Let M be such a standard Levi subgroup of G, and let AM be the maximal F-split torus in Z(M ). There is a canonical decomposition a0 = aM ⊕ a M ,

(42)

where aM = X ∗ (AM ) ⊗Z R and aM = {χ ∈ X ∗ (A0 ) : χ|AM = 1} ⊗Z R. Let R(G, M ) be the set of roots of G with respect to AM . For a parabolic subgroup P = M U of G, we let R(P, M ) be the set of roots of (G, M ) that appear in (the Lie algebra of) P. M When M1 ⊂ M is another standard Levi subgroup of G, we write aM M1 = aM1 ∩ a . M Every parabolic subgroup P1 = M1 U1 of M determines an obtuse cone in aM1 : + M aP1

=

 α∈R(P1 ,M1 )

  cα αAM : cα > 0 ∀α . 1

Here we would obtain the same cone if we used only the simple roots for (P1 , M1 ). M The closure of + aM P1 in aM1 is + aM P1

=

 α∈R(P1 ,M1 )

  cα αAM : cα ≥ 0 ∀α . 1

It is easy to see that the normalized Jacquet restriction functor

On Completions of Hecke Algebras

229

JPG : Rep(G) → Rep(M ) does not preserve temperedness. Fortunately, the normalized parabolic induction functor IPG : Rep(M ) → Rep(G) does and also respects non-temperedness: Proposition 3.1 Let π ∈ Rep(M ) be of finite length. Then IPG (π) is tempered if and only if π is tempered. Proof. The if-part is well known, see [43, Lemme III.2.3] or [28, Lemme VII.2.2]. By conjugating P, M and π, we can achieve that P ⊃ P0 and M ⊃ M0 . Recall from [43, Proposition III.2.2] that π is tempered if and only if, for every parabolic subgroup P1 = M1 U1 of M with M1 ⊃ M0 and every AM1 -weight χ of JPM1 (π): log |χ| ∈ + aM P1 .

(43)

Moreover, it is equivalent to impose this condition for all P1 such that P1 = M or P1 is a standard maximal parabolic subgroup of M . To show that IPG preserves non-temperedness, it suffices to consider the case that P is a standard maximal parabolic subgroup of G. Namely, there exists a chain of parabolic subgroups P ⊂ P1 = M1 U1 ⊂ · · · ⊂ Pn = Mn Un ⊂ G = Mn+1 such that every Pi−1 ∩ Mi is a maximal parabolic subgroup of Mi . If we can prove i that each IPMi−1 ∩Mi preserves non-temperedness, the transitivity of parabolic induction [28, Lemme VI.1.4] implies that IPG does so as well. Since IPG is an exact functor [28, Théorème VI.1.1], we may, furthermore, assume that π is irreducible. So, we suppose that π is irreducible and not tempered, and (contrary to what we want to prove) that IPG (π) is tempered. Let us consider the Z(G)-character of π. It is also the Z(G)-character of IPG (π). Since IPG (π) is tempered, its central character is unitary [28, Corollaire VII.2.6]. We claim that the AM -character ζ of π must also be unitary. Suppose it is not, and consider its absolute value |ζ| ∈ Xnr (M ) \ {1}. Let α ∈ R(G, M0 ) be the unique simple root of (G, M ) and let sα ∈ W (G, M0 ) be the associated reflection. The length of sα in W (G, M0 ) is one, so it is a minimal length representative for a double coset in W (M , M0 ) \ W (G, M0 )\W (M , M0 ). By the Geometric Lemma [28, Théorème VI.5.1], both ζ and sα ζ occur as AM -weights of JPG (IPG (π)). Since M is a maximal Levi subgroup of G and |ζ|AG = |ζ|Z(G) = 1, both G . The reflection sα log |ζ| and log |sα ζ| lie in the one-dimensional vector space aM G −1 acts as −1 on aM , so |sα ζ| = |ζ| . As |ζ| = 1, it is not possible that both log |ζ| and log |sα ζ| lie in the cone + aPG . From (43), we see that this contradicts the temperedness of IPG (π). Consequently, |ζ| must be 1, and ζ must be unitary. Now we invoke the non-temperedness of π. By [43, Proposition III.2.2(iii)], there exists a standard parabolic subgroup P = M U of G such that: • M = M or M is a maximal Levi subgroup of M ;

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• JPM ∩M (π) has an AM -weight χ with log |χ| ∈ aM not in + aM P ∩M . When M = M , then χ = ζ and the above claim says that |χ| = 1. That would be in contradiction with the second bullet. Hence M  M . As χ|AM = ζAM is unitary, χ is of the form cβ β, where β ∈ R(M , M0 ) is the unique simple root for (M , M ). Then the second bullet says that cβ < 0. By the Geometric Lemma [28, Théorème VI.5.1] χ also an AM -weight / + aPG . of JPG (IPG (π)). As β ∈ + aPG (a potentially larger cone than + aM P ∩M ), cβ β ∈ G  Together with (43), this shows that IP (π) cannot be tempered. Let L = L(F) be a Levi subgroup G and let (σ, Vσ ) ∈ Irr(L) be an irreducible tempered supercuspidal L-representation. Let Xnr (L) be the group of unramified characters L → C× and let Xunr (L) be the subgroup of unitary unramified characters. Recall that the inertial equivalence class of the pair (L, σ) consists of all pairs of the form (gLg −1 , (g · σ) ⊗ χ), where g ∈ G and χ ∈ Xnr (gLg −1 ). We write s = [L, σ]G and call this an inertial equivalence class for G. It gives rise to a subset Irr(G)s ⊂ Irr(G), namely all those irreducible smooth G-representations whose supercuspidal support lies in s. This in turn is used to define a subcategory Rep(G)s of Rep(G), namely those smooth G-representations all whose irreducible constituents lie in Irr(G)s . The Bernstein blocks Rep(G)s have better finiteness properties than Rep(G): Theorem 3.2 [18, §1.6] (a) Let M ⊂ G be a Levi subgroup containing L. There exist tempered πM ,i ∈ Irr(M )[L,σ]M (i = 1, . . . , κM ), such that {πM ,i ⊗ χM : i = 1, . . . , κM , χM ∈ Xunr (M )} is the collection of irreducible tempered representations in Rep(M )[L,σ]M that are not isomorphic to the normalized parabolic induction of representation of a proper Levi subgroup of M . (b) Let M run through a set of representatives for the conjugacy classes of Levi subgroups of G containing L. Then the set  M

{IPG (πM ,i ⊗ χM ) : i = 1, . . . , κM , χM ∈ Xnr (M )}

spans the Grothendieck group of the category of finite length representations in Rep(G)s . Let B(G) be the set of all inertial equivalence classes s for G. By [5, Corollaire 3.9], it is countably infinite (unless G = 1). The Bernstein decomposition [5, Theorem 2.10] says that

On Completions of Hecke Algebras

 Rep(G) = s∈B(G) Rep(G)s , H(G) = s∈B(G) H(G)s ,

231

(44)

where H(G)s is the largest two-sided ideal of H(G) which annihilates Rep(G)s for all s = s. Alternatively, it is characterized by H(G)s · V = V for all V ∈ Rep(G)s . Let S(G)s (resp. Cr∗ (G)s ) be the two-sided ideal of S(G) (resp. Cr∗ (G)) generated by H(G)s . Upon completion, (44) yields further Bernstein decompositions

S(G) = s∈B(G) S(G)s , Cr∗ (G) = s∈B(G) Cr∗ (G)s .

(45)

The latter must be interpreted as a direct sum in the Banach algebra sense: It is the completion of the algebraic direct sum with respect to the operator norm of Cr∗ (G). For a compact open subgroup K of G, we let K be the corresponding idempotent of H(G). Then H(G, K) := KH(G) K (46) is the subalgebra of K-bi-invariant functions in H(G). We define S(G, K) and Cr∗ (G, K) analogously. For every compact open subgroup K of G, S(G, K) is a Fréchet algebra [42, Theorem 29]. The Schwartz algebra S(G) is their union (over all possible K), so it is an inductive limit of Fréchet algebras. We will focus on one Bernstein block Rep(G)s of Rep(G). By [5, Corollaire 3.9], there exists a compact open subgroup Ks of G such that every representation in Rep(G)s is generated by its Ks -fixed vectors. This leads to Morita equivalences H(G)s ∼M H(G, Ks )s := H(G)s ∩ H(G, Ks ) S(G)s ∼M S(G, Ks )s := S(G)s ∩ S(G, Ks ) Cr∗ (G)s ∼M Cr∗ (G, Ks )s := Cr∗ (G)s ∩ Cr∗ (G, Ks ).

(47)

3.1 The Plancherel Isomorphism We will describe the structure of S(G)s and S(G, Ks )s in more detail. Let [L, σ]L = Ts ⊂ Irr(L) be the set of isomorphism classes of L-representations of the form σ ⊗ χ with χ ∈ Xnr (L). Thus there is a finite covering of complex varieties Xnr (L) → Ts : χ → σ ⊗ χ.

(48)

Let Ts,un be the subset of unitary representations in Ts , it is covered by Xunr (L) via (48). We write Xnr (L, σ) = {χ ∈ Xnr (L) : σ ⊗ χ ∼ = σ}. This is a finite subgroup of Xunr (L). The map (48) induces an isomorphism of algebraic varieties Xnr (L)/Xnr (L, σ) → Ts .

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The group W (G, L) = NG (L)/L acts on Irr(L) by (gL · π)(l) = π(glg −1 ).

(49)

(The representation gL · π is only determined up to isomorphism.) This action stabilizes Xnr (L), the unitary representations in Irr(L), and the supercuspidal Lrepresentations. Let Ws be the stabilizer of Ts in NG (L)/L. This group will play the same role as W  did in Sect. 2. The theory of the Bernstein centre [5, Théorème 2.13] says that the centre of H(G, Ks )s is naturally isomorphic with O(Ts )Ws = O(Ts /Ws ). It will be convenient to lift everything from Ts to Xnr (L). However, Ws does not act naturally on Xnr (L). To overcome this and similar issues, we need the following lemma. Lemma 3.3 Let p : T → T be a surjection between complex tori, with finite kernel K = ker p. Let  be a finite group acting on T by automorphisms of algebraic varieties (so  need not fix 1 ∈ T ). Then there exists a canonical short exact sequence 1 → K →  →  → 1 and a canonical action of  on T which extends the multiplication action of K on T and lifts the action of  on T . Proof. Let X be the character lattice of T . Then O(T ) ∼ = C[X ] and  acts on O(T ) by (γ · f )(t) = f (γ −1 t). Since O(T )× = {zθx : z ∈ C× , x ∈ X } ∼ = C× × X ,  also acts naturally on X ∼ = O(T )× /C× . For γ ∈ , we denote this action by lγ : X → X . Notice that it defines an action of  on T = HomZ (X , C× ) by algebraic group automorphisms. The given action on O(T ) can now be written as γ(zθx ) = zzγ−1 (lγ (x))θlγ (x) , for a unique zγ ∈ T . Consequently the original action of  on T can be expressed as γ(t) = zγ lγ (t).

(50)

The character lattice X of T contains X with finite index (namely |K|). Thus lγ induces a canonical linear action of  on X , which we also denote by lγ . For every γ ∈  we choose a zγ ∈ p−1 (zγ ), and we define φγ : T → T , φγ (t ) = zγ lγ (t ).

Clearly, φγ is a lift of (50), so for every γ, γ ∈  there exists a unique zγ,γ

∈ K with

On Completions of Hecke Algebras

233



φγ ◦ φγ ◦ φ−1 ∀t ∈ T . γγ (t ) = zγ,γ t

(51)

Let  be the subgroup of Aut(T ) generated by the φγ (γ ∈ ) and K. Then (51) gives a canonical isomorphism  /K ∼ = . The only unnatural steps in the above argument are the choices of the zγ . Different

choices would lead to different zγ,γ

in (51), but to the same group  . Hence  is

canonically determined by the data T , T and . Next we recall the Plancherel isomorphism for S(G)s , as discovered by HarishChandra and worked out by Waldspurger. As induction data for G we take quadruples (P, M , ω, χ), where • P is a parabolic subgroup of G with a Levi factor M ; • (ω, Vω ) ∈ Irr L2 (M ), the set of (isomorphism classes of) irreducible smooth squareintegrable modulo centre representations of M ; • χ ∈ Xnr (M ). To such a datum, we associate the smooth G-representation IPG (ω ⊗ χ). When χ is unitary, the M -invariant inner product on (ω ⊗ χ, Vω ) induces a G-invariant inner product on IPG (Vω ), so IPG (ω ⊗ χ) is pre-unitary [10, Proposition 3.1.4]. However, IPG (Vω ) is only complete with respect to the associated metric if P = G and dim(Vω ) is finite. Let (ω, ˇ Vˇω ) be the smooth contragredient of ω and put G×G (ω ⊗ ω) ˇ = IPG (ω) ⊗ IPG (ω). ˇ L(ω, P) = IP×P

ˇ can be identified with the smooth contragredient of IPG (ω) [10, ProposiSince IPG (ω) tion 3.1.2], L(ω, P) can be regarded as the algebra of finite rank linear operators on IPG (Vω ). Notice that for every χ ∈ Xnr (M ), we can identify L(ω ⊗ χ, P) with L(ω, P) as algebras. The inner product on IPG (Vω ) induces a *-operation on this algebra. That makes O(Xnr (M )) ⊗ L(ω, P) to a *-algebra with ˇ ∗. f ∗ (χ) = f (χ) There is a natural *-homomorphism H(G) → O(X  nr (M )) ⊗ L(ω, P),  f → χ → IPG (ω ⊗ χ)(f ) .

(52)

We put Tω = {ω ⊗ χ : χ ∈ Xnr (M )} and we record the covering map Xnr (M ) → Tω : χ → ω ⊗ χ. The group Xnr (M , ω) = {χ ∈ Xnr (M ) : ω ⊗ χ ∼ = ω}

(53)

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M. Solleveld

is finite, because all its elements must be trivial on Z(M ). All the fibres of (53) are cosets of Xnr (M , ω) in Xnr (M ). For every k ∈ Xnr (M , ω), there exists a unitary M -intertwiner ω → ω ⊗ k, unique up to scalars. The same map Vω → Vω also intertwines ω ⊗ χ with ω ⊗ χk, for any χ ∈ Xnr (M ). Applying IPG , we get a family a G-intertwiners π(k, ω, χ) : IPG (ω ⊗ χ) → IPG (ω ⊗ χk),

(54)

independent of χ and unitary when χ ∈ Xunr (M ). Let ˇ ˇ → IPG (ωˇ ⊗ χk) π(k, ˇ ω, χ) : IPG (ωˇ ⊗ χ)) be the inverse transpose π(k, ω, χ). Since π(k, ω, χ) is unique up to scalars,   I (k, ω, χ) := π(k, ω, χ) ⊗ π(k, ˇ ω, χ) ∈ Hom G×G L(ω ⊗ χ, P), L(ω ⊗ χk, P) (55) is canonical. Moreover, it is unitary for χ ∈ Xunr (M ) and independent of χ as map between vector spaces. Let W (Tω ) be the stabilizer of Tω in W (G, M ) = NG (M )/M , with respect to the action on Irr(M ) as in (49). Then W (Tω ) acts naturally on Tω . From Lemma 3.3, we get a group extension 1 → Xnr (M , ω) → W (Tω ) → W (Tω ) → 1

(56)

and an action of W (Tω ) on Xnr (M ) compatible with the covering (53). In [43], the representations ω ⊗ χ and ω ⊗ χk are often not distinguished. The introduction of W (Tω ) and of the I (k, ω, χ) allows us to compare IPG (ω ⊗ χ) and IPG (ω ⊗ χk) in a systematic way. From [43, §VI.1], one can see that actually our setup is just another way to keep track of all the ingredients of [43]. The following results are proven in [43, Paragraphe V]. For w ∈ W (Tω ), there exist unitary G-intertwining operators π(w , ω, χ) : IPG (ω ⊗ χ) → IPG (ω ⊗ w (χ)) χ ∈ Xunr (M ),

(57)

unique up to scalars. These give canonical unitary intertwiners   I (w , ω, χ) = π(w , ω, χ) ⊗ π(w ˇ , ω, χ) ∈ Hom G×G L(ω ⊗ χ, P), L(ω ⊗ w (χ), P)

(58) with the following properties [43, Lemme V.3.1]: • as functions of χ, π(w , ω, χ) and I (w , ω, χ) are continuous with respect to the Zariski topology on the real algebraic variety Xunr (M ); • I (w2 , ω, w1 (χ)) ◦ I (w1 , ω, χ) = I (w2 w1 , ω, χ) for w1 , w2 ∈ W (Tω ).

On Completions of Hecke Algebras

235

The properties of the intertwining operators (57) imply that, for every g ∈ G, ω ∈ Irr L2 (M ), χ ∈ Xnr (M ) and every parabolic subgroup P ⊂ G with Levi factor gM g −1 , the representations IPG (ω ⊗ χ) and IPG (g · ω ⊗ g · χ) have the same irreducible subquotients, counted with multiplicity [37, Corollary 2.7]. We remark that I (w , ω, χ) is called ◦ cP|P (w , ω ⊗ χ) in [43]. The intertwining operators (58) give rise to an action of W (Tω ) on the algebra C ∞ (Xunr (M )) ⊗ L(ω, P)

by

(w · f )(w χ) = I (w , ω, χ)f (χ).

We fix a parabolic subgroup PL with Levi factor L, and we recall that s = [L, σ]G . To study representations in the Bernstein block Rep(G)s , it suffices to consider induction data such that P ⊃ PL , M ⊃ L and the cuspidal support of ω lies in [L, σ]M . Then W (Tω ) can be regarded as a subgroup of Ws . Choose representatives for the G-association classes of parabolic subgroups P containing PL . Notice that every such P has a unique Levi factor M containing L. We also choose representatives ω for the action of Ws  Xunr (M ) on Irr L2 (M ) ∩ Irr(M )sM , where sM = [L, σ]M . We denote the resulting set of representative triples by (P, M , ω)/ ∼. Harish-Chandra established the following Plancherel isomorphism (see [33, Theorem 8.9] for an alternative proof). Theorem 3.4 [43, Théorème VII.2.5] The maps (52) induces isomorphisms of topological *-algebras  ∞ W (Tω ) C (Xunr (M )) ⊗ L(ω, P) ,

 G  W (Tω )

s ∞ Ks S(G, Ks ) → (P,M ,ω)/∼ C (Xunr (M )) ⊗ EndC IP (Vω ) .

S(G)s

→

(P,M ,ω)/∼

Plymen [26] showed that Theorem 3.4 has a natural extension to C ∗ -algebras. Let H(ω ⊗ χ, P) be the Hilbert space completion of IPG (Vω⊗χ ) = IPG (Vω ) and let K(ω ⊗ χ, P) be the C ∗ -algebra of compact operators on H(ω ⊗ χ, P). Theorem 3.5 The maps (52) induces isomorphisms of C ∗ -algebras  W (Tω ) C Xunr (M ); K(ω, P) ,

 G  W (Tω )

∗ s Cr (G, Ks ) → (P,M ,ω)/∼ C(Xunr (M )) ⊗ EndC IP (Vω )Ks .

Cr∗ (G)s

→

(P,M ,ω)/∼

Proof. First we note that we have intertwining operators associated with the group W (Tω ), instead of W (Tω ) in [26, 43]. The reason for this is explained after (56). In view of Theorem 3.4, it only remains to prove that completing with respect to the operator norm of Cr∗ (G) boils down to replacing C ∞ (Xunr (M )) ⊗ L(ω, P) by  C Xunr (M ); K(ω, P) . This is shown in [26, Theorem 2.5].

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3.2 The Space of Irreducible Representations As in Paragraph 2.2, we need more information about the space of all irreducible smooth G-representations (tempered or not). Suppose that π ∈ Irr(G) has supercuspidal support σ ⊗ χ, where σ ∈ Irr L2 (M ) and χ ∈ Xnr (M ). Then log |χ| ∈ aM , and its image in a0 is uniquely determined, up to W (G, M0 ), by π. In other words, cc(π) := W (G, M0 ) log |χ|

(59)

is an invariant of π. Since the norm on a0 comes from a W (G, M0 )-invariant inner product, cc(π) := log |χ| is well defined. Theorem 3.6 (a) For every π ∈ Irr(G) there exists an induction datum (P, M , ω, χ), unique up to conjugation, such that π is a constituent of IPG (ω ⊗ χ) and cc(ω) is maximal for this property. In this case we call π a Langlands constituent of IPG (ω ⊗ χ). (b) π is tempered if and only if χ ∈ Xunr (M ). (c) For any induction datum (P, M , ω, χ), every constituent of IPG (ω ⊗ χ) is either  a Langlands constituent or a constituent of some IPG (ω ⊗ χ ) with cc(ω ) > cc(ω). (d) Suppose that L is a standard Levi subgroup and that π ∈ Irr(G)s , where s = [L, σ]G . Then we can choose (P, M , ω, χ) from part (a) such that P ⊃ P0 , M ⊃ L and ω ∈ Irr(M )[L,σ]M . Proof. (a) See [37, Theorem 2.15(b)]. (b) This is a direct consequence of [37, Proposition 2.14(b) and Theorem 2.15(a)]. (c) By [37, Lemma 2.13] (P, M , ω, χ) is equivalent to an induction datum ξ + in positive position. By [37, Corollary 2.7], IPG (ω ⊗ χ) has the same irreducible subquotients, counted with multiplicity, as the parabolically induced representation associated with ξ + . Therefore we may assume that (P, M , ω, χ) is in positive position, that is, P ⊃ P0 and log |χ| lies in the closed positive cone in a0 (determined by P0 ). Then [37, Theorem 2.15(a)] says that the Langlands constituents of IPG (ω ⊗ χ) are precisely its irreducible quotients. Furthermore, by [37, Proposition 2.15(a)] IPG (ω ⊗ χ) is a direct sum of representations of the form IQG (τ ⊗ |χ|), where (Q, τ , log |χ|) is a datum for the Langlands classification of Irr(G). Suppose that π is a constituent of IQG (τ ⊗ |χ|), but not a quotient. By [37, Lemma 2.11(a) and Lemma 2.12], π is the Langlands quotient of IQG (τ ⊗ ν ), for a Langlands datum (Q , τ , log ν ) with   Q ⊃ Q and cc(τ ) > cc(τ ). By [37, Proposition 2.15(a)] π is a Langlands constituent of IPG (ω ⊗ χ ), for some induction datum (P , M , ω , χ ) with     cc(ω ) = cc(τ ) > cc(τ ) = cc(ω) . (d) Let Pi = Mi Ui be a standard parabolic subgroup and let Pi be the unique parabolic subgroup with Levi factor Mi that is opposite to Pi . Let JPG : Rep(G) → Rep(Mi ) be i the normalized Jacquet restriction functor.

On Completions of Hecke Algebras

237

From [28, §VII.4.2], we recall how π can be realized as a Langlands quotient. Namely, we take P1 such that JPG (π) contains a representation of the form τ ⊗ ν, 1 where (P1 , τ , log ν) is a Langlands datum. By [43, Proposition III.4.1], there exists a parabolic subgroup P2 with P0 ⊂ P2 ⊂ P1 , and a ω ∈ Irr L2 (M2 ), such that τ ⊗ ν is a direct summand of IMM11∩P2 (ω ⊗ ν). From the proof of part (c), we see that π is a Langlands constituent of IPG2 (ω ⊗ ν). By the second adjointness theorem 0 = Hom G (IPG2 (ω ⊗ ν), π) ∼ = HomM2 (ω ⊗ ν, JPG (π)). 2

(60)

The cuspidal support of JPG (π) equals that of π, so ω ⊗ ν also has cuspidal support 2 in [L, σ]G . More precisely, the cuspidal support of ω ⊗ ν is of the form [L , σ ]M2 , where L is a standard Levi subgroup of G conjugate to L. Since every Levi subgroup containing L is G-conjugate to a standard Levi subgroup of G containing L, we may replace (P2 , M2 , ω , ν) by a G-conjugate (P, M , ω, χ) with M standard. Thus, we can arrange that the cuspidal support becomes [L, σ

]M , for some cuspidal σ

∈ Irr(L). Then (60) is also valid for IPG (ω ⊗ χ), since IPG2 (ω ⊗ ν) is not affected by Gconjugation of (P2 , M2 , ω , ν). Second adjointness tells us that JPG (π) ∈ Irr(M )[L,σ]M , so also ω ⊗ χ ∈ Irr(M )[L,σ]M . Finally, we may replace P by a standard parabolic subgroup with Levi factor M , for this does not change the collection of constituents  of IPG (ω ⊗ χ). In [37], Theorem 3.6 was used to study the geometry of Irr(G), and the relation with the subspace of tempered irreducible representations. For our purposes, we need some aspects of that, and we need to know that for almost all induction data every constituent is a Langlands constituent. Proposition 3.7 Let (P, M , ω, χ) be an induction datum for G. (a) For r ∈ R>−1 , the number of inequivalent Langlands constituents of IPG (ω ⊗ χ |χ|r ) does not depend on r. (b) For all but finitely many r ∈ R>−1 , IPG (ω ⊗ χ |χ|r ) is completely reducible. Then all its irreducible subquotients are Langlands constituents. Proof. (a) All the induction data under consideration have the same stabilizer in W (Tω ). As W (Tω ) is by construction the stabilizer of Tω in the W from [37], the statement is a special case of [37, Lemma 2.16]. (b) As noted in the proof of Theorem 3.6(c), IPG (ω ⊗ χ |χ|r ) is a direct sum of representations of the form IQG (τ ⊗ |χ|r+1 ), where (Q = LUQ , τ , log |χ|r+1 ) is a Langlands datum. Hence it suffices to show that IQG (τ ⊗ |χ|r+1 ) is irreducible for almost all r ∈ R>−1 . The conditions of a Langlands datum say that τ ∈ Irr(L) is tempered and that log |χ|r+1 ∈ aL is strictly positive with respect to the roots for (Q, L). This implies that, for every r ∈ R>−1 and every root α for (G, L), α , log |χ|r+1  = 0. Now [31, Théorème 3.2] says that, for r ∈ R>−1 close enough to −1, IQG (τ ⊗ |χ|r+1 ) is irreducible. On the algebraic family of finite length representations IQG (τ ⊗ |χ|r+1 ) with r ∈ R, irreducibility is an Zariski-open condition [28, Proposition VI.8.4]. Hence the locus of r’s for which this representation is reducible is a finite set. 

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4 Morita Equivalences In this section, we will first formulate a long list of conditions for the objects we want to compare. Assuming these conditions, we will prove a comparison theorem. In the next sections, we will check that these conditions are fulfilled in cases of interest.

4.1 Conditions and First Consequences We keep the notations from the previous section. Condition 4.1 For every parabolic subgroup P with PL ⊂ P ⊂ G and Levi factor M ⊃ L, an algebra HM and a Morita equivalence M : Rep(M )sM → Mod(HM ) are given. When P ⊃ P is another such parabolic subgroup, an algebra injection

λMM : HM → HM is given, with the below properties. (i) The following diagram commutes: M

Rep(M )sM

Mod(HM ) M M MM (H )

indH λ

M IP∩M

M

Rep(M )sM

Mod(HM )

(ii) Let P be the parabolic subgroup of G which has Levi factor M and is opposite to P. Let pr sM : Rep(M ) → Rep(M )sM be the projection coming from the Bernstein decomposition for M . The following diagram commutes: M

Rep(M )sM

M M MM (H )

ResH λ

M pr sM ◦JP∩M

Rep(M )sM

Mod(HM )

M

Mod(HM )

(iii) If P ⊂ P ⊂ P

⊂ G, then λMM

= λM M

◦ λMM . The Conditions 4.1 are quite general, in the sense that they do not involve the structure of the algebras HM . We will see later that in many cases these conditions hold already by abstract functoriality principles. The next series of conditions is more specific. For P = M U , let R(M , L) be the set of roots of M with respect to the maximal F-split torus AL in the centre of L. This

On Completions of Hecke Algebras

239

is a root system when L is a minimal F-Levi subgroup of G. In general, it is only an orthogonal projection of such a root system (in many cases encountered in the literature it is nevertheless a root system). For P ⊃ PL , we define the set of positive roots as R+ (M , L) = R(M ∩ PL , L), and we call the minimal elements of this set the simple roots of (M , L). Condition 4.2 Assume Condition 4.1. (i) HG (or (HG )op ) is an extended affine Hecke algebra H(R, q)  . (ii) All the HM (or all the (HM )op ) are parabolic subalgebras and the λMM are inclusions of parabolic subalgebras. (iii) Consider the bijection L : Xnr (L)/Xnr (L, σ) ∼ = Irr(L)sL → Irr(HL ) ∼ =T and its differential dL : X ∗ (L) ⊗Z C → Y ⊗Z C. ∨+ for H(R, q) to R+ (G, L), and Then d−1 L maps the positive coroots R −1 ∨ simple dL (QR ) has a Q-basis consisting   roots of (G, L). (iv) Suppose that Q ⊂  and dL QR(M , L) ∩ R∨ = R∨Q . Then HM = HQ  M   for some M ⊂ (Q, Q). If, moreover, dL QR(M , L) = QQ∨ , then M satisfies Condition 2.1 for Q. In practice, the positivity part of Condition 4.2(iii) is innocent. Namely, usually one starts by fixing a minimal parabolic subgroup and proves statements with that parabolic as the standard one. Suppose that R(G, L) is a root system and that all the above conditions hold, except the positivity part of Condition 4.2(iii). Then ∨ d−1 L (QR ) ∩ R(G, L) is a parabolic root subsystem of R(G, L), so it is conjugate under W (G, L) to a standard parabolic root subsystem, say R(M , L). Applying an element of W (M , L), we can, moreover, arrange that the image of R∨+ consists of positive roots. Equivalently, with respect to a different parabolic subgroup PL of G with Levi factor L, Condition 4.2(iii) is fulfilled. Then we restart the whole procedure with PL instead of PL , and the same arguments as before will also prove the required positivity statements. This applies to all the examples discussed in Sects. 5 and 6. We draw some first consequences from the above conditions. Lemma 4.3 Assume Conditions 4.1 and 4.2. (a) There exists a canonical surjective homomorphism of complex tori nr : Xnr (L) → T , with finite kernel Xnr (L, σ). (b) When dL (QR(M , L)) ∩ R∨ = R∨Q , the image of Xnr (M ) under the map from part (a) is contained in T Q . When, moreover, dL (QR(M , L)) = QQ∨ , the image of Xnr (M ) equals T Q . (c) For all ω ∈ Rep(M )sM and χ ∈ Xnr (M ): M (ω ⊗ χ) = M (ω) ⊗ nr (χ). Proof. (a) The map χ → σ ⊗ χ induces an isomorphism of algebraic varieties

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Xnr (L)/Xnr (L, σ) → Irr(L)sL = {σ ⊗ χ : χ ∈ Xnr (L)}. By Condition 4.2(ii), L gives a bijection Irr(L)sL → Irr(HL ) = T . This lifts to a surjective group homomorphism nr : Xnr (L) → T with L (σ ⊗ χ) = L (σ) ⊗ nr (χ).

(61)

(b) For every Q ⊂ , we defined the subtorus T Q = {t ∈ T : t(x) = 1 ∀x ∈ QQ ∩ X } of T . Using Condition 4.2(iv), we can write Xnr (M ) = {χ ∈ Xnr (L) : χ = 1 on QR(M , L)∨ ∩ X ∗ (Xnr (L))}. Q The relation between M and Q shows that the preimage of −1 nr (T ) contains Xnr (M ). ∨ When dL (QR(M , L)) = QQ , L also induces a bijection between QR(M , L)∨ and QQ, and Xnr (M ) is the full preimage of T Q . (c) The kernel of nr : Xnr (M ) → T Q is

Xnr (M , σ) := Xnr (L, σ) ∩ Xnr (M ). Then Xnr (M , σ) acts on Xnr (M ) by translations and GQ,σ acts on T Q ∼ = Xnr (M )/ Xnr (M , σ). By Lemma 3.3 there exists a canonical short exact sequence

→ GQ,σ → 1, 1 → Xnr (M , σ) → GQ,σ

(62)

such that the action of GQ,σ on Xnr (M ) lifts that of GQ,σ on T Q . For ω ∈ Rep(M )sM Condition 4.1(ii) and (61) imply that M ResH λLM (HL ) (M (ω ⊗ χ)) = L (JP ∩M (ω ⊗ χ)) M

L

=

L (JPM ∩M (ω) L

⊗ χ) =

L (JPM ∩M (ω)) L

(63)

⊗ L (χ)

H = ResH λLM (HL ) (M (ω)) ⊗ nr (χ) = ResλLM (HL ) (M (ω) ⊗ nr (χ)). M

M

When ω is irreducible, M (ω ⊗ χ) lies in the same connected component of Irr(HM ) as M (ω), so (63) shows that it is an unramified twist of M (ω). Hence M (ω ⊗ χ) = M (ω) ⊗ nr (χ) when ω is irreducible.

(64)

Using the invertibility of M , both sides of (64) define a group action of Xnr (M ) on Mod(HM ), by exact functors which commute with inductive (and projective) limits. Since these actions agree on irreducible representations, they agree on all representations. 

On Completions of Hecke Algebras

241

Lemma Conditions 4.1 and 4.2 and suppose that  4.4 Assume  dL QR(M , L) ∩ R∨ = R∨Q . (a) The map L induces a group isomorphism WsM → W (RQ)M .  (b) WsM fixes Xnr (M ) pointwise and, when, moreover, dL QR(M , L) = QQ∨ , W (RQ )M fixes T Q pointwise. Proof. (a) It suffices to prove this when M = G. By [5, Théorème 2.13], the centre of the category Rep(G)s is O(Irr(L)sL )Ws = O(Irr(L)sL /Ws ). The pointwise fixator of Xnr (L) in NG (L) is ZG (AL ) = ZG (Z(L)◦ ) = L. Since Ws ⊂ NG (L)/L, it acts faithfully on Xnr (L) by algebraic group automorphisms. Hence Ws also acts faithfully on Irr(L)sL . By (13) the centre of Mod(H  ) is Z(H  ) = O(T )W  = O(T /W ),

(65)

provided that W  acts faithfully on T . Clearly, W acts faithfully on T . By assumption, every γ ∈  acts on R by a diagram automorphism, so it cannot act on T as any nontrivial element of W . Hence to check that W  acts faithfully on T , it suffices to do so for . In view of (15), the isomorphism L : Irr(L)sL → T implies that ∅ is trivial. We recall from [31, Théorème 3.2] that IPGL (σ ⊗ χ) is irreducible for χ in a Zariski-open nonempty subset of Xnr (L). If γ ∈  \ {1} would act trivially on T , then so would the cyclic group γ generated by it. In that case Ct = indH (Ct ⊗C C γ) indH H∅ H∅  γ would be reducible for all t ∈ T (as C γ is reducible). That would contradict Condition 4.1(i). So W  acts faithfully on T and (65) holds. Now Condition 4.1 says that Z(H  ) = O(T /W ) ∼ = O(Irr(L)sL /Ws ). From this and Condition 4.1(i), we deduce that L : Irr(L)sL → T induces a bijection Irr(L)sL /Ws → T /W . On both sides, the finite groups act faithfully by automorphisms of complex algebraic varieties. Consider the open subvariety of T (resp. of Irr(L)sL ) where the stabilizers in W  (resp. in Ws ) are trivial. For such a t ∈ T and γ ∈ W , the equation L (w−1 L t) = γt holds for a unique w ∈ Ws . This defines the group isomorphism W  → Ws .

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(b) The first claim is trivial, because WsM ⊂ NM (L)/L. The second claim follows directly from the first claim, part (a) and Lemma 4.3(b).  Lemma 4.5 Assume

  M → dL QR(M , L) ∩ ∨

induces a bijection between: • Ws -association   classes of Levi subgroups M ⊂ G such that L ⊂ M and dL QR(M , L) equals the Q-span of a subset of ∨ ; • subsets of ∨ , modulo W -association. Proof. Suppose that M and M are such Levi subgroups, and that they are conjugate by an element w ∈ Ws . Then the functors IPG : Rep(M )sM → Rep(G)s and IPG : Rep(M )sM → Rep(G)s have the same image, for IPG ◦ Ad(w)∗ = Ad(w)∗ ◦ IPG ∼ = IPG . H With Condition 4.1(i), this implies that indH λM G (HM ) and ind λ G

G

M M G (H )

have the same

image. Condition 4.2(ii) says that H = H  M and H = HQ  M are parabolic subalgebras of HG , and then Proposition 2.6 shows that they must be G-associate. Condition 4.2(iv) implies that M

Q

M

    Q∨ = dL QR(M , L) ∩ ∨ and Q ∨ = dL QR(M , L) ∩ ∨ are W -associate. Hence the map of the lemma is well defined on the given equivalence classes.

∨ W Using the as above, suppose that Q∨ and  same notations   Q are  -associate. ∨ Then dL QR(M , L) = QQ is W -associate with dL QR(M , L) = QQ ∨ . By Lemma 4.4, QR(M , L) is Ws -associate with QR(M , L). Hence M and M are conjugate by an element of Ws , showing that the map of the lemma is injective. By Condition 4.2(iii), the subgroup Ps ⊂ G generated by PL and the root sub∨ groups for roots in d−1 L (QR ) is a parabolic subgroup of G. The map of the statement sends the standard Levi factor Ms of Ps to ∨ . ˜ ∨ ⊂ ∨ which does not lie in Suppose that the map is not surjective. Choose Q ˜ =  the image and is maximal for that property. Since ∨ belongs to the image, Q ˜ ∨ ∪ {α∨ } does lie in the image. We ˜ such that Q∨ := Q and we can find α ∈  \ Q ˜ Q). ˜ write Q˜ = Q ∩ (Q, For every Levi subgroup M ⊂ M , we choose finitely many representations πM ,i ∈ Irr(M )sM as in Theorem 3.2. Then the representations IPM ∩M (πM ,i ⊗ χM ) with χM ∈ Xnr (M ), for all M and all possible i, span the Grothendieck group of Rep(M )sM . Applying Condition 4.1, we find that the representations

On Completions of Hecke Algebras

243

indH λ

M

M M M (H )

M (πM ,i ⊗ χM )

(66)

span the Grothendieck group Gr(HM ) of Mod(HM ). By Lemma 4.3 M (πM ,i ⊗ χM ) = M (πM ,i ) ⊗ nr (χM )



and nr (χM ) ∈ T Q . For M = M and t ∈ T Q , M (πM ,i ) ⊗ t is a representation of

˜ Hence the collection of representations HQ  M with Q not W -associate with Q. M (πM ,i ) ⊗ t with t ∈ T Q

(67)

spans the quotient Gr(HQ  M )



 Q Q,Q

By Theorem 2.8(a), indH λ

Q

˜ not associateQ

M ˜ Q ˜ (H Q ˜) QM Q

M indH λ (HQ  M M

M )

Gr(HQ  M ).

(68)

˜

Gr(HQ  Q˜ ) contributes an entire T W (RQ˜ )Q˜ -

orbit of representations to (68). By Lemma 4.4 T Q ⊂ T W (RQ )Q , which shows in particular that the translation part zγ of γ is trivial for all γ ∈ Q . As W (RQ˜ )Q˜  W (RQ )Q , we have T W (RQ˜ )Q˜ ⊃ T Q . We want to see that the left-hand side has higher dimension than the right-hand ˜ ˜ side. By construction, W (RQ˜ ) fixes T Q pointwise. The torus T1 = (TQ ∩ T Q )◦ is ˜ = 1. For the same reason α : T1 → C× is a surone-dimensional, because |Q \ Q| ˜ Q) ˜ stabilizes both Q and Q, ˜ jection with finite kernel. The group Q˜ = Q ∩ (Q, so fixes α. Therefore Q˜ fixes T1 pointwise. It follows that W (RQ˜ )Q˜ fixes the torus ˜

T Q = T Q T1 pointwise. ˜ Returning to (67) and (68), we see now that the contribution from HQ  Q˜ en˜

compasses at least one T Q -orbit. But that is impossible, because the i’s in (67) belong

to a finite set and dimC (T Q ) > dimC (T Q ). This contradiction entails that the map from the statement is surjective. 

4.2 Preservation of Temperedness We will show that the above conditions imply that the Morita equivalences preserve temperedness and (under an extra condition) discrete series. For −1 M this is relatively easy.

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Lemma 4.6 Assume Conditions 4.1 and 4.2, and let P = M U be a parabolic subgroup containing PL such that dL (QR(M , L)) = QQ∨ . (a) −1 M preserves temperedness of finite length representations. M (b) −1 M sends finite dimensional essentially discrete series H -representations to essentially square-integrable M -representations. (c) Suppose that π ∈ Mod(HM ) has finite dimension, is tempered, essentially discrete series and factors through ψt : HQ  M → HQ  M for some t ∈ T Q .

Then the M -representation −1 M (π ) is square-integrable modulo centre. Proof. (a) Since every irreducible HM -module has finite dimension, M restricts to an equivalence between finite length representations on the one hand, and the finite dimensional modules on the other hand. Let π ∈ Rep(M )sM be of finite length, and recall the criterion for temperedness from [43, Proposition III.2.2] and (43). As the supercuspidal support of π is contained in [L, σ]G , it is equivalent to impose these conditions only with respect to the parabolic subgroup PL = LUL [16, Proposition 1.2(i)]. Let PL be the parabolic subgroup opposite to PL . Then PL ∩ M is opposite to PL ∩ M . The above condition (for PL ) is equivalent to: log |χ| ∈ + aM = P ∩M

 α∈R(PL ∩M ,L)

L

cα α|AL : cα ≤ 0



(69)

for every AL -weight χ of JMM∩P π. L

∨ By the assumption on M and Condition 4.2(iii), d−1 L : QQ → QR(M , L) is a linear bijection which sends

a−Q =

 α∈Q

λα α ∨ : λα ≤ 0



to

+ aM . PL ∩M

(70)

Suppose that M (π) is tempered. By definition, this means that all HL -weights t M −Q . By Condition 4.1(ii) and (70), all AL of ResH λLM (HL ) (M (π)) satisfy log |t| ∈ a weights χ of JMM∩P π have log |χ| ∈ + aM . Thus (69) says that π is tempered. PL ∩M L (b) By [43, Proposition III.1.1] and arguments analogous to the above, π is squareintegrable modulo centre if and only if = log |χ| ∈ + aM P ∩M L

 α∈R(PL ∩M ,L)

cα α|AL : cα < 0



(71)

for every AL -weight χ of JMM∩P π. The criterium for essential square-integrability L then becomes + aM . (72) log |χ| ∈ + aM P ∩M L

for every AL -weight χ of JMM∩P π. Since the rank |Q| of RQ equals the rank of R(M , L) L

−−Q and d−1 (the interior of a−Q ) to aM . By Lemma L preserves positivity, it maps a P ∩M

Q 4.3(b) −1 L (T ) = Xnr (M ).

L

On Completions of Hecke Algebras

245

Suppose that M (π) is essentially discrete series. By definition, this means that M −−Q )T Q . By the above and all HL -weights t of ResH λLM (HL ) (M (π)) satisfy |t| ∈ exp(a Condition 4.1(ii), (72) holds for all AL -weights χ of JMM∩P π. Hence π is essentially L square-integrable. (c) Recall that a M -representation is essentially square-integrable if its restriction to the derived subgroup of M is square-integrable. If a M -representation with a central character is tempered, then Z(M ) acts on it by a unitary character. Hence all tempered essentially square-integrable representations with a central character are square-integrable modulo centre.

By parts (a) and (b), −1 M (π ) is tempered and essentially square-integrable. Since −1 M (π ) lies in one Bernstein component Rep(M )sM , the maximal compact subgroup of Z(M ) acts on it by a single character χ0 , which is automatically unitary. Then Xnr (M ) (resp. Xunr (M )) parametrizes the extensions of χ0 to a character (resp. unitary

character) of Z(M ). Lemma 4.3(c) says that Z(M ) acts on −1 M (π ) by the character Q −1 determined by χ0 and nr (t). Lemma 2.3(b) shows that t ∈ Tun and −1 nr (Tun ) =  Xunr (L), so Z(M ) acts by a unitary character. Part (a) of Lemma 4.6 admits a quick generalization to all Levi subgroups that we encounter. On the other hand, that is not possible for parts (b) and (c). In fact, for Levi subgroups M ⊂ G containing L but not of the form as in Lemma 4.6, Rep(M )sM contains no essentially square-integrable representations. We delay the proof of that claim to Proposition 4.10. We note that in those cases Irr L2 (HM ) can still be nonempty. Lemma 4.7 Assume Conditions 4.1 and 4.2, and let P = M U be a parabolic ∨

subgroup containing PL such that QR(M , L) ∩ d−1 L (R ) does not span QR(M , L). −1 Then M preserves temperedness of finite length representations. Proof. By Condition 4.2(iii) dL (QR(M , L)) ∩ R∨ is a standard parabolic root subsystem of R∨ , that is, of the form R∨Q for a unique Q ⊂ . By Lemma 4.5, there exists a Levi subgroup M ⊂ M such that L ⊂ M and dL (QR(M , L)) = QQ∨ .

By Condition 4.2(iv) HM = HQ  M and HM = HQ  M . Further, by Condition 4.2(ii) M ⊃ M and λMM is just the inclusion. The cone T −Q ⊂ Trs is the

same for HM , HQ and HM . For any finite dimensional HM -module V :   M

Wt indH HM (V ) = {γ(t) : t ∈ Wt(V ), γ ∈ M }.

(73)

M

Since M preserves T −Q , indH HM (V ) is tempered if and only if V is tempered.

Similarly, (73) shows that a finite dimensional HM -module V is tempered if and

M

only if ResH HM (V ) is tempered. We note also that



M HM

∼

∼

indH HM ResHM (V ) = C[M ] ⊗C[M ] V = C[M / M ] ⊗C V ,

(74)

a HQ  M -module for the diagonal action. Then (74) contains V as the direct summand C[M / M ] ⊗C V , and the restriction of (74) to HM is a direct sum of M

copies of ResH HM (V ).

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M. Solleveld

M We recall from [28, Lemme VII.2.2] that IP∩M

always preserves temperedness.

Consider a finite dimensional tempered module V ∈ Mod(HM ). By Lemma 4.6(a)

M

−1 H M

the M -representation IP∩M

◦ M ◦ ResHM (V ) is tempered. By Condition 4.1 it is isomorphic to M

M

−1 H H

∼ M M

−1 M ◦ ind HM ◦ ResHM (V ) = IP∩M ◦ pr sM ◦ JP∩M ◦ M (V ),



(75)

−1

Since −1 M is an equivalence, (74) shows that (75) contains M (V ) as a direct summand. So the latter is tempered as well. 

Let M ⊃ L be a Levi subgroup of G and write dL (QR(M , L)) ∩ R∨ = R∨Q . As family of parabolic subalgebras of HM we take HM and the HM1 where M  M1 ⊃ L and dL (QR(M1 , L)) = QQ1 for some Q1  Q. By Lemma 4.5 every (proper) subset of Q is obtained in this way. Recall the notion of Langlands constituents from Theorems 2.10 and 3.6. Lemma 4.8 Let M be a Levi subgroup of M containing L, such that dL (QR(M , L)) = QQ ∨ for a subset Q ⊂ Q. Let σ ∈ Irr L2 (HQ  M ) and t ∈

T Q . Then −1 −1 −1 −1 −1

sM . −1 M (σ ⊗ t) = M (σ ⊗ t |t| ) ⊗ nr (|t|) with M (σ ⊗ t |t| ) ∈ Irr L2 (M )

The Morita equivalence −1 M restricts to a bijection between the Langlands conHM stituents of indHM (σ ⊗ t) and those of  HM  −1   −1 −1 ∼ M −1 M ind HM (σ ⊗ t) = I(M ∩PL )M M (σ ⊗ t |t| ) ⊗ nr (|t|) . Remarks By Lemma 4.3(a), nr becomes injective when restricted to unramified

characters with values in R>0 . Therefore −1 nr (|t|) ∈ Xnr (M ) is well defined.

When Q = Q, Langlands constituents are not defined on the Hecke algebra M = side. In that case, the lemma must be interpreted differently. The functor indH HM M preserves complete reducibility (by Clifford theory, as M is finite). By indH HQ M M Condition 4.1(i), so does I(P

. In view of Proposition 3.7(b), the lemma beL ∩M )M

comes true in the case Q = Q, provided we declare that all irreducible subquotients M (σ ⊗ t) are Langlands constituents. of indH HM Q

Proof. The alternative expression for −1 M (σ ⊗ t) comes from Lemma 4.3(c). By

−1 Lemma 2.3 σ ⊗ t |t|−1 ∈ Irr L2 (HQ  M ), and by Lemma 4.6(c) −1 M (σ ⊗ t |t| ) ∈

Irr L2 (M ). M (σ ⊗ t) is completely reducible. By Proposition Case I. Suppose that indH HM 2.11(b), all its irreducible subquotients are Langlands constituents. Since M is  HM  ind (σ ⊗ t) is also completely reducible. By Proposition an equivalence, −1

M HM 3.7(b) all its irreducible subquotients are Langlands constituents. Hence −1 M provides a bijection between these two collections of Langlands constituents.

On Completions of Hecke Algebras

247

Case II. Suppose that indH (σ ⊗ t) is not completely reducible. By Proposition HM M 2.11(b), there exists an r ∈ R>−1 such that indH (σ ⊗ t |t|r ) is completely reducible. HM By Proposition 2.11(a), Case I and Proposition 3.7(a), the four representations M

 HM   M M −1  r HM indH (σ ⊗ t), indH (σ ⊗ t |t|r ), −1 M indHM (σ ⊗ t |t| ) and M indHM (σ ⊗ t) HM HM

have the same number of inequivalent Langlands constituents. M (σ ⊗ t). By Theorem 2.10(c), Let π be a non-Langlands constituent of indH HM there exists a Levi subgroup M1 ⊂ M , with M1 = M or dL (QR(M1 , L)) = QQ1∨  M (σ1 ⊗ t1 ) for some t1 ∈ T Q1 and σ1 ∈ Q∨ , such that π is a constituent of indH HM1 Irr L2 (HQ1  M1 ) with cc(σ1 ) > cc(σ). When M1 = M , Lemma 2.3 shows that the same condition on π is also fulfilled for the unique Levi subgroup M2 ⊂ M with dL (QR(M2 , L)) = QQ∨ . In that case, we replace M1 by M2 . −1

Now −1 M1 (σ1 ⊗ t1 |t1 | ) ∈ Irr L2 (M1 ) and π is a constituent of  −1  M −1 −1 I(M ∩PL )M1 M (σ1 ⊗ t1 |t1 | ) ⊗ nr (|t1 |) .

(76)

  Recall that the invariant ccM1  on Irr(M1 )sM1 is defined via a Ws -invariant inner product on aL = X ∗ (AL ) ⊗Z R. Via Lemmas 4.3(a) and 4.4(a) this can be transferred (canonically) to a W -invariant inner product on a. The supercuspidal support (which is involved in ccM1 ) on Irr(M1 )sM1 is (up to conjugation) given by JPM1∩M . Then L 1 Condition 4.1(ii) shows that      ccM −1 (σ1 ⊗ t1 |t1 |−1 )  = cc(σ1 ⊗ t1 |t1 |−1 ) = cc(σ1 ) > 1 M1      −1  cc(σ) = cc(σ ⊗ t |t|−1 ) = ccM −1 . (77) M (σ ⊗ t |t| )

Now Theorem 3.6(c) says that −1 M (π ) is not a Langlands constituent of  HM  −1 M indHM (σ ⊗ t) . Summarizing, we know that:

of inequivalent irreducible sub• −1 M provides a bijection between the collections  HM  −1 HM quotients of indHM (σ ⊗ t) and of M indHM (σ ⊗ t) ; • these two collections have the same number of Langlands constituents and the same number of non-Langlands constituents; HM • −1 M maps non-Langlands constituents of indHM (σ ⊗ t) to non-Langlands con  HM stituents of −1 M ind HM (σ ⊗ t) . Consequently, −1 M also provides a bijection between the collections of inequivalent Langlands constituents on both sides.  Now we are ready for the proof of main result of this paragraph. Theorem 4.9 Assume Conditions 4.1 and 4.2, and let P = M U be a parabolic subgroup containing PL .

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M. Solleveld

(a) M restricts to an equivalence between the category of finite length tempered representations in Rep(M )sM and the category of finite dimensional tempered HM -modules. (b) Suppose that dL (QR(M , L)) = QQ∨ for some Q ⊂ . Then M sends finite length essentially square-integrable M -representations to essentially discrete series HM -representations, and −1 M does the converse. Proof. (a) In view of Lemmas 4.6(a) and 4.7, it suffices to prove that M preserves temperedness of finite length representations. Suppose, on the contrary, that there exists a finite length tempered π ∈ Rep(M )sM such that M (π) ∈ Mod(HM ) is not tempered. Since M (π) has finite length, it has a composition series with finite dimensional irreducible quotients. It follows directly from the definition of temperedness for HM = HQ  M that at least one of these irreducible subquotients, say ρ1 , is not tempered. Then we may replace M (π) by ρ1 and π by −1 M (ρ1 ). Hence it suffices to prove the claim for irreducible representations. We take the same family of parabolic subalgebras of HM as in Lemma 4.8. By Theorem 2.10(c), there exists a Levi subgroup M1 ⊂ M , with M1 = M or dL (QR(M1 , L)) = QQ1∨  Q∨ , such that M (π) is a Langlands constituent of M (σ1 ⊗ t1 ) for some t1 ∈ T Q1 and σ1 ∈ Irr L2 (HQ1  M1 ). When M1 = M , indH HM1 Lemma 2.3 shows that the same condition on π is also fulfilled for the unique Levi subgroup M2 ⊂ M with dL (QR(M2 , L)) = QQ. In that case we replace M1 by M2 . By Lemma 4.8, π is a Langlands constituent of  HM   −1  −1 −1 ∼ M −1 M ind HM1 (σ ⊗ t) = I(M ∩PL )M1 M1 (σ ⊗ t |t| ) ⊗ nr (|t|) .

(78)

Suppose that M (π) is not tempered, so t ∈ / Tun by Theorem 2.10(d). Then −1 nr (|t|) ∈

Xnr (M ) \ Xunr (M ), and by Theorem 3.6(b) π is not tempered. With Lemma 4.6(a), we see that −1 M preserves both temperedness and non-temperedness of irreducible representations. Hence so does M . (b) For −1 M this is Lemma 4.6(b), so we only have to consider the claim for M . Up to (78) we can follow the proof of part (a), only replacing tempered by essentially discrete series everywhere. Suppose that M (π) is not essentially discrete series. By the uniqueness in Theorem 2.10(b), Q is a proper subset of Q. Then M is a proper Levi subgroup of M , so by the uniqueness in Theorem 3.6(a) π is not essentially square-integrable. With Lemma 4.6(b), we conclude that −1 M sends those irreducible representations which are essentially discrete series to essentially square-integrable representations, and those which are not essentially discrete series to representations that are not essentially square-integrable. Now it is clear that M also respects these properties.  For essentially square-integrable representations, we can be more precise than Theorem 4.9. We write dL (QR(M , L)) ∩ R∨ = R∨Q .

On Completions of Hecke Algebras

249

Proposition 4.10 Assume Conditions 4.1 and 4.2, and let P = M U be a parabolic subgroup containing PL . ∨ sM (a) Suppose that QR(M , L) ∩ d−1 L (R ) does not span QR(M , L). Then Rep(M ) contains no finite length essentially square-integrable representations. (b) Suppose that π ∈ Rep(M )sM is square-integrable modulo centre and has finite length. Then M (π) is tempered, essentially discrete series and factors through Q ψt : HQ  M → HQ  M for some t ∈ Tun . (c) Suppose that dL (QR(M , L)) = QQ∨ . Then M gives a bijection between Irr L2 (M )sM and Irr L2 (HM ).

Proof. (a) Suppose, contrary to what we need to show, that Rep(M )sM does contain a representation of the indicated kind. Since it has finite length, it has an irreducible subrepresentation, say π. Let ζ be the central character of π, and let |ζ| ∈ Xnr (M ) be its absolute value. Then π ⊗ |ζ|−1 ∈ Rep(M )sM is an irreducible essentially square-integrable representation with a unitary central character. Hence it is square-integrable modulo centre and in particular tempered. By Theorem 4.9(a) M (π ⊗ |ζ|−1 ) is also tempered. Let P1 = M1 U1 ⊂ G be the parabolic subgroup such that M ⊃ M1 ⊃ L and ∨ dL (QR(M1 , L)) is spanned by QR(M , L) ∩ d−1 L (R ). From (73) we know that HM −1 ResλMM (HM1 ) ◦ M (π ⊗ |ζ| ) is tempered and nonzero. By Lemma 4.6(a) 1

H −1 M1 ◦ ResλMM M

1

(HM1 )

◦ M (π ⊗ |ζ|−1 ) = JPM1 ∩M (π ⊗ |ζ|−1 )

is also tempered and nonzero. But [43, Lemme III.3.2] says that this contradicts the square-integrability (modulo centre) of π ⊗ |ζ|−1 . (b) This follows from Theorem 4.9, in the same way as Lemma 4.6(c) followed from parts (a) and (b) of that lemma. (c) This follows from Theorem 4.9(b), Lemma 4.6(c) and part (b). 

4.3 Comparison of Completions In this paragraph, we will show that the equivalences M induce Morita equivalences between the appropriate Schwartz algebras. In Proposition 2.6, we described the Plancherel isomorphism for the Schwartz completion of an affine Hecke algebra, in terms of the following data: • • • • •

the set of parabolic subalgebras HQ  Q of H  , up to W -equivalence, Q the tori Tun , Q the sets Irr L2 (HQ  Q ), up to the actions of Tun and W (Q, Q), the groupoid G, the intertwining operators I (g, Q, σ, t) for g ∈ GQ,σ .

These data depend mainly on the categories Mod(HQ  Q ). In Condition 4.2, we included the possibility that not the HM , but the (HM )op are affine Hecke algebras,

250

M. Solleveld

  so that M becomes an equivalence between Rep(G)s and Mod (HQ  Q )op . Then we use Lemma 2.7 to describe the Plancherel isomorphism of S(R, q)   in terms of right modules of its subalgebras HQ  Q , that is in terms of the categories Mod(HM ). With this in mind, it suffices to consider the case where each HM is an (extended) affine Hecke algebra. On the other hand, in Theorem 3.4 the Plancherel isomorphism for S(G)s was formulated in terms of: • • • • •

the set of parabolic subgroups P ⊃ PL , up to conjugation by Ws , the tori Xunr (M ), the sets Irr L2 (M )sM , up to the actions of Xunr (M ) and StabWs (M ), the groups W (Tω ), the intertwining operators I (w , ω ⊗ χ) for w ∈ W (Tω ).

We will compare these two data sets and manipulate them until we get a nice bijection from one side to the other. By Proposition 4.10(a), only the P with dL (QR(M , L)) of the form QQ∨ occur in the Plancherel isomorphism, since for the other P the set Irr L2 (M )sM is empty. Given Q ⊂ , we define Q as M , where QR(M , L) = QQ∨ . By Condition 4.2(iv), there is a canonical bijection from the set of parabolic subgroups P = M UP , with P ⊃ PL , M ⊃ L and dL (QR(M , L)) of the form QQ∨ and modulo conjugation by elements of Ws , to the parabolic subalgebras HQ  Q of HG , up to association by W . From Theorem 3.4, one sees that two such Levi subgroups M ⊂ G are Ws -conjugate if and only if the tempered parts of the two subsets IPG (Rep(M )sM ) coincide. By Condition 4.1(i) and Theorem 4.9(a), this means G M G precisely that two subsets indH λM G (HM ) (Mod(H )) of Mod(H ) coincide. By Theorem 2.4, that happens if and only if the two HM are W -equivalent. Thus we can pick of representatives for such P modulo Ws -conjugacy, and then the corresponding HM form representatives for W -equivalence classes of parabolic subalgebras HM = HQ  Q of HG . By Proposition 4.10,b M gives a bijection between Irr L2 (M )sM and Irr L2 (HM ). Q Upon parabolic induction, every Xunr (M )-orbit in Irr L2 (M )sM (resp. every Tun -orbit M s in Irr L2 (H )) gives rise to a family of tempered representations in Rep(G) (resp. in Mod(HG )). From Theorem 3.4, we see that IPG (ω) and IPG (ω ) belong to the same such family if and only if ω = w(ω ⊗ χ) for some w ∈ StabWs (M ) and χ ∈ Xunr (M ). Similarly, by 4.2(ii) and Proposition 2.6 H

indH λM G (HM ) (σ) and ind λM G (HM ) (σ ) G

G

belong to the same family in Mod(HG ) if and only if σ = g(σ ◦ φt ) for some g ∈ GQQ Q and t ∈ Tun . Applying G and Condition 4.1(i), we see that the respective equivalence relations on Irr L2 (M )sM and Irr L2 (HM ) agree via M . Let the set of representatives (Q, σ)/ ∼ be as in (34). Let (P, M , ω)/ ∼ be its image under Lemma 4.5 and the −1 M . Then (P, M , ω)/ ∼ is a set of representatives as in Theorem 3.4. Lemma 4.3(c) and Condition 4.1(i) guarantee that

On Completions of Hecke Algebras

251

G (IPG (ω ⊗ χ)) = indH λM G (HM ) (σ ⊗ nr (χ)) = π(Q, σ, nr (χ)). G

(79)

Hence G matches the finite length tempered elements of Rep(G) associated with (P, M , ω) (via Theorem 3.4) with the finite dimensional tempered HG -modules associated with (Q, σ) (via Proposition 2.6). By Theorem 3.4 IPG (ω ⊗ χ) and IPG (ω ⊗ χ ) are isomorphic if and only χ = w χ for some w ∈ W (Tω ). Analogously, Proposition 2.6 entails that π(Q, σ, t) and π(Q, σ, t ) are isomorphic if and only if t = g(t) for some g ∈ GQ,σ . From this and (62), we see that nr (from Lemma 4.3a) induces a bijection Q

/GQ,σ ∼ . (80) Xunr (M )/W (Tω ) → Tun = Xunr (M )/GQ,σ In the proof of Lemma 4.4, we checked that Ws (resp. W ) acts faithfully on Xnr (L) (resp. on T ). Then we see from (56) and (21) that the group actions in (80) are faithful. Comparing the outer sides of (80) and using the same method as in the

as subgroups of Aut(Xunr (M )). proof of Lemma 4.4, we deduce that W (Tω ) = GQ,σ Now we come to the intertwining operators. Recall from (57) and (58) that I (w , ω ⊗ χ) for w ∈ W (Tω ) comes from a unitary operator π(w , ω, χ) : IPG (ω ⊗ χ) → IPG (ω ⊗ w (χ)).

(81)

, at the same For bookkeeping purposes, we replace T Q by Xnr (M ) and GQ,σ by GQ,σ time defining

π(Q, σ, χ) := π(Q, σ, nr (χ)) and π(g , Q, σ, χ) = π(g, Q, σ, nr (χ))

is a lift of g ∈ GQ,σ . In particular, for k ∈ Xnr (M , σ), the interwiner when g ∈ GQ,σ π(k, Q, σ, χ) is the identity as map on the underlying vector spaces, it only changes

in Proposition 2.6 and (34) comes χ to kχ. Then (26) says that the action of GQ,σ from unitary intertwiners

π(g , Q, σ, χ) ∈ HomHG (π(Q, σ, χ), π(Q, σ, g (χ)).

(82)

Lemma 4.11 The intertwining operators (81) and (82) can be normalized so that G (π(w , ω, χ)) = π(g , Q, σ, nr (χ))

from (80). whenever w corresponds to g under the identification W (Tω ) = GQ,σ

Proof. Both (81) and (82) are unique up to scalars, because they depend algebraically on χ and because for generic χ ∈ Xunr (M ) the involved representations are irreducible. (The latter follows for example from the Plancherel isomorphisms.) Therefore, if w = g in the indicated way, G (π(w , ω, χ)) equals π(g , Q, σ, nr (χ)) up to a complex number  of absolute value 1.  To make this scalar 1, we simply replace π(g , Q, σ,  (χ)) .  π(w , ω, χ) by −1 nr G

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M. Solleveld

We remark that the normalization from Lemma 4.11 is harmless, because it does not change I (w , ω ⊗ χ). Theorem 4.12 Under the Conditions 4.1, 4.2 and 2.1, G : Rep(G)s → Mod(HG ) induces Morita equivalences S(G)s ∼M S(R, q)   and Cr∗ (G)s ∼M Cr∗ (R, q)  . Proof. In view of Proposition 2.6 and Theorem 3.4, we have to compare the Schwartz algebras 

 ∞ W (Tω ) C (Xunr (M )) ⊗ EndC (IPG (Vω )Ks and    G   (83) G Q C ∞ Tun ; EndC (VQ,σ ) Q,σ = C ∞ Xunr (M ); EndC (VQ,σ ) Q,σ . (P,M ,ω)/∼

(Q,σ)/G

(Q,σ)/G

To justify the equality in the second line, we note that a section of the algebra bundle over Xunr (M ) is Xnr (M , σ)-invariant if and only if it descends to a section of the Q analogous algebra bundle over Tun . By the above constructions, the M provide a bijection between the indexing sets for the sums in (83), so it suffices to compare  W (Tω ) with A1 := C ∞ Xunr (M ); EndC (IPG (Vω )Ks )

 GQ,σ ∞ A2 := C Xunr (M ); EndC (VQ,σ )

(84)

when (P, M ) corresponds to Q via Lemma 4.5 and M (ω) = σ. The Morita equivalences S(G, Ks )s ∼M S(G)s and G send IPG (ω ⊗ χ)Ks to π(Q, σ, χ) and by Lemma 4.11 this is compatible with the intertwining operators. Identifying W (Tω )

via (80), we consider the following bimodules for A1 and A2 : and GQ,σ  W (Tω ) B1 := C ∞ Xunr (M ); HomC (IPG (Vω )Ks , VQ,σ ) ,

  W (T ) ω . B2 := C ∞ Xunr (M ); HomC (VQ,σ , IPG (Vω )Ks ) Here the W (Tω )-actions are (w · f1 )(w χ) = π(w , ω ⊗ χ)f1 (χ)π(w , Q, σ, χ)−1 (w · f2 )(w χ) = π(w , Q, σ, χ)f2 (χ)π(w , ω ⊗ χ)−1

f1 ∈ B1 , f2 ∈ B2 .

Notice that by Lemma 4.11, these are honest group actions, not just up to some scalars. We claim that B1 ⊗A1 B2 ∼ = A2 and B2 ⊗A2 B1 ∼ = A1

(85)

On Completions of Hecke Algebras

253

as bimodules over A2 , respectively, A1 . Since all these algebras and modules are

of finite rank over C ∞ (Xunr (M ))W (Tω ) , it suffices to check this locally, at any χ ∈ Xunr (M ). Then the proof of the first half of (85) reduces to checking that



HomC (IPG (Vω )Ks , VQ,σ )W (Tω )χ ⊗EndC (IPG (Vω )Ks )W (Tω )χ HomC (VQ,σ , IPG (Vω )Ks )W (Tω )χ

∼ = EndC (VQ,σ )W (Tω )χ ,

(86)

and the other way round for B2 ⊗A2 B1 ∼ = A1 . By the uniqueness of π(w , Q, σ, χ) up to scalars, w → π(w , Q, σ, χ) defines a projective representation of W (Tω )χ . Let W

be a finite central extension of W (Tω )χ , such that this lifts to a linear representation of W

. By (85), the map w → π(w , ω ⊗ χ) also lifts to a linear representation of W

. Then W

and W (Tω ) have the same invariants in the all involved modules, so we can rewrite (86) as HomC[W

] (IPG (Vω )Ks , VQ,σ ) ⊗EndC[W

] (IPG (Vω )Ks ) HomC[W

] (VQ,σ , IPG (Vω )Ks ) ∼ (87) = EndC[W

] (VQ,σ ). This is a statement about finite dimensional representations of the finite group W

. One can verfiy (87) by reducing it to the case of irreducible W

-representations, where it is obvious. This also proves (86) and (85), and shows that the algebras in (83) are Morita equivalent. Combining that with Theorem 3.4 and (34), we find the desired Morita equivalences of Schwartz algebras. To prove that Cr (G)s and Cr (R, q)   are Morita equivalent, we can use exactly the same argument. We only have to replace C ∞ by continuous functions everywhere, and to use Theorem 3.5 instead of Theorem 3.4. 

5 Hecke Algebras from Bushnell–Kutzko Types Let L ⊂ G be a Levi subgroup and let σ ∈ Irr(L) be supercuspidal. Recall from [9, §4] that a type for s = [L, σ]G consists of a compact open subgroup J ⊂ G, and a λ ∈ Irr(J ), such that Rep(G)s is precisely the category of smooth G-representations which are generated by their λ-isotypical subspace. To such a type one associates the algebra H(G, J , λ) = End G (indJG λ), which (by definition) acts from the right on indJG λ. Then there is a Morita equivalence Mod(H(G, J , λ)) G : Rep(G)s → π → HomJ (λ, π) ∼ = Hom G (indJG λ, π).

(88)

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For a Levi subgroup M ⊂ G containing L, Bushnell and Kutzko [9, §8] developed the notion that (J , λ) covers a [L, σ]M -type (JM , λM ). Roughly speaking, this means that JM = J ∩ M , that λM = ResJJM λ and that H(G, J , λ) contains invertible “strongly positive” elements. Under these conditions, writing sM = [L, σ]M , there is a Morita equivalence M : Rep(M )sM → Mod(H(M , JM , λM )) as in (88), which is in several ways compatible with G . Lemma 5.1 Suppose that (J , λ) is a cover of a [L, σ]L -type (JL , λL ). Then Condition 4.1 is fulfilled, with HM = H(M , JM , λM )). Proof. Let P and P be as in the condition. By [9, Proposition 8.5] (JM , λM ) is an sM -type, (JM , λM ) is an sM -type and the former covers the latter. By [9, Corollary 8.4] there exists a unique algebra monomorphism tP∩M : H(M , JM , λM ) → H(M , JM , λM ) such that H(M ,J

,λ )

M M M

RestP∩M (H(M ,JM ,λM )) ◦ M = M ◦ pr sM ◦ RP∩M .

Here RM means the unnormalized parabolic restriction functor. To obtain the P∩M M version with the normalized Jacquet functor JP∩M

, we must adjust tP∩M by the square root of a modular character. This yields our λMM . The uniqueness of λMM and the transitivity of normalized Jacquet restriction entail that λM M

◦ λMM = λMM

when P ⊂ P ⊂ P

⊂ G. H(M ,J

H(M ,J

,λ )

,λ )

M M M M On general grounds indλMM (H(M ,JM ,λM )) is the left adjoint of ResλMM (H(M ,JM ,λM )) . M By Bernstein’s second adjointness theorem IP∩M : Rep(M ) → Rep(M ) is the left M adjoint of JP∩M

. Hence

M sM → Rep(M )sM IP∩M

: Rep(M )

M is the left adjoint of pr sM ◦ JP∩M

. By the uniqueness of adjoints

H(M ,J

,λ )

M M M M ◦ IP∩M ◦ M .

= ind λ MM (H(M ,JM ,λM ))



Having checked Condition 4.1 in a general framework, we turn to more specific instances where Condition 4.2 holds. In most cases, the intermediate algebras HM are not mentioned explicitly in the literature. One can obtain them by applying the same references to the group M instead of G. Using the canonical construction of λMM as in the proof of Lemma 5.1, Condition 4.2(ii) will be satisfied automatically in those cases. We will check the remaining conditions, mainly by providing relevant references. Recall that to achieve Condition 4.2(iii), we can use the method described on page 239.

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255

Iwahori–spherical representations. This is the classical case. Let oF be the ring of integers of the non-Archimedean local field F, let pF be its maximal ideal, and let kF = oF /pF be the residue field. Choose an apartment A of the Bruhat–Tits building of G and let L be the correponding minimal F-Levi subgroup of G. Let I be an Iwahori subgroup of G associated with a chamber of A. For convenience of the exposition, we define I here as the maximal compact subgroup of the pointwise fixator of that chamber. Let PL be the parabolic subgroup of G with Levi factor L such that PL (kF ) is the neutral component (in the sense of kF -groups) of the reduction of I modulo pF . Borel [7] showed that the trivial representation of I is an s-type where s = [L, trivL ]G . Borel assumes that G is semisimple, but it is easy to generalize his arguments to reductive G. By [17, §3], there is a *-algebra isomorphism   Cc (I \G/I ) ∼ = H(G, I , triv) ∼ = H X∗ (L), R∨ (G, L), X ∗ (L), R(G, L), , qI , (89) where the basis  is determined by PL and qI ,α = vol(Isα I )/vol(I ) for a simple reflection sα . From [7, §3.1], one sees that Conditions 4.2(iii) and (iv) hold. Here M = 1 for all M , so Condition 2.1 is vacuous. Of course Theorem 4.9 was already known for irreducible Iwahori-spherical representations. Indeed, by [19, Sect. 8] and [1, Theorem 15.1.(2) and Proposition 16.6], the bijection Irr(G)s → Irr(H(G, I , triv)) preserves temperedness and essential square-integrability. Moreover, Theorem 4.12 has been proven for Schwartz algebras in [12, Theorem 10.2]: (89) extends to an isomorphism of Fréchet *-algebras   S(I \G/I ) ∼ = S X∗ (L), R∨ (G, L), X ∗ (L), R(G, L), , qI . Principal series representations of split groups. Suppose that G is F-split and let T be a maximal split torus of G. Fix a smooth character χs ∈ Irr(T ) and put s = [T , χs ]G , so that Xnr (T ) → Ts : χ → χχs is a homeomorphism. Notice that χs restricted to the unique maximal compact subgroup Tcpt of T is a type for [T , χs ]T . By [29, Lemma 6.2], there exist a root subsystem Rs ⊂ R∨ (G, T ) and a subgroup Rs ⊂ Ws such that Ws = W (Rs )  Rs . Theorem 5.2 [29, Theorem 6.3] There exists a type (J , λ) for s and a *-algebra isomorphism H(G, J , λ) ∼ = H(Ts , Rs , q)  Rs , where qα = |kF | for all α ∈ Rs . Moreover, (J , λ) is a cover of (Tcpt , χ).

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Furthermore, Conditions 4.2(iii) and (iv) hold by construction. If QR(M , T ) = QQ∨ , then X  WQ Q ⊂ X∗ (T )  W (M , T ), so by Remark 2.2 Condition 2.1 holds as well. We note that for these Bernstein components Theorem 4.9(b) was already proven in [29, Theorem 10.7], while Theorem 4.9(a) follows from [12, Theorem 10.1], using [29, Sect. 8]. Level-zero representations. These are G-representations which are generated by non-zero vectors fixed by the pro-unipotent radical of a parahoric subgroup of G. Iwahori-spherical representations constitute the most basic example of this kind. A type (J , λ) for any Bernstein component s consisting level-zero representations was exhibited in [23], while it was proven in [24, Theorem 4.9] that it actually is a type. More precisely, by [24, §3.8] (J , λ) is a cover of a type for the underlying supercuspidal Bernstein component of a Levi subgroup L of G. By [23, Theorem 7.12] (see also [21]) H(G, J , λ) ∼ = H(R, q)  C[, s ]

(90)

for suitable R, q and . In all examples of level-zero Bernstein blocks which have been worked out, the 2-cocycle s of  is trivial. But even if it were non-trivial, we could easily deal with it. There always exists a finite central extension φ

→→1 1 → 1 → 2 − such that the pullback of s to 2 splits. Then H(G, J , λ) can be regarded as the direct summand of H(R, q)  2 associated with a minimal central idempotent p s ∈ C[1 ]. The algebra H(R, q)  2 , with the parabolic subalgebras HQ  φ−1  (Q ), is of the kind studied in Sect. 2. In this situation, the Conditions 4.1 and 4.2 must be adjusted slightly, now each HM should be p s HQ  φ−1  (M ) for some Q ⊂ . With these minor modifications, all the arguments in Sect. 4 remain valid. Conditions 4.2(iii) and (iv) follow from the setup in [23, §3.12–3.14] and [24, §1.10], combined with the description of R in [23, Proposition 7.3]. The groups Q for QQ∨ = QR(M , L) satisfy Condition 2.1 because they are contained in X  W (M , S), where W (M , S) is the Weyl group of M with respect to a maximal F-split torus S ⊂ L. As in the above examples, there is previous work on temperedness also. It is claimed in [12, Theorem 10.1] that Theorem 4.9(a) holds here. For this, one needs to know that (90) preserves the traces (maybe up to a positive factor) and the natural *-operations. The former follows from the support of the basis elements Tw of H(G, J , λ) constructed in [23] (only the unit element Te is supported on J ). For a simple (affine) reflection s, both Ts and Ts∗ have support JsJ , so they differ only by a scalar factor. They also satisfy the same quadratic relation, so Ts∗ = Ts . This implies that (90) is an isomorphism of *-algebras.

On Completions of Hecke Algebras

257

Inner forms of GLn (F). Let D be a division F. Every Levi subgroup of G = GLm (D) is of  algebra with centre the form L = i GLmi (D)ei , where i mi ei = m. Fix a supercuspidal ω ∈ Irr(L), of  the form ω = ki=1 ωi⊗ei , where ωi ∈ Irr(GLmi (D)) is supercuspidal and not inertially k k equivalent with ωj if i = j. Then Ts ∼ = i=1 (C× )ei , Rs is of type i=1 Aei −1 and the  k stabilizer of s = [ω, L]G in W (G, L) is W (Rs ) ∼ = i=1 Sei . Theorem 5.3 [35, 36] There exists a type (J , λ) for s, which is a cover of a [ω, L]L -type. There exists a parameter function qs : Rs → qN such that there is an isomorphism of *-algebras H(G, J , λ) ∼ = H(X ∗ (Ts ), Rs , X∗ (Ts ), R∨s , qs ), where the right-hand side is a tensor product of affine Hecke algebras of type GLe with e ≤ m. Moreover, this isomorphism sends the natural trace of H(G, J , λ) to a positive multiple of the trace of the right-hand side. We remark that the claims about the * and the traces are not made explicit in [35, 36]. They can be deduced in the same way as for level-zero representations, see above. With [12, Theorem 10.1] that proves Proposition 4.10 for these groups. Via the tensor product factorization Condition 4.2(iii) reduces to the case of a supercuspidal representation σ ⊗e of GLr (D)e . There it is a consequence of the constructions involved in [35, Théorème 4.6], which entail that the same notion of positivity in real tori is used for (GLr (D)e , GL1 (D)re ) and for H(GLe , q). Condition 4.2(iv) is irrelevant because all the groups M are trivial. For the Schwartz algebras of these groups, Theorem 4.12 can be found in [2, Theorem 6.2]. The proof over there is similar but simpler, because not all complications from Sect. 4 arise. Inner forms of SLn (F). Let G be the kernel of the reduced norm map GLm (D) → F × . It is an inner form of SLn (F), and every inner form looks like this. It was shown in [1] that for every inertial equivalence class s, H(G)s is Morita equivalent with an algebra which is closely related to affine Hecke algebras of type GLe (yet is of a more general kind). It is not known whether there exists an s-type for every s, but in any case the constructions in [1] are derived from the work of Sécherre and Stevens on inner forms of GLn (F), so types are not far away. Condition 4.1(i) is [2, Theorem 1.5(b)], the maps λMM are simply inclusions, and Condition 4.1(ii) follows by uniqueness of adjoints. Condition 4.2 does not hold precisely for the algebras HM obtained in this setting (in fact the Plancherel isomorphism for these HM has not been worked out), so we cannot apply Proposition 4.10 or Theorem 4.12. Nevertheless the conclusions of these results hold, see [2].

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Let us summarize the conclusions from this section. Corollary 5.4 Let s = [L, σ]G be an inertial equivalence class of the kind discussed in this paragraph (principal series of split group, level zero, inner form of GLn (F) or SLn (F)). Then S(G)s is Morita equivalent to the Schwartz completion of an extended affine Hecke algebra and Cr∗ (G)s is Morita equivalent to the C ∗ -completion of the same extended affine Hecke algebra. Proof. Except for the last case, this follows by applying Theorem 4.12. We just checked that all its assumptions are fulfilled. For the inner forms of SLn (F), [2, Theorem 6.4] gives the result in the case of Schwartz algebras. As in the proof of Theorem 4.12, the method in [2, §6.2] also works for the C ∗ -algebras, with minor modifications.

6 Hecke Algebras from Bernstein’s Progenerators We return to the notations from Sects. 3 and 4. Let s = [L, σ]G be any inertial equivalence class for G. Bernstein [6, §3] constructed a projective generator s for the category Rep(G)s . By [28, VI.10.1], for any Levi subgroup M ⊂ G containing L:

sM = IPML ∩M ( sL ), and this is a progenerator of Rep(M )sM . In other words, the map M : V → HomM (IPML ∩M ( sL ), V ) is an equivalence between Rep(M )sM and the category of right modules of EndM (IPML ∩M sL ). For PL ⊂ P = M UP ⊂ G we put HM = EndM (IPML ∩M sL )op = EndM ( sM )op . Then M provides an equivalence of categories Rep(M )sM → Mod(HM ). Lemma 6.1 In the above setting Condition 4.1 is fulfilled. Proof. The functoriality of normalized parabolic induction gives natural injections λMM : HM → HM

for P ⊂ P ⊂ G.

By naturality, the λMM satisfies Condition 4.1(iii). By Bernstein’s second adjointness theorem, for V ∈ Rep(M )sM :

On Completions of Hecke Algebras

259



M

M M

M (JP∩M

V ) = Hom M (IPL ∩M sL , JP∩M V )

∼ = HomM (I M I M s , V ) P∩M PL ∩M

L

∼ = HomM (IPML ∩M sL , V ) = M (V )

as HM -modules (via λMM ). This establishes the first commutative diagram in Condition 4.1. As in the proof of Lemma 5.1, the second commutative diagram follows from that by invoking the uniqueness of left adjoints. In the remainder of this section, we assume that G is: • either a symplectic group, not necessarily split, • or a special orthogonal group, not necessarily split, • or an inner form of GLn (F). Besides the discussion of inner forms of GLn (F) in the previous section, we point out that types for Bernstein components of symplectic or special orthogonal groups have been constructed in [22]. However, as far as we know the Hecke algebras associated with these types are in only few cases known explicitly. For the groups listed above, Heiermann has subjected (HG )op = End G (IPGL sL ) to a deep study. In [15] he proved that it is an extended affine Hecke agebra with positive parameters. The constructions in [15, §5] are such that every EndM (IPML ∩M sL ) arises as a parabolic subalgebra. For Condition 4.2(iii), see [16, §3]. It served as a step towards Theorem 4.9 for these groups [16, Théorème 5]. By [15, Proposition 1.15], the groups WQ Q are always contained in W (R˜ Q ) where R˜ Q ⊂ QRQ is a larger root system. In view of Remark 2.2, Condition 4.2(iv) holds. In fact, a more precise description of the root data and the groups M is available. By [15, 1.13], the root datum underlying the affine Hecke algebra End G (IPGL sL ) is a tensor product of root data of four types: GLn , Sp2n , SO2n+1 and SO2n . The groups M are described in [15, 1.15], but unfortunately some elements were overlooked (for a complete picture, we refer to [13]). The only nontrivial M come from the type D factors, it can happen that for a root datum of type (SO2n )e we have (extended) Weyl groups WM ∼ = W (Dn )e ,

W M M ∼ = W (Dne ) ∩ W (Bn )e .

(91)

Then |M | = 2e−1 . In the above setting, Theorem 4.12 says: Theorem 6.2 Let G be a symplectic group or a special orthogonal group over F (not necessarily split), or an inner form of GLn (F). Let s be any inertial equivalence class for G. Then S(G)s is Morita equivalent with the Schwartz completion of an extended affine Hecke algebra. The underlying root datum is a tensor product of root data of type GLn , Sp2n , SO2n+1 and SO2n , and the group  is a direct product of groups M as in (91). Furthermore Cr∗ (G)s is Morita equivalent with the C ∗ -completion of that extended affine Hecke algebra.

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Theorem 6.2 was one of the motivations for the author to write a paper about the K-theory of C ∗ -completions of (extended) affine Hecke algebras [41]. It enables us to show that the K-groups of the reduced C ∗ -algebras of the above groups are torsion-free. Theorem 6.3 Let G be as in Theorem 6.2. Then K∗ (Cr∗ (G)) is a free abelian group. It is countably infinite (unless G = 1). Proof. Recall the Bernstein decomposition from (45): Cr∗ (G) ∼ =

 s∈B(G)

Cr∗ (G)s .

Since topological K-theory is a continuous functor on the category of Banach algebras, it commutes with direct sums. This reduces the theorem to one factor Cr∗ (G)s . By Morita invariance and Theorem 6.2, it suffices to show that the K-theory of the C ∗ -completion of an extended affine Hecke algebra as in Theorem 6.2 is a finitely generated free abelian group. It was checked in [41, (62)] that the Künneth theorem for topological K-theory [34] applies to such algebras. Thus we only need to prove the result when the underlying root datum is of type GLn , Sp2n or SO2n+1 and  is trivial, and when the root datum is of type (SO2n )e and  is as in (91). These K-groups were computed in [41], see respectively (99), Theorem 3.3, (122), and Proposition 3.5. They are free abelian and have finite rank (given explicitly in terms of partitions).  Acknowledgements We thank the referee for suggestions and a careful reading.

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39. M. Solleveld, On the classification of irreducible representations of affine Hecke algebras with unequal parameters. Represent. Theory 16, 1–87 (2012) 40. M. Solleveld, Hochschild homology of affine Hecke algebras. J. Algebra 384, 1–35 (2013) 41. M. Solleveld, Topological K-theory of affine Hecke algebras. Ann. K-theory 3(3), 395–460 (2018) 42. M.-F. Vignéras, On formal dimensions for reductive p-adic groups, in Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Israel Mathematical Conference Proceedings vol. 2 (Weizmann, Jerusalem, 1990), pp. 225–266 43. J.-L. Waldspurger, La formule de Plancherel pour les groupes p-adiques (d’après HarishChandra). J. Inst. Math. Jussieu 2(2), 235–333 (2003)

On Relatively Tempered Representations for p-adic Symmetric Spaces Shuichiro Takeda

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . θ-split Parabolic Subgroups and Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 θ-split Parabolic Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Coroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Canonical Pairing for a∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Canonical Pairing for s∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Restriction from a∗ to s∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Positive Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Real Parts of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Casselman Criteria for Distinguished Representations . . . . . . . . . . . . . . . . . . . . . . . . 3.1 H -distinguished Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Map rP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Relative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 H -relatively Tempered Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Casselman Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Subrepresentation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Subrepresentation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Weak Jacquet Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem and Its Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

264 266 266 267 269 269 271 272 273 274 277 277 278 278 280 282 283 283 284 285 287 289

Abstract We generalize a well known subrepresentation theorem for tempered representations to the context of relatively tempered representations for p-adic symmetric spaces, assuming p = 2.

S. Takeda (B) Mathematics Department, University of Missouri-Columbia, 202 Math Sciences Building, Columbia, MO 65211, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 A.-M. Aubert et al. (eds.), Representations of Reductive p-adic Groups, Progress in Mathematics 328, https://doi.org/10.1007/978-981-13-6628-4_9

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1 Introduction Let F be a non-Archimedean local field whose residue characteristic is odd. Let G be a connected reductive group over F with an F-involution θ, and let H be the subgroup of θ-fixed points of G, so that the quotient H \G is a symmetric space. An admissible (not necessarily irreducible) representation (π, V ) of G is said to be H -distinguished if there exists a nonzero H -invariant linear form λ : V → C such that λ(π(h)v) = λ(v) for all h ∈ H and v ∈ V , namely λ ∈ HomH (π, 1). (Note that λ is not necessarily smooth.) In [6], Kato and Takano defined the H -matrix coefficient of an H -distinguished representation (π, V ) as follows. Let v ∈ V and λ ∈ HomH (π, 1). The H -matrix coefficient ϕλ,v is the function on the symmetric space H \G given by ϕλ,v (g) = λ, π(g)v for g ∈ G. Further in [7], they defined an H -relatively square-integrable representation as an H -distinguished representation (π, V ) which admits a unitary central character and has the property that ϕλ,v is in L2 (ZG H \G) for all λ ∈ HomH (π, 1) and all v ∈ V , namely    ϕλ,v (g)2 d g < ∞, ZG H \G

where ZG is the center of G. They also obtained the analogue of the Casselman criterion for square-integrability in terms of exponents. Now, we can naturally define an H -relatively tempered representation as an H distinguished representation (π, V ) which admits a unitary central character and has the property that for all ε > 0 the H -matrix coefficient ϕλ,v is in L2+ε (ZG H \G) for all λ ∈ HomH (π, 1) and all v ∈ V , namely 

  ϕλ,v (g)2+ε d g < ∞.

ZG H \G

A minor modification of [7] gives the analogue of the Casselman criterion for H relative temperedness. In this paper, we will prove the following subrepresentation theorem, which is a generalization of a well known subrepresentation theorem for tempered representations. Theorem 1.1 Let (π, V ) be an (H , λ)-relatively tempered representation for some λ ∈ HomH (π, 1). Then there is a nonzero G-intertwining map π −→ IndPG τ , where P = M U is θ-split and τ is M θ -relatively square-integrable. Further if π is irreducible, it is a direct summand of IndPG τ .

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This has an immediate corollary. Corollary 1.2 If (π, V ) is irreducible and (H , λ)-relatively tempered for some λ ∈ HomH (π, 1), then it is unitary and its contragredient π ∨ is also (H , λ∨ )-relatively tempered for some λ∨ ∈ HomH (π ∨ , 1). Let us emphasize that we can prove our theorem only when the residue characteristic is odd. This is because the important papers of Kato and Takano [6, 7] have this assumption. The structure of the paper is as follows. In Sect. 2, we will recall the notion of θsplit parabolic subgroup, which one might as well consider as the parabolic subgroup relevant to distinguished representations. We then establish numerous facts on the associated root systems. In Sect. 3, we will introduce the Casselman criterion for relative temperedness. In Sect. 4, we will prove our main theorem. Notation and Assumptions Throughout, F is a non-archimedean local field of residue characteristic different from 2, but the characteristic of F can be positive. We let qF be the number of elements in the residue field and | · |F the absolute value on F. Let G be a connected reductive group over F. We often identify G with its Frational points. For admissible representations π and  of G, we write π ≤  when π is a subquotient of . If P ⊆ G is a parabolic subgroup, we denote the normalized Jacquet module of π along P by πP . Also we often use the notation P = M U , which means that M is a Levi factor of P and that the unipotent radical is U . For a representation τ of M , we write IndPG τ for the representation of G obtained from τ via normalized parabolic induction. The following form of Frobenius reciprocity holds: Hom G (, IndPG τ ) = HomM (P , τ ). Note that the representations τ and  need not be irreducible. Throughout the paper, our reductive group G is equipped with an F-rational involution θ unless otherwise stated. For each subgroup G 1 ⊆ G, we write G θ1 := {g1 ∈ G 1 : θ(g1 ) = g1 }. Also we set H := G θ and hence G θ1 = G 1 ∩ H . By a representation, we always mean an admissible representation which is not necessarily irreducible, need not admit a central character and is not necessarily of finite length, unless stated otherwise. For a representation π of a subgroup P ⊆ G and for a fixed g ∈ G, we define the g-twist g π to be the representation of g P := gPg −1 whose space is the same as that of π with action g π(gpg −1 ) = π(p) for p ∈ P.

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2 θ-split Parabolic Subgroups and Root Systems A parabolic subgroup P of G with Levi decomposition P = M U is said to be θsplit if P ∩ θ(P) = M . The θ-split parabolic subgroups are precisely the ones relevant to distinguished representations. In this section, we will recall some results on θ-split parabolic subgroups and related issues, and establish some necessary facts. Though this section is essentially expository, we will be as thorough as possible for convenience of the reader.

2.1 Root Systems An F-split torus S ⊆ G is said to be θ-split if θ(s) = s−1 for all s ∈ S. We let S0 be a maximal θ-split torus. Further we fix a maximal F-split torus A containing S0 . Note that we have S0 = {s ∈ A : θ(s) = s−1 } and Aθ = {s ∈ A : θ(s) = s}. All of A, S0 , Aθ are θ-stable, and hence θ naturally acts on the groups of rational characters X ∗ (A) := Hom(A, Gm ), X ∗ (S0 ) := Hom(S, Gm ), X ∗ (Aθ ) := Hom(Aθ , Gm ) of the respective tori. Define a∗ := X ∗ (A) ⊗ R,

s∗ := X ∗ (S0 ) ⊗ R,

s∗ θ := X ∗ (Aθ ) ⊗ R.

The involution θ acts on a∗ with eigenvalues ±1, and their respective eigenspaces are s∗ θ and s∗ , so we have the decomposition a∗ = s∗ ⊕ s∗ θ . In this paper, the projection

(2.1)

→ s∗ p : a∗ −

will be used frequently, and for each ν ∈ a∗ we often write ν := p(ν). There is a natural splitting of p given by s∗ −→ a∗ , ν →

1 (ν − θ(ν)), 2

where ν is any preimage of ν. Hence we have

On Relatively Tempered Representations for p-adic …

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1 (ν − θ(ν)) for all ν ∈ a∗ . 2

Let  = (G, A) ⊆ X ∗ (A) be the set of roots of (G, A), which is θ-stable, and let W be its Weyl group. We choose a set  of simple roots so that the corresponding ordering has the property α > 0 and θ(α) = α

=⇒

θ(α) < 0.

(2.2)

We let + be the set of positive roots with respect to this ordering. We define θ := {α ∈  : θ(α) = α} ⊆ X ∗ (Aθ ) and θ :=  ∩ θ . Let

(2.3)

 = p()\{0} = p(\θ ),

where p is the projection as above. It has been proven by Helminck and Wang [5] that  is a root system with a basis  := p()\{0} = p(\θ ). +

We choose the ordering on  to be the one determined by , and denote by  the set of positive roots in . Let W be the Weyl group of the root system . Though we will not use it in this paper, let us give a more explicit description of it. Let Wθ be the subgroup of W generated by root reflections sα for α ∈ θ , and let W1 = {w ∈ W : w(s∗ θ ) = s∗ θ }. Then each w ∈ W1 acts on a∗ /s∗ θ = s∗ , and the kernel of W1 → Aut(s∗ ) is Wθ , so Wθ  W1 . Then we have W = Wθ \W1 ; (2.4) namely W is a subquotient of W . (See [3, 2.7].)

2.2 θ-split Parabolic Subsets We will describe those subsets of roots which correspond to θ-split parabolic subgroups. Recall that each standard parabolic subgroup corresponds to a subset of . Analogously, each θ-split parabolic subgroup corresponds to what we call a θ-split subset of , which in turn corresponds to a subset of  as we will describe below. First, for each subset I ⊆ , define I = (p−1 (I ) ∩ ) ∪ θ ,

(2.5)

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namely I is the union of θ and the set of all simple roots in  that “lie above” the roots in I . It is shown in [3, Lemma 2.10] that for each α ∈ , there are at most two roots lying above α. To be more precise, let wθ be the longest element in Wθ , and define θ∗ := −θ ◦ wθ , which has been shown to be an involution. (See [3].) Then we have p−1 (α) ∩  = {α, θ∗ (α)}, α ∈ \θ . We call a subset of the form (2.5) a θ-split subset. On the other hand, for each θ-split subset I ⊆ , we define I := {α : α ∈ I }\{0} = {α : α ∈ I \θ }. The association I → I gives an inclusion preserving one-to-one correspondence between the θ-split subsets of  and the subsets of . Further, for a subset I ⊆ , the corresponding parabolic PI is θ-split if and only if I is θ-split. It has been shown that a subset I ⊆  is θ-split if and only if I contains θ and is θ∗ -stable if and only if the subsystem I generated by I is θ-stable. Of course the maximum θ-split subset is  and P = G; and the minimum θ-split subset is θ and Pθ is a minimal θ-split parabolic subgroup. Let us mention that for each α ∈ \θ we have θ(α) = −θ∗ (α) − β

(2.6)

for some β ∈ θ ∩ + . (See [4, 1.7]. Strictly speaking, there it is stated as β ∈ θ , / θ .) but the positivity is clear because θ(α) < 0 by (2.2) and θ∗ (α) ∈ We call a θ-split parabolic subgroup of the form PI a -standard θ-split parabolic subgroup, or when  is understood from the context, simply standard θ-split parabolic subgroup. Recall that the notion of the (usual) standardness is after all completely arbitrary in the sense that any parabolic subgroup can become standard by choosing an appropriate simple roots . The same is true for θ-split parabolic subgroups. Namely Lemma 2.1 Any θ-split parabolic subgroup P is standard with respect to an appropriate choice of . Proof See [6, p. 11].



Of course, once  is fixed, non-standard parabolic subgroups are conjugate to standard ones. And for the case of θ-split parabolic subgroups, they are often H conjugate, but not always. Kato and Takano have shown the following. Lemma 2.2 (Lemma 2.5 of [6]) Let H be the algebraic group for H , and M0 be the one for the Levi M0 of the minimum θ-split standard parabolic Pθ , so H(F) = H and M0 (F) = M0 .

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(a) Any θ-split parabolic subgroup is of the form γ −1 PI γ for some θ-split subset I ⊆  and γ ∈ (M0 H)(F). (b) If (M0 H)(F) = M0 H , then every θ-split parabolic subgroup is H -conjugate to a standard θ-split one. It seems to be sometimes useful to have that every θ-split parabolic subgroup is H -conjugate to a standard one. Accordingly, we make the following definition. Definition 2.3 The involution θ is said to be standard if every θ-split parabolic subgroup is H -conjugate to a standard one. Remark 2.4 We believe that standardness is a fairly mild assumption. For example, all the cases treated in [10] are standard. See [5, §4.12] for a non-standard example.

2.3 Coroots As usual, we define the set of cocharacters of A by X∗ (A) := Hom(Gm , A) and set a = X∗ (A) ⊗ R. For each α ∈ , we denote the corresponding coroot by α∨ and the set of the coroots by ∨ ⊆ X∗ (A), so (X ∗ (A), , X∗ (A), ∨ ) is a root datum. Also we denote the set of simple coroots by ∨ . Similarly we have the root datum ∨ (X ∗ (S0 ), , X∗ (S0 ),  ) and set s = X∗ (S0 ) ⊗ R. For each root α ∈ , we denote ∨ the corresponding coroot by α∨ , and the set of simple coroots by  . By analogy with (2.1), one has (2.7) a = s ⊕ sθ , where sθ and s are, respectively, ±1 eigenspaces of θ.

2.4 Canonical Pairing for a∗ Let P = MP UP be a (not necessarily) θ-split parabolic subgroup, where MP is the Levi factor and UP is the unipotent radical as usual. Let AP be the split component of the center of MP . Then there exists a choice of simple roots  with respect to which P is standard, so that P = PI for some I ⊆ , in which case we have AP =

 

◦ ker α

and

MP = ZG (AP ).

α∈I

We let X ∗ (MP ) := Hom(MP , Gm ) and X ∗ (AP ) := Hom(AP , Gm ) be the groups of F-rational characters of MP and AP , respectively, and we let X∗ (AP ) = Hom(Gm , AP ) be the group of F-rational cocharacters of AP . Further we define a∗P := X ∗ (AP ) ⊗ R = X ∗ (MP ) ⊗ R and aP := X∗ (AP ) ⊗ R,

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where the equality X ∗ (AP ) ⊗ R = X ∗ (MP ) ⊗ R is given by the natural homomorphism X ∗ (MP ) → X ∗ (AP ), which is injective with finite cokernel. We have the canonical pairing X∗ (AP ) × X ∗ (AP ) −→ End(Gm ) ∼ = Z, which gives rise to a pairing ·, · : a∗P × aP −→ R.

(2.8)

Let Q ⊆ P, so that we have the inclusions AP ⊆ AQ ⊆ MQ ⊆ MP , which induce

X ∗ (MP ) ⊆ X ∗ (MQ ) and X∗ (AP ) ⊆ X∗ (AQ ).

By tensoring with R, we have the natural inclusions a∗P ⊆ a∗Q and aP ⊆ aQ . Let us denote the annihilators of aP in a∗Q and a∗P in aQ , respectively, by aPQ ∗ and aPQ , namely aPQ ∗ ={ν ∈ a∗Q : ν, x = 0 for all x ∈ aP } aPQ ={x ∈ aQ : ν, x = 0 for all ν ∈ a∗P }. Then we have the orthogonal direct sum decompositions a∗Q = a∗P ⊕ aPQ ∗ and aQ = aP ⊕ aPQ .

(2.9)

When Q is minimal, we write the decomposition (2.9) as a∗ = a∗P ⊕ aP0 ∗ and a = aP ⊕ aP0 ,

(2.10)

and via these decompositions we usually view a∗P and aP as subspaces of a∗ and a, respectively. We let (2.11) ρP : a∗ −→ a∗P be the projection on a∗P . Also we obtain a∗P = a∗G ⊕ aPG ∗ and aP = aG ⊕ aPG by setting Q to be P and P to be G in (2.9)

(2.12)

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If we write P = PI for some I ⊆  with an appropriate , it is helpful to keep in mind that a∗G = {ν ∈ a∗ : ν, α∨  = 0 for all α ∈ } aPG ∗ = spanR {ρP (α) : α ∈ \I } aP0 ∗ = spanR (I ), and similarly for a. Finally, since the decomposition (2.9) is orthogonal, by taking Q to be minimal, one can see that the pairing for a∗ × a restricts to the pairing for a∗P × aP , and hence we use the same notation for the pairing as P varies. We should mention Lemma 2.5 The canonical pairing is θ-invariant, θ∗ -invariant and W -invariant, namely for ν ∈ a∗ and x ∈ a, one has θ(ν), θ(x) = ν, x, θ∗ (ν), θ∗ (x) = ν, x, and w(ν), w(x) = ν, x for all w ∈ W . Proof The θ-invariance and W -invariance follow from the definition of how θ and  W act. The θ∗ -invariance follows from those two. Hence we have Proposition 2.6 The decomposition (2.1) is an orthogonal decomposition with respect to the canonical pairing, namely (s∗ )⊥ = sθ and (s∗ θ )⊥ = s. Proof This follows from the θ-invariance of the canonical pairing.



2.5 Canonical Pairing for s∗ Next assume P = MP UP is θ-split, and let S0 ⊆ P be the maximum θ-split torus contained in P. We set SP := S0 ∩ AP . Let X ∗ (SP ) = Hom(SP , Gm ) and X∗ (SP ) = Hom(Gm , SP ), and define s∗P := X ∗ (SP ) ⊗ R and sP := X∗ (SP ) ⊗ R. When P is minimal, we have s∗P = s∗ and sP = s. The canonical pairing X ∗ (SP ) × X∗ (SP ) → Z gives rise to a pairing ·, ·, s : sP × s∗P −→ R. Let Q ⊆ P be another θ-split parabolic subgroup. Then we have the orthogonal sum decompositions

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s∗Q = s∗P ⊕ sPQ ∗ and sQ = sP ⊕ sPQ ,

(2.13)

which are analogous to the decomposition (2.9). So by taking Q to be minimal, one can see that the pairing for s∗ × s restricts to the pairing for s∗P × sP , and hence we use the same symbol for the pairing as P varies. Also one can see that the pairing s∗ × s → R is invariant under the Weyl group W as in (2.4). As before, when Q is minimal, we write the decomposition (2.13) as s∗ = s∗P ⊕ sP0 ∗ and s = sP ⊕ sP0 .

(2.14)

Via these decompositions we usually view s∗P and sP as subspaces of s∗ and s, respectively. Also let (2.15) ρP : s∗ −→ s∗P be the projection on s∗P . Here we use the same symbol ρP as in (2.11), but this will be justified later. The decomposition (2.13) also gives s∗P = s∗G ⊕ sPG ∗ and sP = sG ⊕ sPG .

(2.16)

Now for any θ-split P, one can choose  in such a way that the corresponding ordering is as in (2.2) and P is standard with respect to , in which case we can write P = PI for some θ-split I ⊆ . Then again it is helpful to keep in mind that s∗G = {ν ∈ s∗ : ν, α∨ s = 0 for all α ∈ } sPG ∗ = spanR {ρP (α) : α ∈ \I } sP0 ∗ = spanR (I ), and similarly for s.

2.6 Restriction from a∗ to s∗ Since we have the natural inclusion X∗ (S0 ) ⊆ X∗ (A), the pairing X ∗ (A) × X∗ (A) → Z restricts to X ∗ (A) × X∗ (S0 ) → Z. Hence we have ·, · : a∗ × s → R, which factors through ·, ·s : s∗ × s −→ R, namely for each ν ∈ a∗ and x ∈ s, we have ν, x = ν, rs .

(2.17)

With this said, the following is elementary. Proposition 2.7 Let Q ⊆ P be both θ-split. (a) s∗P and sP are the −1-eigenspaces of a∗P and aP under the involution θ. Namely we can write

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a∗P = s∗P ⊕ s∗P θ and aP = sP ⊕ sP θ , where s∗P θ and sθP are, respectively, the +1-eigenspaces of a∗P and aP under θ. Further this direct sum is an orthogonal direct sum. (b) The projection map p : a∗Q → s∗Q and the injection map sQ → aQ preserve the orthogonal decompositions (2.9) and (2.13). Namely the projection a∗Q = a∗P ⊕ aPQ ∗ −→ s∗Q = s∗P ⊕ sPQ ∗ maps a∗P to s∗P and aPQ ∗ to sPQ ∗ , and the injection sQ = sP ⊕ sPQ −→ aQ = aP ⊕ aPQ maps sP to aP and sPQ to aPQ . Proof The proof is elementary linear algebra and left to the reader.



This immediately implies the following, which justifies the use of the same symbol ρP for both (2.11) and (2.15). Corollary 2.8 For each θ-split P, the projection ρP : a∗ → a∗P restricts to the projection ρP : s∗ → s∗P . In other words, we have the commutative diagram a∗  s∗

ρP

/ a∗

P

ρP

 / s∗

P

where the vertical arrows are the projection maps as in the previous proposition. Proof Apply the above proposition with Q minimal.



The following is also immediate. Corollary 2.9 The projection a∗ → s∗ maps a∗G to s∗G . Proof Apply the above proposition with Q minimal and P = G.



2.7 Positive Cones We define the positive cones for both a∗P and s∗P as usual. For this assume that  is so chosen that the corresponding ordering is as in (2.2) and P is standard with respect to , so that P = PI for some I ⊆ . Then we define

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+a∗P := {ν ∈ a∗P : ν =



cα ρP (α), cα > 0}

α∈\I

+a∗P := {ν ∈ a∗P : ν =



cα ρP (α), cα ≥ 0},

α∈\I

and if P = PI is θ-split, we define +s∗P := {ν ∈ s∗P : ν =



cα ρP (α), cα > 0}

α∈\I¯

+s∗P

:= {ν ∈

s∗P

: ν=



cα ρP (α), cα ≥ 0},

α∈\I¯

where ρP is as in Corollary 2.8. These cones are related to each other as follows. Lemma 2.10 Let P be θ-split. (a) The projection p : a∗ → s∗ gives rise to the surjections +a∗P −→ +s∗P and +a∗P −→ +s∗P . (b) The injection s∗ → a∗ gives rise to the inclusions +s∗P −→ +a∗P and +s∗P −→ +a∗P . Proof (a) This follows from Corollary 2.8. (b) By (2.6), for each α ∈ \θ we have α=

1 1 (α − θ(α)) = (α + θ∗ (α) + β) 2 2

for some β ∈ θ . Note that θ∗ (α) ∈ , and ρP (β) = 0 because β ∈ θ . The lemma follows. 

2.8 Real Parts of Characters In this subsection, we will review the notion of the real part of a quasi-character of a reductive group with enough details to ensure that our later arguments will be rigorous enough. (See, for example, [11, p. 239].) In this subsection (and only in this subsection), we will let G be an arbitrary connected reductive group over F (not necessarily our fixed G) and H ⊆ G an arbitrary connected reductive subgroup group of G (not necessarily our H = G θ ). (The cases of interest to us are actually when G is a torus or a Levi factor of a parabolic.) We set

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a∗G := X ∗ (G) ⊗ R and a∗G,C := a∗G ⊗ C. Here it should be mentioned that this a∗G is indeed equal to the previously defined a∗G and hence the notation. Further we set un = Homcont (G/G 1 , C× ),  = Homcont (G, C× ) and G G  is the group of quasi-characters of G and where G 1 = χ∈X ∗ (G) ker |χ|F . Namely G un  G is the group of unramified quasi-characters of G. It is well-known that there is a surjective group homomorphism un , a∗G,C −−− G



χi ⊗ si → (g →

i



|χi (g)|si ),

(2.18)

i

for χi ∈ X ∗ (G) and si ∈ C, and the kernel of this homomorphism is the lattice L of the form (2πi/ log qF )L0 for some lattice L0 ⊆ X ∗ (G) ⊗ Q, so we have the isomorphism ∼ un . a∗G,C /L −−→ G

Note that the lattice L is “purely imaginary”, and hence we have a well-defined map Re : a∗G,C /L −→ a∗G , ν → Re(ν), where Re(ν) is the real part of ν, namely Re(χ1 ⊗ s1 + · · · + χk ⊗ sk ) = χ1 ⊗ Re(s1 ) + · · · + χk ⊗ Re(sk )  we have |χ| ∈ G un . Since it is real for χi ∈ X ∗ (G) and si ∈ C. Now for each χ ∈ G, ∗ valued, there exists a ν ∈ aG which maps to |χ| under the map (2.18). Then the real part Re(ν) is uniquely determined by χ, and we call it the real part of χ and denote it by Re(χ). Namely, Re(χ) is the image of the composite ∼ Re  −→ G un −− → a∗G,C /L −−−→ a∗G , G

where the first map is given by χ → |χ|, and the last map is the Re map as above. Hence the map χ → Re(χ) is a group homomorphism. Moreover, the kernel of the ◦ . Also map χ → |χ| is the group of unitary characters of G, which we denote by G un+  we denote its image by G , which is the set of positive real valued unramified characters, namely un+ = Homcont (G/G 1 , R+ ). G

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Then the map χ → Re(χ) gives isomorphisms ∼ ∼ un+ −−  G ◦ −− →G → a∗G G/

of abelian groups. ∼ un+ −− → a∗G , which we denote by exp, is The inverse of the isomorphism G described by the Harish-Chandra homomorphism. Recall that the Harish-Chandra homomorphism for G is the homomorphism defined by HG : G −→ Hom(X ∗ (G), R), g → (χ → logqF |χ(g)|). This is characterized by the property that χ,HG (g)

qF

= |χ(g)|

for all χ ∈ Hom(X ∗ (G), R). Then for each ν ∈ a∗G , we define the map exp(ν) : G → R+ ,

ν,HG (g)

g → qF

,

(2.19)

so in particular if χ ∈ X ∗ (G), then we have exp(χ)(g) = |χ(g)|, or more in general, if ν = χ1 ⊗ r1 + · · · + χk ⊗ rk ∈ a∗G , where χi ∈ X ∗ (G) and ri ∈ R, then we have exp(ν)(g) = |χ1 (g)|r1 · · · |χk (g)|rk . One can easily verify that exp is indeed the inverse of |χ| → Re(|χ|). Next let H ⊆ G be a reductive subgroup. (As we mentioned at the beginning, this ∼  un+ − → a∗H for H does not have to be our G θ .) Then we also have the “real part map” H un+ → H  un+ and a∗G → a∗H both given by restriction. H . We have the obvious maps G (To be more precise, one needs to use H 1 ⊆ G 1 to show that the restriction indeed gives the map for the unramified characters.) Then by disentangling the definitions, one can see that the diagram  G   H

//G un+ o

Re / exp

a∗G

 Re /  //H  un+ o a∗H exp

commutes, where all the vertical arrows are given by “restriction”.

(2.20)

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3 The Casselman Criteria for Distinguished Representations In this section, after reviewing some of the basics of distinguished representations, we will elucidate the Casselman type criteria for H -relative square integrability and H -relative temperedness, which are essentially proven by Kato-Takano in [7].

3.1 H-distinguished Representation A (not necessarily irreducible) representation (π, V ) of G is said to be H -distinguished if HomH (π, 1) = 0. If λ ∈ HomH (π, 1) is nonzero, we sometimes say (π, V ) is distinguished by λ. More generally, we make the following definition. Definition 3.1 Assume (π, V ) is an H -distinguished representation of G. Let λ ∈ HomH (π, 1) be nonzero. We say a (not necessarily irreducible) subquotient τ ≤ π is distinguished by λ or (H , λ)-distinguished if λ naturally gives rise to an H -invariant functional on (τ , Vτ ). Namely, there exist G-invariant subspaces V1 ⊆ V0 ⊆ V such that V1 \V0 = Vτ , λ|V0 = 0 and λ|V1 = 0, so that λ gives a natural nonzero H -invariant linear functional on τ . Definition 3.2 Assume (π, V ) is an H -distinguished representation of G. For each nonzero λ ∈ HomH (π, 1), there is a nonzero G-intertwining map → C ∞ (H \G), v → ϕλ,v , Tλ,π : V − where ϕλ,v (g) = λ(π(g)v). We call ϕλ,v an H -matrix coefficient or (H , λ)-matrix coefficient. Let

ev : C ∞ (H \G) − → C, ϕ → ϕ(1),

be the “evaluation at 1” map. This map makes every subrepresentation of C ∞ (H \G) H -distinguished. Moreover, for a smooth H -distinguished representation (π, V ) and for nonzero λ ∈ HomH (π, 1), we have ev ◦ Tλ,π = λ; namely λ factors through the image Tλ,π (V ) of V under Tλ,π , making Tλ,π (V ) distinguished by λ. Note that every subrepresentation of Tλ,π (V ) is distinguished by λ, and ker(Tλ,π ) is the largest subrepresentation of π on which λ is identically zero. Of course, if π is irreducible, Tλ,π is one-to-one, giving a concrete realization of π and λ.

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3.2 The Map rP Next, we recall the map rP defined by Kato-Takano in [6]. (Also see the work [9] of Lagier.) Let (π, V ) be an H -distinguished representation, and let λ ∈ HomH (π, 1) be nonzero. For each θ-split parabolic P = M U , there exists a linear map rP : HomH (π, 1) −→ HomM θ (πP , 1) of invariant linear forms such that if v ∈ V is a canonical lift (in the sense of Casselman) of v¯ ∈ VP , then ¯ = λ, v rP (λ), v for all λ ∈ HomH (π, 1). Moreover, this equality is independent of the choice of the canonical lift. We need to mention that the map rP has the expected transitivity property with respect to Jacquet modules. Namely Proposition 3.3 (Proposition 5.9 of [6]) Let Q ⊆ P be θ-split, and (π, V ) be an H -distinguished representation of G. Then we have rQ = rQ∩M ◦ rP , where M is the Levi factor of P. Let us note that the map rP : HomH (π, 1) −→ HomM θ (πP , 1) might fail to be surjective, and hence even if τ ≤ πP is not distinguished by rP (λ) for any λ ∈ HomH (π, 1), it might still be distinguished. And of course if rP (λ) is non-zero, at least one of the constituents of πP is distinguished by rP (λ). (Here “distinguished by rP (λ)” is defined as in Definition 3.1; namely “distinguished by rP (λ)” means (M θ , rP (λ))-distinguished.)

3.3 Relative Exponents In this subsection, we will recall the notion of “relative exponents” introduced by Kato and Takano in [7]. First we need to recall some basic concepts. Let (π, V ) be a finitely generated Z representation of G. Let Z ⊆ ZG be a closed subgroup of the center ZG of G. Let  be the set of all quasi-characters of Z. For each χ ∈  Z, we define  Vχ := v ∈ V :

There exists d ∈ N such that (π(z) − χ(z))d v = 0 for all z ∈ Z

.

Namely Vχ is the generalized χ-eigenspace. It is well-known that (π, V ) has the direct sum decomposition

On Relatively Tempered Representations for p-adic …

(π, V ) =

279

(πχ , Vχ ),

χ∈ Z

Z. Note that each Vχ is indeed G-invariant, giving where Vχ = 0 for almost all χ ∈  rise to a subrepresentation (πχ , Vχ ) of (π, V ). Note that if Z = ZG , every subquotient of (πχ , Vχ ) that admits a central character has χ as the central character. Next, let us first recall the usual notion of exponents. For a finitely generated representation (π, V ) of G and a parabolic subgroup P, we define AP : VP, χ = 0}, ExpAP (πP ) = {χ ∈    ExpAP (πP ) = Re(χ) ∈ a∗P : χ ∈ ExpAP (πP ) , where VP, χ is the generalized χ-eigenspace as above, and Re(χ) is the real part of χ as defined previously. Note that ν ∈ ExpAP (πP ) if and only if exp(ν) = |χ| for some χ ∈ ExpAP (πP ) if and only if exp(ν) ⊗ τ ≤ πP for some irreducible representation τ with a unitary central character. We call each χ ∈ ExpAP (πP ) a central exponent of π along P and each ν ∈ ExpAP (πP ) an exponent of π along P. Let Q ⊆ P be parabolic subgroups. We have maps ExpAQ (πQ ) → ExpAP (πP ) and ExpAQ (πQ ) → ExpAP (πP ) given by restriction. Indeed, by using the diagram (2.20) with G = AQ and H = AP , one can obtain the commutative diagram of sets ExpAQ (πQ )

Re / ExpAQ (πQ )

 ExpAP (πP )

 Re / ExpAP (πP )

where the horizontal arrows are surjective. Next assume (π, V ) is H -distinguished and P = M U is a θ-split parabolic with SP being the θ-split component. By following Kato-Takano [7, Section 4], we first define SP : VP, χ = 0}, ExpSP (πP ) = {χ ∈    ExpSP (πP ) = Re(χ) ∈ s∗P : χ ∈ ExpSP (πP ) . By applying the diagram (2.20) with G = AP and H = SP , one can obtain the commutative diagram of sets ExpAP (πP )

Re / ExpAP (πP )

  Re / ExpSP (πP ) ExpSP (πP )

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where all the arrows are surjective. Now let λ ∈ HomH (π, 1) be fixed. By following [7], we define   ExpSP (πP , λ) = χ ∈ ExpSP (πP ) : rP (λ) = 0 on VP, χ ,   ExpSP (πP , λ) = Re(χ) ∈ s∗M : χ ∈ ExpSP (πP , λ) , where of course P = M U is assumed to be θ-split. We call each element in ExpSP (πP , λ) an (H , λ)-relative central exponent of π along P, and each element in ExpSP (πP , λ) an (H , λ)-relative exponent of π along P. So χ ∈ ExpSP (πP , λ) if and only if there exists a constituent τ ≤ πP distinguished by rP (λ) with the central character ωτ such that ωτ |SP = χ. Proposition 3.4 Let Q ⊆ P ⊆ G be θ-split. Then we have the commutative diagram of sets ExpSQ (πQ , λ)

Re / ExpSQ (πQ , λ)

 ExpSP (πP , λ)

 Re / ExpSP (πP , λ)

where the horizontal arrows are surjective, and the vertical arrows are given by restriction. Proof It has been shown in [7, Lemma 4.5] that the restriction map ExpSQ (πQ ) → ExpSP (πP ) gives rise to ExpSQ (πQ , λ) → ExpSP (πP , λ). This gives the vertical arrows of the diagram. The commutativity of the diagram can be shown by using (2.20) with  G = SQ and H = SP .

3.4 H-relatively Tempered Representations Though we already briefly recalled the notion of H -relative temperedness in the introduction, let us define it here more precisely. Definition 3.5 Let (π, V ) be an H -distinguished representation of G which admits a unitary central character. (a) For each nonzero λ ∈ HomH (π, 1), we say (π, V ) is (H , λ)-relatively tempered if for all  > 0, the (H , λ)-matrix coefficients ϕλ,v are in L2+ (ZG H \G) for all v ∈ V. (b) We say (π, V ) is H -relatively tempered if it is (H , λ)-relatively tempered for all λ ∈ HomH (π, 1).

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Remark 3.6 Assume π is irreducible. One can see that if ϕλ,v0 is in L2+ (ZG H \G) for one nonzero v0 ∈ V , then ϕλ,v are in L2+ (ZG H \G) for all v ∈ V , because any v ∈ V is a finite linear combination of vectors of the form π(g)v0 for some g. This definition is motivated by the following definition introduced by Kato-Takano in [6, 7]. Definition 3.7 Let (π, V ) be an H -distinguished representation of G. 1. (a) For each nonzero λ ∈ HomH (π, 1), (π, V ) is said to be (H , λ)-relatively cuspidal if the (H , λ)-matrix coefficients ϕλ,v are compactly supported modulo ZG H for all v ∈ V . (b) (π, V ) is called H -relatively cuspidal if it is (H , λ)-relatively cuspidal for all λ ∈ HomH (π, 1). 2. Assume π admits a unitary central character. (a) For each nonzero λ ∈ HomH (π, 1), (π, V ) is said to be (H , λ)-relatively square integrable if the (H , λ)-matrix coefficients ϕλ,v are square integrable modulo ZG H for all v ∈ V . (b) (π, V ) is called H -relatively square integrable, if it is (H , λ)-relatively square integrable for all λ ∈ HomH (π, 1). Remark 3.8 For the definition of the relative cuspidality, one does not need to assume π has a central character, though for relative square integrability one does. Of course, a relatively cuspidal representation with a unitary central character is relatively square integrable. Remark 3.9 As with relative-temperedness, if π is irreducible, it is (H , λ)-relatively cuspidal (resp. (H , λ)-relatively square integrable) if and only if ϕλ,v is compactly supported (resp. square integrable) modulo ZG H for one nonzero v ∈ V . Remark 3.10 Strictly speaking, for the square integrability, Kato-Takano used the word “(H , λ)-square integrable” instead of “(H , λ)-relatively square integrable”. Yet, it seems to the author that it is more appropriate and standard to include the word “relatively”. Let us mention that the major achievement of [6] is the following. Theorem 3.11 (Theorem 6.2 of [6]) Let (π, V ) be an H -distinguished representation of G and λ ∈ HomH (π, 1). Then it is (H , λ)-relatively cuspidal if and only if rP (λ) = 0 for all proper θ-split parabolic subgroups P of G. Further, if we assume the involution θ is standard, then the condition has to be checked only for the standard θ-split parabolic subgroups of G.

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3.5 The Casselman Criteria Now we have the following “Casselman criteria” for H -relative square integrability and H -relative temperedness. Theorem 3.12 Let (π, V ) be a finitely generated H -distinguished representation which admits a unitary central character. Fix λ ∈ HomH (π, 1). (a) (π, V ) is (H , λ)-relatively square integrable if and only if for all θ-split parabolic P, we have ν ∈ +s∗P for all ν ∈ ExpSP (πP , λ). (b) (π, V ) is (H , λ)-relatively tempered if and only if for all θ-split parabolic P, we have ν ∈ +s∗P for all ν ∈ ExpSP (πP , λ). Further, if we assume that the involution θ is standard, then the conditions have to be checked only for the standard θ-split parabolic subgroups PI . Proof (a) This is the main theorem of [7] (i.e. [7, Theorem 4.7]), though their theorem is stated by using central exponents instead of exponents. To translate their theorem to our “exponent version” is straightforward. (b) The proof for this is essentially the same as (a), with a very small modification at the end of the proof, and left to the reader. (Details are also found in [8].) The case of standard θ can be easily derived from the previous part. See [10, Proposition 4.21] for the detail.  Let us mention a couple of corollaries of this theorem. Corollary 3.13 Every (H , λ)-relatively square integrable representation is (H , λ)relatively tempered. Proof This follows because +s∗P ⊆ +s∗P .



Corollary 3.14 Let (π, V ) be a finitely generated H -distinguished representation of G, which is tempered and admits a unitary central character. Then π is H -relatively tempered. Proof By the Casselman criterion of (usual) temperedness, for all parabolics (hence all θ-split parabolics) P ⊆ G and all χ ∈ ExpAP (πP ), we have Re(χ) ∈ +a∗P . As in Lemma 2.10 the projection a∗P → s∗P maps +a∗P into +s∗P . The  corollary follows by the surjectivity of ExpAP (πP ) → ExpSP (πP ) and the inclusion λ∈HomH (π,1) ExpSP (πP , λ) ⊆ ExpSP (πP ). 

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It is often necessary to consider representations which do not admit a central character. For this reason, let us now broaden our definition of relatively tempered representation as follows. Definition 3.15 Let (π, V ) be a finitely generated H -distinguished representation of G which does not admit a unitary central character. (a) For each nonzero λ ∈ HomH (π, 1), we say (π, V ) is (H , λ)-relatively tempered if for all θ-split parabolic P and for all ν ∈ ExpSP (πP , λ), we have ν ∈ +s∗P . (b) We say (π, V ) is H -relatively tempered if it is (H , λ)-relatively tempered for all λ ∈ HomH (π, 1). By the Casselman criterion (Theorem 3.12), this definition coincides with the previous one if π admits a unitary central character. Remark 3.16 This definition is also used in [1]. Also it is shown in [2] that this is equivalent to saying that ϕλ,v is a tempered function, as in the non-relative case.

4 Subrepresentation Theorem In this section, we will prove the subrepresentation theorem discussed in the introduction.

4.1 Subrepresentation Theorems First let us recall the following theorem by Kato-Takano, which is a relative version of the well-known Jacquet’s subrepresentation theorem. Theorem 4.1 (Theorem 7.1 of [6]) Let (π, V ) be an irreducible H -relatively square integrable representation. Then π is a subrepresentation of an induced representation IndPG τ , where P = M U is a θ-split parabolic and τ is an irreducible M θ -relatively cuspidal representation. Further if the involution θ is standard, then for each choice of , one can always choose P to be standard. Proof The first part is [6, Theorem 7.1]. Now assume θ is standard and fix . By the first part, we already have π ⊆ IndPG τ for some θ-split P = M U . One can always choose h ∈ H such that h P is standard. Then one can see IndPG τ ∼ = Ind Gh P h τ , and h τ h θ h θ h θ  is ( M ) -relatively cuspidal since ( M ) = (M ). Remark 4.2 In this theorem the inducing representation τ is (M θ , λ)-relatively cuspidal for all λ ∈ HomM θ (τ , 1).

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Now the following is our main theorem, which is a relative version of [11, Proposition III.4.1] which asserts that every tempered representation can be embedded into an induced representation whose inducing data is a square integrable representation. Namely we will prove Theorem 4.3 Let (π, V ) be an (H , λ)-relatively tempered representation for some λ ∈ HomH (π, 1) which admits a unitary central character. Then there exists a nonzero G-intertwining map π −→ IndPG τ , where P = M U is θ-split and τ is an irreducible M θ -relatively square integrable representation of M . In particular, if π is irreducible, it is a direct summand of IndPG τ . Further if the involution θ is standard, then for each choice of  one can always choose P to be standard.

4.2 Some Lemmas To prove this theorem, we need some preparations. First we need Lemma 4.4 Let π be a representation of G with a unitary central character, and let ν ∈ a∗G . Then exp(ν) ⊗ π is H -distinguished if and only if exp(ν)|H = 1 and π is H -distinguished. Moreover, we have HomH (exp(ν) ⊗ π, 1) = HomH (π, 1), where the equality is literally given by the equality on the spaces of linear functionals on the space Vexp(ν)⊗τ = Vτ . Proof The if-part is obvious. To show the only-if-part, note that if exp(ν) ⊗ τ is H -distinguished, then its central character is trivial on ZGθ = ZG ∩ H , and hence its absolute value is trivial on ZGθ . Namely exp(ν)|ZGθ = 1. Thus exp(ν) = 1 by [6, Lemma 1.10 (2)]. Once we know this, it is immediate that π is H -distinguished, and the equality of the hom spaces is clear.  Next, we need Lemma 4.5 Let (π, V ) be an H -distinguished representation of G which admits a unitary central character. Assume π is (H , λ)-relatively square integrable for λ ∈ HomH (π, 1). Then λ factors through a nonzero unitary semi-simple quotient (π λ , V λ ) of (π, V ). In particular, (π, V ) has an irreducible unitary quotient distinguished by λ. Proof As discussed in Sect. 3.1, λ always factors through the image of the map Tλ,π : V → C ∞ (H \G). But since π is relatively square integrable, the image of Tλ,π is actually in L2 (ZG H \G), which is unitary. Let us denote this image by (π λ , V λ ).

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Since π is admissible, so is (π λ , V λ ). So (π λ , V λ ) is semi-simple. (Recall that a unitary admissible representation is semi-simple.)  This implies Lemma 4.6 Let (π, V ) be a finitely generated H -distinguished representation of G, and let λ ∈ HomH (π, 1) be fixed. Assume that for every θ-split P one has ν ∈ +s∗P for all ν ∈ ExpSP (πP , λ). (Namely π satisfies the Casselman criterion for (H , λ)relative square integrability.) Then π has an irreducible distinguished quotient σ which is (H , λ )-relatively square integrable for some λ ∈ HomH (σ, 1). (Note that if π admits a unitary central character, this is immediate from the previous lemma. But here we do not assume it.) Proof As in the proof of the previous lemma, λ always factors through the image of the map Tλ,π : V → C ∞ (H \G). Let us call the image , so that  ⊆ C ∞ (H \G). Note that every subrepresentation of  is distinguished by λ. Since  is a quotient of π, it also satisfies the Casselman criterion, and thus every subrepresentation of  that admits a central character is (H , λ)-relatively square integrable. (Note that since  satisfies the Casselman criterion, the central character has to be unitary.) Now by [6, Lemma 1.7],  has a quotient representation σ  isomorphic to a subrepresentation σ  ⊆  which admits a central character. Note that, being an (H , λ)-distinguished subrepresentation of , σ  is (H , λ)-relatively square integrable, and hence by the previous lemma λ|Vσ factors through a semisimple quotient. But being a subrepresentation of the image  of Tλ,π , we know that λ|Vσ does not factor through any proper subquotient of σ  , which implies σ  is already semisimple. Hence σ  is semi-simple, and is (H , λ )-relatively square integrable, where λ is the composite ∼

λ

σ  −→ σ  − → C. So we can take σ to be any irreducible constituent of σ  .



4.3 Weak Jacquet Module Our proof of the subrepresentation theorem is a relative version of [11, Proposition III.4.1], in which one considers “weak Jacquet modules”. To introduce this notion, let (π, V ) be a finitely generated admissible H -distinguished representation of our reductive group G and P = M U be a θ-split parabolic subgroup. Recall that the Jacquet module (πP , VP ) is written as (πP , VP ) =

(πP,χ , VP,χ ),

χ∈ExpSP (πP )

where VP,χ is the generalized χ-eigenspace, and each VP,χ is invariant under M . For a fixed λ ∈ HomH (π, 1) one can further write (πP , VP ) = (πPλ , VPλ ) ⊕ (πP0 , VP0 ),

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VPλ :=



VP,χ and VP0 :=

χ∈ExpSP (πP ,λ)

VP,χ .

χ∈Exp / SP (πP ,λ)

Namely, rP (λ) is nonzero on each summand VP,χ of the space VPλ , but identically zero on VP0 . By following [11, p. 267], let us write (πPλ , VPλ ) = (πPλ,w , VPλ,w ) ⊕ (πPλ,+ , VPλ,+ ), where

VPλ,w :=

VP,χ and VPλ,+ :=

χ∈ExpSP (πP ,λ) Re(χ)=0

VP,χ .

χ∈ExpSP (πP ,λ) Re(χ)=0

We call VPλ,w the weak Jacquet module with respect to λ, and hence the superscript w . We then have Lemma 4.7 Let (π, V ) be a finitely generated (H , λ)-relatively tempered representation which does not necessarily admit a central character. The representation (πPλ,w , VPλ,w ) (if nonzero) is (M θ , rP (λ))-relatively tempered for all θ-split P. Proof We will apply the Casselman criterion for relative temperedness to (πPλ,w , VPλ,w ). Let χ ∈ ExpSP (πP , (λ)), where P  is a θ-split parabolic subgroup such that P  ⊆ P. But since for all χ ∈ ExpSP (πP , λ) such that VP,χ ⊆ VPλ,w we have Re(χ) = ∗

∗

0, the projection of Re(χ ) onto s∗P is zero, and hence Re(χ ) ∈ +sPP := +s∗P ∩ sPP , which shows (πPλ,w , VPλ,w ) is relatively tempered. (Note that the representation πPλ,w is a representation of MP and hence the Casselman criterion has to be with respect ∗  to +sPP .) This implies Lemma 4.8 Let (π, V ) be a finitely generated (H , λ)-relatively tempered representation which does not necessarily admit a central character. Then the following conditions are equivalent. (a) For all proper θ-split parabolic subgroups P, one has ν ∈ +s∗P for all ν ∈ ExpSP (πP , λ). (b) For all proper θ-split parabolic subgroups P, one has VPλ,w = 0. (c) For all proper maximal θ-split parabolic subgroups P, one has VPλ,w = 0. Further if (π, V ) admits a unitary central character, (b), (c) and the following (d) are all equivalent. (d) (π, V ) is (H , λ)-relatively square integrable. Finally, if the involution θ is standard, then all the above P’s may be taken to be standard.

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Proof (a) ⇒ (b): This is immediate. (b) ⇒ (c): This is immediate, too. (c) ⇒ (a): Let ν ∈ ExpSP (πP , λ), where P is a proper θ-split parabolic subgroup. Further let Q ⊇ P be a proper maximal θ-split parabolic subgroup and let νQ ∈ ExpSQ (πQ , λ) be the image of ν under the surjection ExpSP (πP , λ) → ExpSQ (πQ , λ), which is the vertical arrow in Proposition 3.4 with P and Q switched. Since π is (H , λ)-relatively tempered, by definition we have νQ ∈ +s∗Q . Also since we have VQλ,w = 0 by our assumption (c), we must have νQ = 0. Now if νQ ∈ +s∗Q and νQ = 0, then νQ ∈ +s∗Q because +s∗Q is a “one-dimensional ray”. Since Q is arbitrary, we have νQ ∈ +s∗Q for all maximal θ-split Q ⊇ P, which implies ν ∈ +s∗P . The equivalences of (b),(c) and (d) are immediate from the Casselman criterion for relative square integrability. Finally in the case of standard θ, the equivalences follow because the Casselman criterion has to be checked only for standard P. 

4.4 Proof of Theorem and Its Corollaries Now finally we are ready to prove the subrepresentation theorem. Proof of Theorem 4.3. By following the argument adapted by [7, proof of Theorem 7.1], we will argue by induction on the rank of the maximal θ-split torus of AG \G. The base step is when the rank is 0, in which case HAG \G is compact by [5, §4.3], and hence π is already relatively square integrable, and there is nothing to show because one can just take P = G. Assume the rank is > 0. If (π, V ) is already (H , λ)-relatively square integrable, then again there is nothing to show. So assume it is not (H , λ)-relatively square integrable. Then by Lemma 4.8, VPλ,w = 0 for some proper θ-split P. Let us take P = M U to be minimal with respect to this property. Since π is (H , λ)-relatively tempered, VPλ,w is (M θ , rP (λ))-relatively tempered by Lemma 4.7. But since (VPλ,w )rQP (λ),w = 0 for any smaller parabolic Q  P, we know by Lemma 4.8 that all the relative exponents of VPλ,w with respect to rp (λ) satisfy the Casselman criterion for relative square integrability. Hence by Lemma 4.6, VPλ,w has an irreducible distinguished quotient σ which is (M θ , λ )-relatively square integrable for some λ . By Frobenius reciprocity, we have nonzero π → IndPG σ. Here since σ is (M θ , λ )-relatively square integrable and hence is (M θ , λ )relatively tempered, and the rank of the maximal θ-split torus of AM \M is strictly smaller, one can apply the induction hypothesis to σ, and obtain σ → IndM Q∩M τ for some θ-split Q ∩ M = MQ UQ and MQθ -relatively square integrable τ . By inducing in stages, we have the desired π → IndQG τ . By Lemma 4.5, we know that τ is unitary. Hence IndQG τ is unitary, which implies that IndPG τ is semi-simple. So if π is irreducible, then π is indeed a direct summand.

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Finally, if the involution θ is standard, one can use Lemma 4.8 only for standard parabolic subgroups, and hence both P and Q in the above argument can be chosen to be standard.  The following is an immediate corollary. Corollary 4.9 Let (π, V ) be an irreducible H -distinguished representation which is (H , λ)-relatively tempered for some λ ∈ HomH (π, 1). Then π is unitary. Proof By Theorem 4.3, we have π → IndPG τ for some relatively square integrable τ , which is unitary by Lemma 4.5. Hence π is unitary because IndPG τ is unitary.  Now recall that for any representation (π, V ) of G, the complex conjugate representation (π, V ) is defined as follows: The space V is the same as V as an abelian group but the scalar multiplication is via complex conjugate, and π is defined by π(g)v = π(g)v for all v ∈ V = V and g ∈ G. Then there is a natural isomorphism ∼

HomH (π, 1) −→ HomH (π, 1), λ → λ, where λ is the complex conjugate of λ, namely λ(v) = λ(v) for v ∈ V . If (π, V ) is an irreducible unitary representation with a unitary structure −, −π , its contragredient (π ∨ , V ∨ ) is identified with (π, V ) via the map ∼

(π, V ) −→ (π ∨ , V ∨ ), v → −, vπ , which induces the isomorphism ∼

HomH (π, 1) −→ HomH (π ∨ , 1), λ → λ∨ , where we define λ∨ to be the composite ∼

λ

λ∨ : π ∨ −→ π − → C. So in particular, if (π, V ) is unitary, then (π, V ) is H -distinguished if and only if (π ∨ , V ∨ ) is H -distinguished. With this said, we have Corollary 4.10 Let (π, V ) be irreducible. Then (π, V ) is (H , λ)-relatively tempered for some λ ∈ HomH (π, 1) if and only if (π ∨ , V ∨ ) is (H , λ∨ )-relatively tempered for some λ∨ ∈ HomH (π ∨ , 1). Also (π, V ) is H -relatively tempered if and only if (π ∨ , V ∨ ) is H -relatively tempered. Proof By symmetry, we have only to show the only-if part. So assume (π, V ) is (H , λ)-relatively tempered for some λ ∈ HomH (π, 1). By the above corollary, π is unitary, and hence we can identify (π ∨ , V ∨ ) with (π, V ). Then one can readily see that (π, V ) is (H , λ)-relatively tempered. The first assertion follows. ∼ Since we have the natural isomorphism Hom H (π, 1) −→ HomH (π, 1), the second assertion follows. 

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Acknowledgements The author was partially supported by Simons Foundations Collaboration Grant #35952, NSA Young Investigator Grant H98230-16-1-0312, and an AMS Centennial Fellowship for 2017–2018. Part of the paper was completed while he was visiting the National University of Singapore in fall 2017 and spring 2018, and he would like to express thanks for their hospitality. He would like to thank Wee Teck Gan for useful conversations and Kimball Martin for introducing him to the works of Kato and Takano. Finally he would like to thank Kenji Takano for sending him the preprint [8].

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