Eisenstein Cohomology for GL_N and the Special Values of Rankin–Selberg L-Functions

Table of contents :
Contents......Page 7
Preface......Page 9
1 Introduction......Page 15
2 The Cohomology of GLn......Page 26
3 Analytic Tools......Page 42
4 Boundary Cohomology......Page 54
5 The Strongly Inner Spectrum and Applications......Page 65
6 Eisenstein Cohomology......Page 85
7 L-Functions......Page 98
8 Harish-Chandra Modules over Z......Page 140
9 The Archimedean Intertwining Operator for GLN......Page 182
Bibliography......Page 225
Index......Page 231

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Annals of Mathematics Studies Number 203

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Eisenstein Cohomology for GLN and the Special Values of Rankin–Selberg L-Functions

G¨unter Harder A. Raghuram

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2020

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c 2020 Princeton University Press

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved ISBN: 978-0-691-19788-3 ISBN (pbk): 978-0-691-19789-0 British Library Cataloging-in-Publication Data is available Editorial: Vickie Kearn, Susannah Shoemaker, and Lauren Bucca Production Editorial: Nathan Carr Text Design: Leslie Flis Jacket/Cover Design: Leslie Fils Production: Jacquie Poirier Publicity: Matthew Taylor and Katie Lewis The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. This book has been composed in LaTeX. Printed on acid-free paper. Printed in the United States of America

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For Lilo and Nita

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Faust: ...Da muß sich manches R¨atsel l¨osen. Mephistopheles: Doch manches R¨atsel kn¨ upft sich auch... (Faust: Eine Trag¨ odie - Kapitel 24, von Goethe)

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Contents

Preface

ix

1 Introduction 1.1 The general context 1.2 Strongly inner cohomology and applications 1.3 The main theorem on Eisenstein cohomology 1.4 The main theorem on arithmetic of Rankin–Selberg L-functions 1.5 Some general remarks

1 3 4 7 8 10

2 The 2.1 2.2 2.3 2.4

Cohomology of GLn The ad`elic locally symmetric space fλ Highest weight modules Mλ and the sheaves M fλ Cohomology of the sheaves M Integral cohomology

12 12 14 21 25

3 Analytic Tools 3.1 Cuspidal parameters and the representation Dλ at infinity 3.2 Square-integrable cohomology 3.3 Cuspidal cohomology

28 28 34 37

4 Boundary Cohomology 4.1 A spectral sequence converging to boundary cohomology 4.2 Cohomology of ∂P S G as an induced representation 4.3 Cuspidal spectrum versus residual spectrum

40 40 42 46

5 The 5.1 5.2 5.3

51 51 54 60

Strongly Inner Spectrum and Applications The strongly inner spectrum: definition and properties Definition of the relative periods The strongly inner cohomology of the boundary

6 Eisenstein Cohomology 6.1 Poincar´e duality and maximal isotropic subspace of boundary cohomology 6.2 The main result on rank-one Eisenstein cohomology 6.3 A theorem of Langlands: the constant term of an Eisenstein series

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71 71 73 75

viii

CONTENTS

7 L-Functions 7.1 Motivic and cohomological L-functions 7.2 Critical points for L-functions and the combinatorial lemma 7.3 The main result on special values of L-functions 7.4 Some remarks

84 84 92 107 119

8 Harish-Chandra Modules over Z (by G¨ unter Harder) 8.1 Introduction 8.2 Harish-Chandra modules over Z 8.3 First examples 8.4 Induction of Harish-Chandra modules

126 126 128 132 143

9 The Archimedean Intertwining Operator (by Uwe Weselmann) 9.1 The general setting 9.2 The intertwining operators 9.3 J-admissible permutations 9.4 Representations and L-functions 9.5 The main theorem on archimedean intertwining operator 9.6 Applications to cohomology

168 169 171 179 181 186 199

Bibliography

211

Index

217

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Preface The first author of this monograph studied Eisenstein cohomology for GL2 over a number field F in [24] and proved rationality results for critical values for the L-functions attached to algebraic Hecke characters of F . With a view to generalizing, he then studied the situation of rank-one Eisenstein cohomology for SL3 /Q to deduce information about ratios of successive critical values for the Hecke L-function L(s, f ) of a holomorphic modular form f . The results of this work appeared in [29]. On attempting a similar study for Eisenstein cohomology for G = GL4 , especially in regard to the contribution from the maximal parabolic P with Levi quotient MP = GL3 × GL1 , he observed that certain maps were always degenerating to the zero map. After he asked Birgit Speh for an explanation, she wrote him a letter to show that a particular standard G(R) intertwining operator (from an induced representation IndP (R) (σ), where σ is a cohomological representation of MP (R), to its companion induced representation that is induced from the parabolic subgroup associate to P ) happens to always give the zero map in relative Lie algebra cohomology. Around this time, the second author was visiting Germany, and both authors discussed Speh’s letter to see what really was going on. Such discussions form the genesis of this project. The main aim of this monograph is to study rank-one Eisenstein cohomology for the group GLN /F, where F is a totally real field extension of Q. This is then used to prove rationality results for ratios of successive critical values for Rankin– Selberg L-functions for GLn × GLn0 over F with the parity condition that nn0 is even. The key idea is to interpret Langlands’s constant term theorem in terms of Eisenstein cohomology. The short announcement [33] contained the main results for GLn × GLn0 over Q and in the case when n is even and n0 is odd. In the meantime it became clear that the methods and results can be extended to the case when both n and n0 are even and also to the situation when the base field is a totally real field. There is a great deal of current work in collaborations involving the authors and others (most notably with Chandrasheel Bhagwat and Muthu Krishnamurthy) about many other Langlands–Shahidi L-functions to which the methods of this monograph are applicable. It is a special pleasure to thank Uwe Weselmann, who gave elegant proofs of two very technical ingredients: his proof of the combinatorial lemma appears in Sect. 7.2.4, and his calculation of the archimedean contribution to our main global results is all of Chap. 9. We also thank Chandrasheel Bhagwat whose

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x

PREFACE

discussions with the second author on the failure of the main results when nn0 is odd, appear in Sect. 7.4.1. It is a pleasure for both authors to acknowledge the support of several institutions and mathematical colleagues over the past decade during which time this project developed. It all started during a hike in the hills in the Black Forest near the Mathematisches Forschugsinstitut Oberwolfach (MFO) in 2008 when the authors met for a conference organized by Steve Kudla and Joachim Schwermer. It evolved further during meetings at the Erwin Schr¨odinger Institute, Vienna; their invitations are gratefully acknowledged. The authors are thankful to Don Blasius, Harald Grobner, Benedict Gross, Michael Harris, and Freydoon Shahidi for their interest and helpful discussions. The second author is grateful to the Max Planck Institut f¨ ur Mathematik (MPIM), an Alexander von Humboldt Fellowship, a National Science Foundation grant (DMS-0856113), and travel grants from Oklahoma State University and IISER Pune that funded his visits to MPIM on innumerable occasions. Finally, both authors are grateful to MFO for hosting them as part of their Research in Pairs program in May, 2014, where a first draft of the manuscript was written. The second author acknowledges funding from the Charles Simonyi Endowment during his stay at the Institute for Advanced Study, Princeton, in spring and summer of 2018, during which time this monograph was prepared for publication in the Annals of Mathematics Studies. The authors are grateful to their wives, Lilo and Nita, for their unconditional support during this arduous journey that started a decade ago and culminated with this book, which in turn opens the doors for so many questions that seem to guarantee long sleepless nights of mathematical wonder. G¨ unter Harder & A. Raghuram August 28, 2019

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xi

The authors (GH on the right and AR on the left) at the Mathematisches Forschungsinstitut Oberwolfach, Germany, in May of 2014, where they spent a week as part of a Research in Pairs program.

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Eisenstein Cohomology for GLN and the Special Values of Rankin–Selberg L-Functions

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Chapter One Introduction

This monograph should be thought of as a disquisition of the general principle: The cohomology of arithmetic groups and the L-functions L(s, π, r) attached to irreducible “pieces” π have a strong symbiotic relationship with each other. The symbiosis goes in both directions: (A) Expressions in the special values L(k, π, r) enter in the transcendental description of the cohomology. Since the cohomology is defined over Q we can deduce rationality (algebraicity) results for these expressions in special values. (B) These special values in turn influence the structure of the cohomology as a Hecke module; prime numbers dividing these values occur in the denominators of Eisenstein classes. Let G be a connected reductive group over Q. A fundamental problem is to G understand the geometry of a locally symmetric space SK attached to G for f some level structure Kf . In particular, one seeks to understand the cohomology G f of S G with coefficients in a local system M f arising from a groups H • (SK , M) Kf f finite-dimensional representation of the algebraic group G. On the other hand, there are families of Langlands–Shahidi L-functions L(s, π, r), attached to an automorphic representation π of a Levi subgroup MP of G and an algebraic representation r of the Langlands dual group of MP . The above dictum takes the G f shape: How do the L-functions L(s, π, r) influence the structure of H • (SK , M)? f Let’s illustrate this with the help of two well-known examples: Let A denote the adele ring of Q. A basic problem in modern number theory is to study the decomposition of the space of automorphic forms A(G(Q)\G(A)) as a representation of G(A). This space has two parts: the discrete spectrum and the continuous spectrum. Within the discrete spectrum, we have the cuspidal spectrum and the residual (discrete, but not cuspidal) spectrum. Since Langlands’s groundbreaking work on Eisenstein series one knows that the holomorphy properties, such as the location of poles, of various L-functions influence the description of these spectra, and this in turn carries over to the cohomology G fC ) after changing the base field to C and then relating them groups H • (SK ,M f to the relative Lie algebra cohomology of the space of automorphic forms. This point plays a decisive role in this monograph when we discuss the implications

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2

CHAPTER 1

of the work of Mœglin–Waldspurger on the discrete spectrum for the structure of the cohomology; see Sect. 4.3. A more subtle influence of L-functions on cohomology is afforded by the study of congruences. Primes that appear in the algebraic part of a special value of some L-function are usuallyPcongruence primes. For example, consider ∞ the Ramanujan ∆-function ∆(z) = n=1 τ (n)e2πinz , the unique cusp form (up to scalars) of weight 12 for Γ = SL2 (Z). Ramanujan had empirically observed that τ (p) ≡ p11 + 1 (mod 691) for all primes p. The prime 691 appears in the numerator of the special value ζ(−11) = 691/32760 of the Riemann zeta function. These ingredients give us information about the cohomology of the fn is the sheaf on quotient of the upper half-space H by Γ. More generally, if M Γ\H corresponding to the irreducible representation of SL2 of dimension n + 1, fn ) is intimately linked then the denominator of an Eisenstein class in H 1 (Γ\H, M to the numerator of ζ(−1−n). For an elaboration of this very beautiful example, see [30, Chap. V]. In this monograph we discuss essentially only results of type (A), but these results allow us to formulate conjectures in the direction of (B) (see [28]). In fact, we study a different kind of influence, albeit somewhat related to the second example given earlier, which we now proceed to describe. Let P/Q be a maximal parabolic subgroup of G with MP its Levi quotient. Let σ f be an irreducible Hecke summand appearing in the cohomology of MP (with coefficients in some local system). Then the algebraically and parabolically induced module a IndG P (σ f ) appears in the cohomology of the Borel–Serre G boundary of SK for some Kf . If Q is the parabolic subgroup associated to P , f w then we may also consider the induced module a IndG Q ( σ f ), which appears in w a different part of the boundary cohomology; here σ f is a certain conjugate of σ f by a Weyl group element w. The machinery of Eisenstein cohomology, which mitigates a relation between total cohomology and the cohomology of the boundary, lets us compare these two induced modules working entirely within G the world of the cohomology of SK . On changing the base to C, and invoking f the relation to automorphic forms, these modules may be compared using the standard intertwining operator, which, by a famous Qm calculation of Langlands, contains within it information about the product i=1 L(is, σ, r˜i ), where on the dual group side we have L UP = r1 ⊕ · · · ⊕ rm as a representation for the adjoint action of L MP .

Let’s simplify to our context: G = GLN and P is the maximal parabolic subgroup with Levi subgroup MP = GLn × GLn0 , with n + n0 = N ; then σ = σ ⊗ σ 0 and w σ = σ 0 ⊗ σ; also m = 1 and L(s, σ, r˜1 ) is nothing but the Rankin–Selberg L-function L(s, σ × σ 0v ), where σ 0v is the contragredient of σ 0 . It turns out that the sum a

IndG P (σ f ) ⊕

a

∗ IndG Q (σ f )

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3

INTRODUCTION

(where σ ∗f is closely related to w σ f ) splits off as a Hecke summand from the cohomology of the boundary, and a certain ratio of L-values L(s0 , σ × σ 0v ) L(s0 + 1, σ × σ 0v ) is a key ingredient in the description of the image of global cohomology in the sum of these two induced modules. Furthermore, this description is possible if and only if s0 and s0 + 1 are both critical (in the sense of Deligne) for the L-function at hand. One expects the primes appearing in the algebraic part of this ratio to have a delicate bearing on the structure of integral cohomology leading to a study of congruences. In other words, the special values of various Rankin–Selberg L-functions, which may be defined as Langlands–Shahidi L-functions using an ambient GLN , influence the structure of the cohomology of GLN . This monograph is an extended cerebration of this idea.

1.1

THE GENERAL CONTEXT

For the moment, let G = ResF/Q (GLn /F ), where F is a totally real field extension of Q. We fix a maximal torus T inside a Borel subgroup B of G. For an open-compact subgroup Kf ⊂ G(Af ), where Af is the ring of finite ad`eles G of Q, let SK be the ad`elic locally symmetric space; see Sect. 2.1. Let E be a f Galois extension of Q that contains a copy of F. For a dominant integral weight λ ∈ X ∗ (T ) we let Mλ,E be the algebraic irreducible representation of G × E fλ,E be the associated sheaf on S G ; see Sect. 2.2. The fundamental and let M Kf objects of interest are the sheaf-theoretic cohomology groups M G G fλ,E ) = fλ,E ). H • (SK ,M H q (SK ,M f f q G G The main tool that we use is the Borel–Serre compactification S¯K of SK . If f f G G ∂SKf is the boundary of S¯Kf , then we have the following long exact sequence (Sect. 2.3.4): •

•

i r G G G fλ,E ) −→ fλ,E ) −→ fλ,E ) −→ · · · −→ Hci (SK ,M H i (S¯K ,M H i (∂SK ,M f f f G fλ,E ) −→ · · · −→ Hci+1 (SK ,M f

The image of cohomology with compact supports inside the full cohomology is called inner or interior cohomology and is denoted H •! := Image(i• ) = Im(Hc• → H • ). (See Sect. 2.3.4.) All these cohomology groups are finite dimensional vector spaces over E. They are Hecke-modules; i.e., there is an action Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 1:59 PM

4

CHAPTER 1

G of π0 (G(R)) × HK . The inner cohomology is a semisimple module under the f Hecke algebra and if E/Q is large enough then we get an isotypical decomposition: M G G fλ,E ) = fλ,E )(πf ), H •! (SK ,M H •! (SK ,M f f πf ∈Coh! (G,Kf ,λ)

where Coh! (G, Kf , λ) is the finite set of isomorphism types of any absolutely G irreducible HK module that occurs with (a finite) nonzero multiplicity in this f decomposition. We may also pass to the limit over all open-compact Kf and fλ,E ), and to retrieve the coget an action of π0 (G(R)) × G(Af ) on H • (S G , M homology group for a particular level-structure Kf , we can take invariants: fλ,E )Kf = H • (S G , M fλ,E ); the definitions of the Hecke algebra and H • (S G , M Kf such Hecke actions are reviewed in Sect. 2.3.2. We may pass to a transcendental level by taking an embedding ι : E → C, G fιλ,C ). and then use the theory of automorphic forms on GLn to study H • (SK ,M f It is well known that inner cohomology over C contains cuspidal cohomology and is contained in square-integrable cohomology; i.e., we have a chain of inclusions: • G G fιλ,C ) ⊂ H!• (SK fιλ,C ) ⊂ Hcusp (SK ,M ,M f f • G G fιλ,C ) ⊂ H • (SK fλ,E ) ⊗E,ι C; H(2) (SK ,M ,M f f

see Sect. 3.3.4. The square-integrable cohomology, via Borel–Garland [8] and Borel–Casselman [7], is captured by the discrete spectrum of GLn ; see Sect. 3.2.3. Furthermore, cuspidal cohomology is understood using results about the possible infinite components of cohomological cuspidal representations; we use here the fact that cuspidal representations are globally generic (i.e., have a Whittaker model) and hence locally generic; the local components at infinity are reviewed in Sect. 3.1.

1.2

STRONGLY INNER COHOMOLOGY AND APPLICATIONS

Our first theorem in this monograph is an arithmetic characterization of cuspidal G fλ,E ) of inner cohomology, which cohomology. We identify a subspace H!!• (SK ,M f we call strongly inner, that by definition is spanned by all those Hecke modules inside inner cohomology whose isotypic component in global cohomology is captured already by the isotypic component in inner cohomology; see Sect. 5.1. Strongly inner cohomology splits off in global cohomology via a Manin–Drinfeld principle, and we get a canonical decomposition G fλ,E ) = H • (S G , M fλ,E ) ⊕ H • (S G , M fλ,E ), H • (SK ,M !! Kf Eis Kf f

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5

INTRODUCTION

which gives an arithmetic definition of Eisenstein cohomology inside global cohomology; see (5.3). If we go to a transcendental level via ι : E → C then we have G fιλ,C ) = H • (S G , M fιλ,C ). H!!• (SK ,M cusp Kf f See Thm. 5.2, where we summarize all the known characterizations of cuspidal cohomology. The proofs of the assertions about strongly inner cohomology involve unG fλ,E ), and especially derstanding the cohomology of the boundary H • (∂SK ,M f being able to pick out the Hecke modules that do not appear in boundary coG homology; this is the subject matter of all of Sect. 4. The boundary ∂SK is f G stratified as ∪P ∂P SKf , with a stratum corresponding to every G(Q)-conjugacy class of parabolic subgroups. There is a spectral sequence built from the cohoG mology of ∂P SK that converges to the cohomology of boundary; this is briefly f G reviewed in Sect. 4.1. Furthermore, for a single stratum ∂P SK , its cohomology f • G f H (∂P SKf , Mλ,E ) may be described in terms of the algebraic induction of the cohomology of the Levi quotient MP of P with coefficient systems depending on λ and the set W P of Kostant representatives for P in the Weyl group of G; see Prop. 4.3. The proofs about strongly inner cohomology also make use of multiplicity one and strong multiplicity one results of Jacquet–Shalika [38], [39] and Mœglin–Waldspurger [52].

1.2.1

The relative periods

As a first application of strongly inner cohomology, we describe how to attach certain periods that play an important role in the results on special values of L-functions. For this paragraph we take n to be even and let πf ∈ Coh!! (G, λ), i.e., πf is an irreducible Hecke module contributing to strongly inner cohomology. Taking E to be large enough, we know that for every character ε of ˜F π0 (G(R)), and for the cohomology degree q being an extremal degree bF n or tn G fλ,E ) with multiplicity (see Prop. 3.4), the module πf × ε appears in H!!q (SK ,M f ε one. We fix an arithmetic identification Tarith (λ, πf ) between the occurrences of πf ⊗ ε and πf ⊗ −ε in degree bF n ; see (5.4). Then, we go to a transcendental level ε via ι : E → C, and fix an identification Ttrans (ιλ, ιπf ) between the occurrences ι ι of πf ⊗ ε and πf ⊗ −ε in cuspidal cohomology in degree bF n by a map described purely in terms of the relative Lie algebra cohomology groups at infinity; see Sect. 5.2.2. We define a nonzero complex number, which we call a relative period, as the discrepancy between these two identifications: ε ε Ωε (ιλ, ιπf ) Ttrans (ιλ, ιπf ) = Tarith (λ, πf ) ⊗E,ι 1.

Varying ι, the family of relative periods attached to πf gives a well defined element of (E⊗C)× /E × . Sometimes we suppress λ and write Ωε (ιπf ) for Ωε (ιλ, ιπf ).

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CHAPTER 1

1.2.2

Manin–Drinfeld principle for boundary cohomology

As a second application of strongly inner cohomology, we go back to boundary cohomology and prove a strong form of the Manin–Drinfeld principle, by showing G fλ,E ) splits off as an isotypical Hecke that a certain E-subspace of H • (∂SK ,M f module; see Thm. 5.12. To explain this result, take N = n + n0 and now let G = ResF/Q (GLN /F ), and P = ResF/Q (P0 /F ) = MP UP be the standard (i.e., contains the standard Borel subgroup of upper triangular matrices) maximal parabolic subgroup with Levi quotient MP = ResF/Q (MP0 ) = Gn × Gn0 := ResF/Q (GLn /F ) × ResF/Q (GLn0 /F ). Let Q = ResF/Q (Q0 ) = MQ UQ be the standard associate parabolic subgroup of P with Levi quotient MQ = Gn0 ×Gn . There is a distinguished element wP ∈ W that conjugates P into the opposite Q− of Q and induces an identification ∼ wP∗ : MP −→ MQ . Take pure weights µ and µ0 for Gn and Gn0 ; only pure weights can support cuspidal or strongly inner cohomology; see § 3.1.2. Let σf ∈ Coh!! (Gn , µ) and σf0 ∈ Coh!! (Gn0 , µ0 ). We make a crucial combinatorial assumption on the weights µ and µ0 , that there is a Kostant representative w ∈ W P such that if we write w = (wτ )τ :F →E then the length l(wτ ) of each component is dim(UP0 )/2 = nn0 /2 and w−1 ·(µ+µ0 ) is dominant as a weight for G. An obvious necessary condition for the existence of such a w is that dim(UP0 ) = nn0 is even. Without loss of generality we will take n even and n0 of any parity. This condition on µ and µ0 has other equivalent formulations that are captured by our combinatorial lemma; see Sect. 7.2.3. Sect. 7.2.4 has a proof of this lemma by Uwe Weselmann. A consequence is that the representation algebraically (un-normalized) parabolically induced from σf ⊗ σf0 appears in boundary cohomology: a

G(A )

F

fλ,E ), IndP (Aff ) (σf ⊗ σf0 ) ,→ H bN (∂P S G , M

F F −1 where bF · (µ + µ0 ). N = bn + bn0 + dim(UP )/2 and λ = w

Pick a level structure Kf ⊂ G(Af ), which satisfies wP Kf wP−1 = Kf and so that the induced representation has Kf -fixed vectors to get a Hecke-stable F fλ,E )Kf . We denote this kk-dimensional subspace (for some k) in H bN (∂P S G , M S 0 0 Kf dimensional space as Ib (σf , σf , ε )P,w , where w is coming from the combinatorial lemma, and ε0 is a signature discussed in (5.21). The element w ∈ W P has an associate w0 ∈ W Q that is also balanced in the sense that l(w0τ ) = dim(UQ0 )/2 = nn0 /2. Another consequence is that a

G(A )

F

fλ,E ), IndQ(Aff ) (σf0 (n) ⊗ σf (−n0 )) ,→ H bN (∂Q S G , M

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7

INTRODUCTION

K

F

f bN fλ,E )Kf . The Manin–Drinfeld prindenoted IbS (σf , σf0 , ε0 )Q,w (∂Q S G , M 0 , in H ciple, stated in Thm. 5.12, says that the space

K

K

f f IbS (σf , σf0 , ε0 )P,w ⊕ IbS (σf0 (−n), σf (n0 ), ε0 )Q,w 0,

which is of dimension 2k, splits off as an isotypical Hecke summand inside F fλ,E )Kf . Furthermore, the sum of the duals (or contragredients) of H bN (∂S G , M F fλv ,E )Kf , these modules splits off as an isotypical summand inside H t˜N −1 (∂S G , M F where the degree t˜N − 1 is coming from the top degrees; see Sect. 5.3.7. For this introduction, we let F fλ,E )Kf H bN (∂S G , M

R

 Kf S 0 0 Kf S 0 Ib (σf , σf , ε )P,w ⊕ Ib (σf (−n), σf (n0 ), ε0 )Q,w 0 denote this Hecke projection in the bottom degree.

1.3

THE MAIN THEOREM ON EISENSTEIN COHOMOLOGY

We now come to our main result on rank-one Eisenstein cohomology which is stated as Thm. 6.2. It states that the image of full cohomology under the composition of the maps R ◦ r∗ as in F

G fλ,E ) H bN (SK ,M f

F



r∗

G fλ,E ) H bN (∂SK ,M f R

 Kf S 0 0 Kf S 0 Ib (σf , σf , ε )P,w ⊕ Ib (σf (−n), σf (n0 ), ε0 )Q,w 0 is a k-dimensional E-subspace of the 2k-dimensional target space. There are two aspects to the proof: 1. One aspect is purely cohomological and says that the Eisenstein part of boundary cohomology is a maximal isotropic subspace under Poincar´e duality (Prop. 6.1) bounding the dimension of the image by k; see Sect. 6.2.2.2. 2. The other aspect of the proof is transcendental and appeals to the theory of

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8

CHAPTER 1

automorphic forms; it also gives information about the internal structure of this image. Let’s elaborate on (2): Base change to C via an embedding ι : E → C, and then use Langlands’s constant term theorem, recalled in Thm. 6.3, which says that the constant term relative to Q of an Eisenstein series built from a section f of an induced representation from P is the same as the standard intertwining operator Tst applied to f . For the interpretation of Langlands’s theorem in cohomology, the reader should compare the diagrams (6.17) and (6.18). This interpretation implies in particular that the required image contains all classes of the form (ξ, Tst∗ ξ); see (6.14); i.e., the image is at least a k-dimensional subspace in a 2k-dimensional vector space. Putting both the aspects together, we conclude that n o
Kf P Image(R ◦ r∗ ) = ξ, TEis (ξ) : ξ ∈ IbS (σf , σf0 , ε0 )P,w , K

K

f f P for an operator TEis : IbS (σf , σf0 , ε0 )P,w → IbS (σf0 (−n), σf (n0 ), ε0 )Q,w 0 defined over E. It is of course possible that the Eisenstein series constructed from a section f picks up a pole at the point of evaluation that happens to be s = −N/2; but in this case we record in Prop. 7.10 that if we are to the right of the unitary axis then this Eisenstein series is in fact holomorphic. If, on the other hand, we are to the left of the unitary axis, then we start from Q and, in this case, the image Kf Q would consist of classes of the form (TEis (ψ), ψ) with ψ ∈ IbS (σf , σf0 , ε0 )Q,w 0 . It is helpful to think about a situation when k = 1; then the image is a line in a two-dimensional space, and we will see later that the slope of this line carries arithmetic information about ratios of L-values.

1.4

THE MAIN THEOREM ON ARITHMETIC OF RANKIN–SELBERG L-FUNCTIONS

Our main result on special values of L-functions (see Thm. 7.21) follows from considering the slope mentioned earlier; i.e., we analyze classes of the form Q P (ξ, TEis (ξ)), or of the form (TEis (ψ), ψ). Passing to a transcendental level, we rewrite such a class in terms of the standard intertwining operator, which is given by an integral. Normalizing the local standard intertwining operator using appropriate L-values, we get an operator denoted Tloc . At finite places one checks that the operator is nonzero and holomorphic using results of Mœglin–Waldspurger [52]. Furthermore, at every finite place, one observes that the intertwining operator has nice rational properties. At an archimedean place, using Speh’s results (see, for example, [51, Theorem 10b]) on reducibility for induced representations for GLN (R), one sees that Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 1:59 PM

9

INTRODUCTION

the induced representations at hand are irreducible; next, using Shahidi’s results [64] on local factors and the fact that −N/2 and 1 − N/2 are critical, we deduce that the standard intertwining operator is both holomorphic and nonzero, and therefore induces an isomorphism at the level of relative Lie algebra cohomology groups that are one-dimensional, and after making certain careful choices of generators, this scalar turns out to be a power of (2πi) as shown in Chap. 8, and independently in Chap. 9. Then we descend to an arithmetic level via the relative periods and the arithmetic identification mentioned earlier. This exercise gives us a rationality result for such a ratio of L-values divided by the relative periods. Before stating the result, let us mention that the Rankin–Selberg L-functions at hand may be thought of from different points of view: 1. As the motivic L-function attached to the tensor product of the pure motives M(σf ) and M(σf0 )v that are conjecturally attached to σf and σf0 . 2. As the cohomological L-function attached to σf and σf0 , and this is entirely from the perspective of Hecke action on the cohomology of arithmetic groups; we will denote this as Lcoh (ι, σ × σ 0v , s) for any embedding ι : E → C. 3. As the automorphic Rankin–Selberg L-functions attached to a pair of cuspidal automorphic representations. The interplay among these three points of view is reviewed in Sect. 7.1. Suppose n is even and n0 is odd then Thm. 7.21 says: 1 Lcoh (ι, σ × σ 0v , m0 ) ∈ ι(E), Ω (ισf ) Lcoh (ι, σ × σ 0v , 1 + m0 ) ε0

(1.1)

and moreover this ratio of L-values divided by the period is well behaved under the absolute Galois group of Q. On the other hand, if both n and n0 are even, then we have Lcoh (ι, σ × σ 0v , m0 ) ∈ ι(E), (1.2) Lcoh (ι, σ × σ 0v , 1 + m0 ) which is also well behaved under Galois automorphisms. The point of evaluation m0 corresponds to the point −N/2 for the automorphic L-function. The combinatorial lemma mentioned earlier has an important consequence: we may replace σf , say, by Tate twists σf (k) for k ∈ Z, and try to get other ratios of L-values; the lemma, however, imposes some restrictions on the set of such permissible k, and it turns out that we get a rationality result for exactly all the successive pairs of critical values, no more and no less!

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10 1.5

CHAPTER 1

SOME GENERAL REMARKS

We may refine this entire discussion and consider integral structures on the 0 cohomology. Such a refinement of our reasoning implies that the periods Ωε (ισf ) × are well defined modulo a group of units OE,S where we inverted primes out of a small well-controlled set S of primes. Then we can speak of the prime factorizations of the expressions in (1.1) and (1.2) and ask whether the primes appearing in such expressions are visible in the structure of the cohomology. A first instance of such an event is discussed in Harder [28], where this choice of the periods was simply an educated guess. Such and related issues are also addressed in Harder [27] and [31]. The preceding results on critical values are compatible with Deligne’s conjecture [15] on the critical values of motivic L-functions. This compatibility is proved by Bhagwat and Raghuram [3] by proving an appropriate period relation for the ratio c+ /c− of Deligne’s periods for the tensor product motive M(σf ) ⊗ M(σf0 ). These period relations were later generalized in Bhagwat [2], and subsequently put into a general framework in Deligne and Raghuram [16]. It is interesting to note that such period relations exist for all possible combinations of parities of n and n0 . However, the methods of this article seem to break down when n and n0 are both odd. In Sect. 7.4.1, Bhagwat and the second author show that the analogue of the combinatorial lemma does not hold: if we ask for a situation in which we have two successive critical values then it is shown in Prop. 7.26 that there does not exist an element of the Weyl group, let alone a Kostant representative, which would be needed to arrange for an induced module to appear in boundary cohomology. (Recall the comment in the Preface about Speh’s letter, which is indeed in such a situation; n = 3 and n0 = 1.) In the papers by Kazhdan, Mazur, and Schmidt [44]; Kasten and Schmidt [43]; and Grenie [21], the authors use the theory of modular symbols to get information on the special values of L-functions for GLn × GLn−1 /Q (and GLn × GLn over a totally imaginary field in [21]). They even get rationality (algebraicity) results for the individual L-values, instead of ratios of L-values; see [43, Thm. A]. The question they raise in their introduction on the nonvanishing of ± their periods P∞,j (j + 21 ) has been settled in a paper by Sun [68]. But it is not ± clear to us whether they have enough control on P∞,j (j + 12 ) to derive our result from their theorem (only in the case of GLn × GLn−1 ). In a sense, we have two entirely different approaches to get an understanding of the nature of special values of these L-functions at critical points, both starting from the study of the cohomology of arithmetic groups. In the present monograph we carry out a step in a larger program that is outlined by Harder [30], where the general program of investigating Eisenstein cohomology is discussed. This includes a detailed study of the restriction map from ri fλ,E ) −→ global cohomology to the cohomology of the boundary: H i (S¯G , M Kf

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INTRODUCTION

11

G fλ,E ). One has to understand the target space and determine the H i (∂SK ,M f image of r• . Our guiding dictum may be phrased as: theorems on the structure of these cohomology groups—as modules under the Hecke algebra—have number theoretic consequences. We also hope to get interesting number theoretic applications if we go deeper into the cohomology, or in other words, to higher filtration steps coming from the spectral sequence that converges to the cohomology of the boundary.

One may also study such number-theoretic applications of Eisenstein cohomology for other reductive groups G. The techniques and methods enunciated in this monograph have already begun to bear results in some other contexts: The reader is referred to Raghuram [58] for an analogous study of Rankin– Selberg L-functions for GLn × GLn0 over a totally imaginary field; to Bhagwat and Raghuram [4] for the case of Eisenstein cohomology for SO(n+1, n+1) over a totally real base field and the special values of L-functions for SO(n, n) × GL1 ; and to Krishnamurthy and Raghuram [48] for the case of Eisenstein cohomology for unitary groups and arithmetic consequences for Asai L-functions attached to GLn over a real quadratic field.

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Chapter Two The Cohomology of GLn

` THE ADELIC LOCALLY SYMMETRIC SPACE

2.1 2.1.1

The base field

Let F be a totally real number field of degree r = rF = [F : Q]. By a number field we mean a finite extension of Q. Let OF be the ring of integers of F . Let ¯ be the field of S∞ = Hom(F, R) denote the set of archimedean places. Let Q algebraic numbers in C. We identify the sets ¯ = Hom(F, C) = Hom(F, R) = S∞ . ΣF := Hom(F, Q)

2.1.2

The groups

Let G0 = GLn /F , and put G = ResF/Q (G0 ) = ResF/Q (GLn /F ). An F -group will be denoted by G0 , H0 , etc., and the corresponding Q-group, via Weil restriction of scalars, will be denoted by the same letter without the subscript. For any Q-algebra A, we have G(A) = G0 (A ⊗Q F ) = GLn (A ⊗Q F ). Let B0 stand for the standard Borel subgroup of G0 consisting of all upper triangular matrices, T0 the standard torus of all diagonal matrices in B0 , and U0 the unipotent radical of B0 . The center of G0 will be denoted by Z0 . These groups define the corresponding Q-groups, and we have G ⊃ B = T U ⊃ T ⊃ Z ⊃ S, where S is the maximal Q-split torus in Z. Let’s identify S ∼ = Gm /Q, by sending x ∈ Gm ( ) to the diagonal matrix with all entries equal to x. We have the norm character NF/Q : Z → Gm , and if we restrict to S then it becomes x 7→ xr . The character group X ∗ (Gm ) = Z, with the character x 7→ xk denoted by [k]. (1)

Let G0

(1)

stand for the group SLn /F and G(1) = ResF/Q (G0 ). The super-

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13

THE COHOMOLOGY OF GLN

script (1) will mean that we have intersected with SLn ; for example, T (1) = T ∩ SLn .

2.1.3

The symmetric space

For any topological group G, let G 0 stand for the connected component of the 0 identity, and πQ connected components. Note 0 (G) = G/G stand for the group of Q that G(R) = v∈S∞ GLn (R). Similarly, Z(R) ∼ = v∈S∞ R× , where each copy of R× consists of nonzero scalar matrices in the corresponding copy Q of GLn (R). The subgroup S(R) of Z(R) consists of R× diagonally embedded in v∈S∞ R× . Q (1) The group K∞ := v∈S∞ SO(n) is a maximal compact subgroup of G(1) (R) and the corresponding Cartan involution Θ on G(1) (R) is given by g 7→ t g −1 on Q each factor. Similarly, C∞ := v∈S∞ O(n) is a maximal compact subgroup of G(R). Let K∞ := C∞ S(R) = C∞ S(R)0 ,

hence

0 (1) K∞ = K∞ × S(R)0 .

The torus T (1) ×Q R is the maximal split torus which is invariant under Θ. Let 0 T [2] denote the 2-torsion subgroup of T (R), then K∞ = K∞ · T [2]. Inclusion induces a canonical identification π0 (K∞ ) = π0 (G(R)) that is isomorphic to an r-fold product of Z/2Z. The (generalized) symmetric space is defined as 0 X := G(R)/K∞ . On this space we have an action of T [2] that acts transitively on the set of connected components.

2.1.4

The ad` elic locally symmetric space

Let A be the ring of ad`eles of Q, which we decompose into its infinite and its finite part: A = R × Af . The group of ad`eles is given by G(A) = G(R) × G(Af ). Elements in the ad`elic group are denoted by underlined letters g, h, etc., and the decomposition of an element Q g into its infinite and its finite part will be denoted g = g∞ × g f . Let Kf = p Kp ⊂ G(Af ) be an open-compact subgroup. The ad`elic symmetric space is 0 0 G(A)/K∞ Kf = X × (G(Af )/Kf ) = G(R)/K∞ × (G(Af )/Kf ). 0 It is a product of the symmetric space X = G(R)/K∞ and an infinite discrete set G(Af )/Kf . On this space G(Q) acts properly discontinuously and we get a quotient 0 G(R)/K∞ × G(Af )/Kf (2.1)



π


0 G(Q)\ G(R)/K∞ × G(Af )/Kf .

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14

CHAPTER 2

The target space, called the ad`elic locally symmetric space of G with level structure Kf , is denoted G 0 SK := G(Q)\G(A)/K∞ Kf . f To get an idea of what this space looks like, consider the action of G(Q) on the discrete space G(Af )/Kf . It follows from classical finiteness results (i) that this quotient is finite; let {g f }i=m i=1 be a finite set of representatives of (i)

G(Q)\G(Af )/Kf . The stabilizer of the coset g f Kf in G(Q) is equal to Γi := Γ

g (i) f

:= G(Q) ∩ g (i) Kf (g (i) )−1 , f f

which is an arithmetic subgroup of G(Q) that acts properly discontinuously on g (i)

X. We say Kf is neat if all the subgroups Γ f are torsion free. For any choice of Kf we can pass to a subgroup Kf0 of finite index in Kf that is neat; we may take Kf0 to be normal in Kf . We have the following decomposition of the adelic locally symmetric space G ∼ SK = f

m a

0 Γi \G(R)/K∞ .

i=1 (i)

Let g = g∞ × g f ∈ G(A); there is a unique index i such that g f Kf = γg f Kf for some γ ∈ G(Q). We will leave it to the reader to check that the map that sends 0 G 0 G(Q) g K∞ Kf in SK to Γi γ −1 g∞ K∞ sets up the required homeomorphism. f

2.2

2.2.1

HIGHEST WEIGHT MODULES Mλ AND THE SHEAVES fλ M The character module of T /Q

Let E/Q be a finite Galois extension and assume that Hom(F, E) 6= 0. We denote this set of embeddings by {τ : F → E}, on which we have a transitive action of the Galois group Gal(E/Q). The base change T ×Q E is a split torus; more precisely, dropping the subscript Q for simplicity, we have Y Y T ×E = T0 ×F,τ E = T0 . τ :F →E

τ :F →E

Often, the field E will be taken to be large enough (so that, for example, some module can be split off over E) and we will analyze the behavior of objects as we change E to a field E 0 always subject to the condition of being Galois over Q and containing an isomorphic copy of F . Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:00 PM

15

THE COHOMOLOGY OF GLN

Consider the group of characters of the torus T over E X ∗ (T × E) = Hom(T × E, Gm ), on which there is a natural action of Gal(E/Q). Since T = RF/Q (T0 ), we have X ∗ (T × E) =

M

M

X ∗ (T0 ×τ E) =

τ :F →E

X ∗ (T0 ),

(2.2)

τ :F →E

and any element λ ∈ X ∗ (T × E) is of the form λ = (λτ )τ :F →E , with λτ ∈ X ∗ (T0 ×τ E) a characterQ of the split torus T0 ×τ E (since, under our hypothesis on E, we have F ⊗Q E = τ :F →E E). Any η ∈ Gal(E/Q) acts on λ ∈ X ∗ (T ×E) by permutations η

λ = ((η λ)τ )τ :F →E = (λη

It is easy to see that

2.2.2

η1 η2

−1

◦τ

)τ :F →E .

(2.3)

λ = η1 (η2 λ) for all η1 , η2 ∈ Gal(E/Q).

The rationality field of λ ∈ X ∗ (T × E)

Define the rationality field E(λ) of any λ ∈ X ∗ (T × E) as E(λ) := E {η∈Gal(E/Q)

:

η

λ=λ}

;

i.e., it is the subfield of E fixed by the subgroup of Gal(E/Q) which fixes λ. We have the following proposition. Proposition 2.1. The field E(λ) is the subfield of E generated by the values of λ on T (Q). Q Proof. Let t ∈ T (Q) = T0 (F ) ,→ T (E) = τ :F →E T0 (τ (F )). Suppose we write t = diag(t1 , . . . , tn ) ∈ T0 (F ); then t 7→ (τ (t))τ ∈ T (E), where τ (t) = diag(τ (t1 ), . . . , τ (tn )) ∈ T0 (τ (F )). Given λ ∈ X ∗ (T × E) written as λ = (λτ )τ , let η ∈ Gal(E/Q) be such that −1 −1 η λ = λ. This means that (λτ )τ = (λη ◦τ )τ or that λτ = λη ◦τ for all τ : F → E. For t ∈ T (Q) as above we have Y Y η(λ(t)) = η( λτ (τ (t))) = λτ (η(τ (t))) τ

τ

=

Y µ

λη

−1

◦µ

(µ(t)) =

Y

λµ (µ(t)) = λ(t).

µ

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16

CHAPTER 2

This gives Q(λ) := Q({λ(t) : t ∈ T (Q)}) ⊂ E {η∈Gal(E/Q)

:

η

λ=λ}

,

or that {η ∈ Gal(E/Q) : η λ = λ} ⊂ Gal(E/Q(λ)). To prove the reverse inclusion, let η ∈ Gal(E/Q(λ)). Then by definition η(λ(t)) = λ(t) for all t ∈ T (Q) or that the algebraic characters η λ and λ agree on T (Q), which is Zariski dense in T (F ); hence η λ = λ, whence {η ∈ Gal(E/Q) : η λ = λ} = Gal(E/Q(λ)). The notation employed in the proof suggests that there is a canonical definition of the rationality field Q(λ) that is independent of the ambient splitting field E.

2.2.3

The Weyl group and the pairing on X ∗ (T (1) )

The normalizer N (T ) of the maximal torus is a subgroup of G/Q whose connected component of the identity is T /Q. The quotient N (T )/T = W is a finite algebraic group scheme over Q. For the base change G × E ⊃ N (T ) × E ⊃ T × E we have W(E) = N (T )(E)/T (E) and W(E) =: W will be the absolute Weyl group, which is a product Y W = W0τ τ :F →E

W0τ

with each isomorphic to W0 the symmetric group in n letters, which is the Weyl group of G0 /F with respect to T0 /F. The Galois group Gal(E/Q) acts on W by permutations as in (2.3). There is a positive definite symmetric pairing h , i : X ∗ (T (1) × E) × X ∗ (T (1) × E) −→ R, which is invariant under the action of W. The direct sum decomposition (2.2) for T (1) is orthogonal with respect to this pairing, and on each summand it is unique up to a scalar. Restriction of characters gives an inclusion X ∗ (T × E) ⊂ X ∗ (T (1) × E) ⊕ X ∗ (Z × E). We extend the form h , i trivially by zero on X ∗ (Z × E); and the Weyl group action is also trivial on this summand.

2.2.4

Standard basis and fundamental basis

For the moment we only consider T0 /F. The character module X ∗ (T0 ) is a free abelian group on {e1 , . . . , en }, where, for any t = diag(t1 , . . . , tn ) ∈ T0 (A), with A an F -algebra, weP have ei (t) = ti ∈ A× . ForQintegers b1 , . . . , bn , the character λ = (b1 , . . . , bn ) := bi ei is given by λ(t) = i tbi i . We will call {e1 , . . . , en } Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:00 PM

17

THE COHOMOLOGY OF GLN

the standard basis for X ∗ (T0 ). For example, the determinant character is given by n X δn := det = (1, . . . , 1) = ei . i=1

The simple roots for SLn or GLn are given by {α1 , . . . , αn−1 },

αi := ei − ei+1 .

The fundamental weights {γ1 , . . . , γn−1 } in XQ∗ (T0 ) := X ∗ (T0 ) ⊗ Q are defined by 2hγi , αj i = δij , and γi |Z0 = 0. hαj , αj i (This makes sense only when γi is in XQ∗ (T0 ).) In terms of the standard basis the fundamental weights are given by   i i i i i γi = (e1 + · · · + ei ) − δn = 1 − , · · · , 1 − , − , · · · , − . n n n n n The basis for XQ∗ (T0 ) given by {γ1 , . . . , γi , . . . , γn−1 , δn } will be called the fundamental basis. This basis respects the decomposition (1)

T0 = T0

· Z0 ,

(up to isogeny); (1)

i.e., restriction of characters gives an inclusion X ∗ (T0 ) ⊂ X ∗ (T0 ) ⊕ X ∗ (Z0 ), which, after tensoring by Q, becomes an isomorphism: XQ∗ (T0 ) ∼ = XQ∗ (T0 (1) ) ⊕ XQ∗ (Z0 ). (1)

(1)

The restriction to T0 of {γ1 , . . . , γn−1 } spans X ∗ (T0 ), and the determinant character δn spans XQ∗ (Z0 ). (1)

For example, half the sum of positive roots for G0 or G0 and is given by ρn :=

n−1 X i=1

 γi =

n−1 n−3 −(n − 1) , , ··· , 2 2 2

(1)

is in XQ∗ (T0 ),

 .

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CHAPTER 2

Given λ ∈ XQ∗ (T0 ) we will write λ =

n−1 X

(ai − 1)γi + d · δn .

i=1

The (ai − 1) might seem strange here, but has the virtue that it will simplify some expressions later on. (See, for example, the discussion that follows on motivic weight.) The above expression for λ is the same as writing λ + ρn = Pn−1 i=1 ai γi + d · δn . Let’s write λ = λ(1) + λab ,

λ(1) :=

n−1 X

(ai − 1)γi ,

λab := d · δn ,

(2.4)

i=1

and call λ(1) the semisimple part of λ, and λab the abelian part of λ. Let’s describe the dictionary betweenPthe standard and the fundamental n−1 bases. Let λ ∈ XQ∗ (T0 ) be written as λ = i=1 (ai − 1)γi + d · δn . Formally, as a character of T0 , it may be written as t = diag(t1 , . . . , tn ) 7→ λ(t) a +a2 +···+an−1 −(n−1)

= t1 1

a +···+an−1 −(n−2)

· t2 2

where rλ := (nd −

n−1 X

a

n−1 · · · tn−1

−1

· (t1 · · · tn )rλ ,

i(ai − 1))/n.

i=1

It’s clear then that in the standard basis, if λ = (b1 , . . . , bn ) then b1 = a1 + a2 + · · · + an−1 − (n − 1) + rλ , b2 = a2 + · · · + an−1 − (n − 2) + rλ , .. .

(2.5)

bn−1 = an−1 − 1 + rλ , bn = rλ . Conversely, ai − 1 = bi − bi+1 ,

for 1 ≤ i ≤ n − 1, and

d = (b1 + · · · + bn )/n.

(2.6)

All the above notations are adapted for characters of T /Q = RF/Q (T0 ).

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19

THE COHOMOLOGY OF GLN

Hence, for λ = (λτ )τ :F →E in XQ∗ (T × E), we have: λτ =

n−1 X

(aτi − 1)γi + dτ · δn = (bτ1 , . . . , bτn ).

(2.7)

i=1

If we define ρn = (ρn )τ :F →E then we can form λ + ρn and get (λ + ρn )τ =

n−1 X

aτi γi + dτ · δn .

(2.8)

i=1

We also put δn = (δn )τ :F →E ; actually we are interested only in the case that dτ = d for all τ, and in this case we write λ = λ(1) + dδn ; see Lem. 2.3.

2.2.5

Integral weights Pn−1 Let λ = i=1 (ai − 1)γi + d · δn = (b1 , . . . , bn ) ∈ XQ∗ (T0 ). We say that λ is an integral weight if and only if λ ∈ X ∗ (T0 ), which is the same as saying that bi ∈ Z for all i. In terms of the fundamental basis: λ ∈ XQ∗ (T0 ) is integral if and only if  ai ∈ Z, 1 ≤ i ≤ n − 1,      nd ∈ Z, λ ∈ X ∗ (T0 ) ⇐⇒ (2.9)     Pn−1  nd ≡ i=1 i(ai − 1) (mod n). The last congruence condition is the same as saying that rλ ∈ Z. A weight λ = (λτ )τ :F →E ∈ XQ∗ (T × E) is integral if and only if each λτ is integral.

2.2.6

Dominant integral weights Pn−1 Let λ = (b1 , . . . , bn ) = i=1 (ai −1)γi +d·δ ∈ X ∗ (T0 ) be an integral weight. For the choice of the Borel subgroup being the standard upper triangular subgroup B0 , λ is dominant if and only if ai ≥ 1 for 1 ≤ i ≤ n − 1 (no condition on d) ⇐⇒ b1 ≥ b2 ≥ · · · ≥ bn . (2.10) A weight λ = (λτ )τ :F →E ∈ XQ∗ (T × E) is dominant-integral if and only if each λτ is dominant-integral. Denote the set of dominant-integral weights for T0 (resp., T ) by X + (T0 ) (resp., X + (T × E)). Remark 2.2. If λ is a dominant-integral weight then the relevant information is given by its semisimple part λ(1) . The number d is uninteresting, the class of nd

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20

CHAPTER 2

mod Z is determined by λ(1) , and if we modify d → d + m we have a canonical identification of the resulting cohomology groups; see Sect. 5.2.4.

2.2.7

The representation (ρλ , Mλ )

Let λ ∈ X + (T × E) be a dominant-integral weight and (ρλ , Mλ ) be the absolutely irreducible finite-dimensional representation of G×E(λ) of highest weight λ. We can get hold of Mλ after going to a large enough Galois extension E/Q and descend to E(λ). Put O Mλ,E = Mλτ , τ :F →E

where Mλτ /E is the absolutely irreducible finite-dimensional representation of G0 ×τ E = GLn /F ×τ E with highest weight λτ . If necessary, we will write (ρλτ , Mλτ ) for this representation. The group G(Q) = GLn (F ) acts on Mλ,E diagonally; i.e., γ ∈ G(Q) acts on a pure tensor ⊗τ mτ via γ · (⊗τ mτ ) = ⊗τ (τ (γ) · mτ ).

(2.11)

We can produce a descent datum and show that Mλ,E is defined over E(λ), and an E(λ)-structure, which will be unique up to homotheties by E × , will be denoted Mλ .

2.2.8

fλ and algebraic dominant-integral weights The sheaf M

For the moment, let M be a finite-dimensional Q-vector space and ρ : G/Q → GL(M) be a rational representation of the algebraic group G/Q. This represenf of Q-vector spaces on S G : the sections over an tation ρ provides a sheaf M Kf G open subset V ⊂ SK are given by f f ) = {s : π −1 (V ) → M | s locally constant and M(V s(γv) = ρ(γ)s(v), γ ∈ G(Q), v ∈ π −1 (V )}, (2.12) where π is as in (2.1). Take M = Mλ as in §2.2.7; if the central character of ρλ fλ is the is not the type of an algebraic Hecke character of F then the sheaf M zero sheaf. (See [24, 1.1.3].) We record this as the following lemma. Pn−1 Lemma 2.3. Let λ ∈ X + (T × E); λ = (λτ )τ :F →E with λτ = i=1 (aτi − 1)γi + fλ = 0. dτ · δ. If there exist τ1 and τ2 in Hom(F, E) such that dτ1 6= dτ2 then M τ1 τ2 f Equivalently, if Mλ 6= 0 then λab = λab .

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21

THE COHOMOLOGY OF GLN

Proof. If some dτ1 6= dτ2 then consider the central character of ρλ on a suitable congruence subgroup of the units in the diagonal center: OF× ' S(OF ) ⊂ GLn (F ) = G(Q) to see that every stalk is zero, and hence the sheaf is the zero sheaf. + Define Xalg (T × E) to be the subset of those dominant-integral weights that satisfy the algebraicity condition that dτ1 = dτ2 for all τ1 and τ2 in Hom(F, E); i.e., + Xalg (T × E) := {λ ∈ X + (T × E) : λτab1 = λτab2 , ∀τ1 , τ2 ∈ Hom(F, E)}.

Henceforth, we will consider only such algebraic, dominant, and integral highest weights λ. Algebraicity means that the central character ωλ ∈ X ∗ (T ×E) factors via the norm character NF/Q ; i.e., ωλ (x) = NF/Q (x)d ,

(2.13)

and its restriction to S = Gm is given by ωλ (y) = y rnd .

(2.14)

fλ is isomorphic to Mλ and the sheaf M fλ If Kf is neat then every stalk of M fλ is a sheaf of E(λ)-vector spaces, and the base is a local system. Note that M fλ,E of E-vector spaces on S G . change Mλ,E = Mλ ⊗E(λ) E gives a sheaf M Kf

2.3

fλ COHOMOLOGY OF THE SHEAVES M

A fundamental problem at the heart of this monograph is to understand the arithmetic information contained in the sheaf-theoretically defined cohomology G fλ,E ). groups H • (SK ,M f 2.3.1

Functorial properties upon changing the level structure

An inclusion K1,f ⊂ K2,f of open-compact subgroups gives a canonical surjecG G tive map SK  SK , which in turn induces a canonical map 1,f 2,f G G fλ,E ) → H • (SK fλ,E ). H • (SK ,M ,M 2,f 1,f

Pass to the limit over all open-compact subgroups Kf and define fλ,E ) := lim H • (S G , M fλ,E ). H • (S G , M Kf −→ Kf

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22

CHAPTER 2

On this limit, there is an action of π0 (G(R)) × G(Af ), which we now describe. An element x∞ ∈ π0 (G(R)) = π0 (K∞ ) is represented by an element of T [2] and so normalizes K∞ . Let xf ∈ G(Af ) and put x = x∞ × xf . Right multiplication G by x, i.e., g 7→ gx, induces a map mx : SK −→ SxG−1 K x . This gives a canonical f f

f

f

map in cohomology: m•x : H • (SxG−1 K

f xf

f

G fλ,E ) −→ H • (SK fλ,E ). , mx,∗ M ,M f

(This is because, for any continuous map f : X → Y of topological spaces, and for any sheaf F of abelian groups (say) on X, we have a canonical map in fλ,E = M fλ,E sheaf cohomology H • (Y, f∗ F) → H • (X, F).) Next, we have mx,∗ M G as sheaves on Sx−1 K x ; this is an easy verification using the definition of the f

f

f

sheaves. Hence, any x gives a canonical map m•x : H • (SxG−1 K f

f xf

G fλ,E ) −→ H • (SK fλ,E ). ,M ,M f

fλ,E ). The Passing to the limit over all Kf gives the action of x on H • (S G , M cohomology for the spaces with level structure can then be retrieved by taking invariants: G fλ,E ) ∼ fλ,E )Kf . H • (SK ,M = H • (S G , M f

2.3.2

Hecke action

We let G HK = Cc∞ (G(Af )//Kf ) f

be the set of all locally constant and compactly supported bi-Kf -invariant Qvalued functions on G(Af ). Take the Haar measure on G(Af ) to be the product of local Haar measures, and for every prime p, the local measure is normalized so G that vol(G(Zp )) = 1. Then HK is a Q-algebra under convolution of functions. f fλ,E ), after taking Kf -invariants, The π0 (G(R))×G(Af ) action on H • (S G , M Kf

induces an action of π0 (G(R)) ×

2.3.3

G HK f

G fλ,E ). on H • (SK ,M f

Functorial properties upon changing the field E

Consider another field E 0 , also Galois over Q with an injection ι : E → E 0 . Then ι induces an isomorphism X ∗ (T × E) −→ X ∗ (T × E 0 ) written as λ 7→ ιλ. The map τ 7→ ι ◦ τ is a bijection from Hom(F, E) onto Hom(F, E 0 ). Hence any τ 0 ∈ Hom(F, E 0 ) is of the form τ 0 = ι ◦ τ for a unique τ ∈ Hom(F, E) and we put τ = ι−1 ◦ τ 0 . If λ = (λτ )τ :F →E then ι

λ = (λτ )ι◦τ :F →E 0 = (λι

−1

◦τ 0

)τ 0 :F →E 0 .

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23

THE COHOMOLOGY OF GLN ∼

We get an identification ι∗ : Mλ,E ⊗E,ι E 0 −→ Mιλ,E 0 . This yields an identifi∼ f fλ,E ⊗E,ι E 0 −→ cation M Mλ,E 0 of sheaves that induces an isomorphism ∼ G G fλ,E ) ⊗E,ι E 0 −→ fιλ,E 0 ). ι• : H • (SK ,M H • (SK ,M f f G The map ι• is π0 (G(R)) × HK -equivariant, since it came from a morphism of f sheaves whereas the Hecke action on these cohomology groups was intrinsic to G the space SK . f

We may assume E = E 0 ; then ι is an element of the Galois group. If η1 , η2 ∈ Gal(E/Q) then it is clear that (η1 ◦ η2 )∗ = η1∗ ◦ η2∗ , and hence (η1 ◦ η2 )• = η1• ◦ η2• , i.e., •

η2 G fη2λ,E ) fλ,E ) / H • (S G , M H • (SK ,M Kf f NNN NNN NNN NNN NN η1• (η1 ◦η2 )• NNN NNN NNN &  G fη1 ◦η2λ,E ) H • (SK ,M f

We say that the system of cohomology groups G fη λ,E )}η∈Gal(E/Q) {H • (SK ,M f

is defined over Q.

2.3.4

The fundamental long exact sequence

G G G G G Let S¯K be the Borel–Serre compactification of SK ; i.e., S¯K = SK ∪ ∂SK , f f f f f G G where the boundary is stratified as ∂SKf = ∪P ∂P SKf with P running through the G(Q)-conjugacy classes of proper parabolic subgroups defined over Q. (See fλ,E on S G naturally extends, using the definiBorel–Serre [10].) The sheaf M Kf G tion of the Borel–Serre compactification, to a sheaf on S¯K that we also denote f G G fλ,E . The inclusion S ,→ S¯ is a homotopy equivalence, and hence the by M Kf Kf restriction from S¯G to S G induces an isomorphism in cohomology Kf

Kf

∼

G G fλ,E ) −→ H • (SK fλ,E ). H • (S¯K ,M ,M f f G fλ,E ) and the cohomology with The cohomology of the boundary H • (∂SK ,M f G fλ,E ) are naturally modules for π0 (G(R)) × HG ; compact supports Hc• (SK ,M Kf f the Hecke action on these other cohomology groups are described exactly as in Sect. 2.3.2. Our basic object of interest is the following long exact sequence of

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24

CHAPTER 2

G π0 (G(R)) × HK -modules: f •

•

i r G G G fλ,E ) −→ fλ,E ) −→ fλ,E ) · · · −→ Hci (SK ,M H i (S¯K ,M H i (∂SK ,M f f f G fλ,E ) −→ · · · . −→ Hci+1 (SK ,M f

The image of cohomology with compact supports inside the full cohomology is called inner or interior cohomology and is denoted   G G G fλ,E ) := Image(i• ) = Image Hc• (SK fλ,E ) → H • (SK fλ,E ) . H •! (SK , M , M , M f f f Complementary to inner cohomology is the theory of Eisenstein cohomology, which is designed to describe the image of the restriction map r• . Let’s note a slight abuse of terminology: inner cohomology is not an honest-to-goodness cohomology theory; for example, a short exact sequence of sheaves would not give a long exact sequence in inner cohomology. However, the terminology is very convenient and helps with our geometric intuition of the nature of the cohomology classes therein. Our considerations in Sect. 2.3.3 about functorial properties on changing the G fλ,E ), where base field E apply verbatim to the cohomology groups H?• (SK ,M f fλ,E ) we mean H • (∂S G , M fλ,E ). ? ∈ {empty, c, !, ∂}; by H • (S G , M ∂

2.3.5

Kf

Kf

Inner cohomology and the inner spectrum Coh! (G, λ)

Inner cohomology is a semisimple module for the Hecke algebra (see Sect. 3.2.3). After taking E/Q to be a sufficiently large finite Galois extension there is an isotypical decomposition: M G G H •! (SK , Mλ,E ) = H •! (SK , Mλ,E )(πf ), (2.15) f f πf ∈Coh! (G,Kf ,λ) G where πf is an isomorphism type of an absolutely irreducible HK -module; i.e., f G there is an E-vector space Vπf with an absolutely irreducible action πf of HK . f G ∞ Let HKp = Cc (G(Qp )//Kp ) be the local Hecke algebra consisting of all locally constant and compactly supported bi-Kp -invariant Q-valued functions on G G(Qp ). The local factors HK are commutative outside a finite set S = SKf p of primes and the factors for two different primes commute with each other. G For p 6∈ S the commutative algebra HK acts on Vπf by a homomorphism p G πp : HKp → E. Let Vπp be the one-dimensional E-vector space E with the distinguished basis element 1 ∈ E and with the action πp on it. Then

Vπf = Vπf ,S ⊗0p6∈S Vπp = ⊗p∈S Vπp ⊗ E,

(2.16)

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25

THE COHOMOLOGY OF GLN

G where the absolutely irreducible HK - module Vπf ,S module is decomposed as f ,S G a tensor product Vπf ,S = ⊗p∈S Vπp of absolutely irreducible HK -modules. We p decompose the Hecke algebra G,S G G G G HK = HK × ⊗p6∈S HK = HK × HK , p f f ,S f ,S f

where the first factor acts on the first factor Vπf ,S and the second factor acts G,S via the homomorphism πfS : HK → E. The set Coh! (G, Kf , λ) of isomorphism f classes that occur with strictly positive multiplicity in (2.15) is called the inner spectrum of G with λ-coefficients and level structure Kf . Taking the union over all Kf , the inner spectrum of G with λ-coefficients is defined to be [ Coh! (G, λ) = Coh! (G, Kf , λ). Kf

Going back to functorial properties on changing the field E via ι : E → E 0 , note that the map ι• , since it came from a morphism of sheaves, preserves inner cohomology: G G fλ,E ) −→ H!• (SK fιλ,E 0 ). ι• : H!• (SK ,M ,M f f Given πf ∈ Coh! (G, Kf , λ), we deduce ι πf ∈ Coh! (G, Kf , ιλ) and that the πf isotypic component is mapped by ι• onto the ι πf -isotypic component. Furthermore, given any character ε of π0 (G(R)) we have   G G fλ,E )(πf × ε) = H!• (SK fιλ,E 0 )(ι πf × ε), ι• H!• (SK ,M ,M f f which, via an abuse of notation, we may write as ι• (πf × ε) = ι πf × ε. If E is large enough so that the πf -isotypic component can be split off from inner cohomology, then we can define the rationality field of πf as E(πf ) := E {η∈Gal(E/Q)

:

η

πf =πf }

.

One may see, using strong multiplicity one for GLn /F , that the field E(πf ) is the subfield of E generated by values of πp for p ∈ / S.

2.4

INTEGRAL COHOMOLOGY

Let’s briefly discuss how to refine many of the foregoing considerations to talk about integral sheaves and their cohomology and why this is interesting. This aspect is not so relevant for the results in this book, but it will become relevant when we apply certain refinements of the results to arithmetic questions. The reader should consult [27] and [30] for more details. Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:00 PM

26

CHAPTER 2

Let OF be the ring of integers of our field F/Q. Instead of starting from algebraic groups over F we start from the split reductive group scheme GLn /OF and consider the affine group scheme G/Z = ROF /Z (GLn /OF ). Accordingly we define the Borel subgroup B/Z and the “torus” T /Z. We take the flag variety of all Borel subgroups X /Z = B\G. (See, for example, Demazure and Grothendieck [17, XXII, 5.8].) Let w0 be the element of the Weyl group of longest length that is represented by an element of G(Z). Let λ ∈ X ∗ (T ×E); the weight w0 (λ) gives a line bundle Lw0 (λ) on X × OE and this line bundle Lw0 (λ) has global sections if and only if λ is dominant. Let A(G) be the algebra of regular functions on G/Z. Then Mλ,O = H 0 (B\G, Lw0 (λ) ) = {f ∈ A(G) ⊗ OE : f (bg) = w0 (λ)(b)f (g), ∀b ∈ B(OE ), ∀g ∈ G(OE )}. This is a module for G/Z, where the ρλ -action of g ∈ G(OE ) on any f ∈ Mλ,O is by right-shifts: (ρλ (g)(f ))(g 0 ) = f (g 0 g),

∀g, g 0 ∈ G(OE ).

This is a finitely generated locally free OE module and Mλ,O ⊗ E is our old module Mλ (the Borel–Weil theorem). The highest/lowest weight vector can be explicitly written down. See [27], and see also Chap. 8. fλ,O of O-modules Starting from Mλ,O we can construct an integral sheaf M G on SK . (The reader is referred to [30, Chap. 6] for a detailed description of f the construction of this sheaf.) We may then study the integral cohomology G fλ,O ) and variations such as inner and boundary cohomology. groups H • (SK ,M f If necessary, to ignore torsion in integral cohomology, one may pass to the image of integral cohomology in rational cohomology; i.e., we may consider • G G fλ,O )/Torsion ⊂ H • (SK fλ ). Hint (SK ,M ,M f f

The Hecke operators described earlier, in general, do not stabilize integral cohomology, but need to be modified; indeed at every finite prime p, a p-optimal modification is possible. This is reviewed in Sect. 7.1.2 on cohomological Lfunctions; see also [30, Sect. 6.3]. We may attempt to refine the rationality results on the ratios of critical L-values that are proven in this monograph by asking for integrality results. The periods that come up at various places are always numbers that are defined only up to an element in E × . If we work with integral cohomology we are × able to fix these periods up to an element in OE . Then it becomes possible to formulate refined versions of the results in Kazhdan, Mazur, and Schmidt [44], Kasten and Schmidt [43], and we may speak about estimates for denominators. We will discuss this partially in Sect. 8.4.7. But integrality results are subtler than rationality results. We do not attempt it here, as it would take us way Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:00 PM

THE COHOMOLOGY OF GLN

27

outside of the original scope of the monograph. The reader should bear in mind that in principle one can go through the entire development of our tools, replacG fλ,E ) by integral cohomology ing at every step rational cohomology H • (SK ,M f • G f H (S , Mλ,O ). Such an exercise, which offers many challenges along the way, Kf

has been carried out in some GL2 situations in [30], and also some foundational considerations in greater generality are explained in [27]. Delicate problems involving the denominators of Eisenstein classes crop up as alluded to in the Introduction and that are discussed in [30]. It would be a very interesting and fruitful exercise to study integrality properties of special values of L-functions and their bearing on congruences for automorphic forms.

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Chapter Three Analytic Tools

We will now go to the transcendental level, i.e., take an embedding ι : E → C, and extend the ground field to C. For all of Chap. 3, we will work over C and therefore we suppress the subscript C. Starting from Chap. 4 we will return to working at an arithmetic level, i.e., over a Galois extension E/Q that contains a copy of the totally real base field F, but for now, we work over C.

3.1

3.1.1

CUSPIDAL PARAMETERS AND THE REPRESENTATION Dλ AT INFINITY Lie algebras and relative Lie algebra cohomology

Let g denote the Lie algebra of G/Q. Similarly, let g(1) , b, t, z, and s be the Lie algebras of G(1) /Q, B/Q, T /Q, Z/Q, and S, respectively. Let g∞ be the Lie 0 algebra of G(R), and k∞ that of K∞ . 0 We consider Harish-Chandra modules, also called (g∞ , K∞ )-modules, (π, V ), 0 where V is the space of smooth K∞ -finite vectors of a reasonable representation (π, V) of G(R). For example, if (π, V) is essentially unitary (i.e., uni0 tary up to a twist) and irreducible, then the space of K∞ -finite vectors is 0 0 automatically smooth. Given a (g∞ , K∞ )-module (π, V ) by H • (g∞ , K∞ ;V ) • 0 (Λ (g∞ /k∞ ), V ). (See Borel we mean the cohomology of the complex HomK∞ and Wallach [9].) If (π, V) is a (reasonable) representation of G(R), then by 0 0 H • (g∞ , K∞ ; V), we mean H • (g∞ , K∞ ; V ) with V as earlier. It will be important later on to stay as far as possible at a rational Q level. Note: g∞ = g ⊗Q R. Similarly, let k(1) be the Lie algebra of the Q-group τ :F →R SO(n); then k∞ = 0 0 (s ⊕ k(1) ) ⊗Q R. Given a (g∞ , K∞ )-module (π, V ), using connectedness of K∞ , • • • 0 (Λ (g∞ /k∞ ), V ) = HomK 0 (Λ (g/k), V ) = Homk (Λ (g/k), V ). we have HomK∞ ∞ We will consider modules (π, V ) with a Q-structure, i.e., V = V0 ⊗Q C, and then 0 0 H • (g∞ , K∞ ; V ) = H • (g, K∞ ; V ) = H • (g, k; V0 ) ⊗ C.

(3.1)

See Chap. 8.

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29

ANALYTIC TOOLS

3.1.2

Pure dominant-integral weights

Not all algebraic dominant-integral weights can support inner cohomology. We have the following purity condition. + Lemma 3.1 (Purity Lemma). Let λ ∈ Xalg (T × C) be an algebraic dominantPn−1 ν ν integral weight; and say λ = (λ )ν:F →C , with λν = i=1 (ai − 1)γi + d · δ. Suppose there is an irreducible essentially unitary representation V of G(R) 0 such that H • (g, K∞ ; V ⊗ Mλ ) 6= 0. Then λ is essentially self-dual; i.e., for each ν ν ν, we have ai = an−i .

This is a well-known consequence of Wigner’s Lemma. For a proof, the interested reader can see the discussion in Wallach [71, Sect. 9.4.1–9.4.6]. An algebraic dominant-integral essentially self-dual weight λ will be called a pure weight. Let’s summarize the restrictions on all the ingredients going into λ if it is a pure weight. ∗ Lemma 3.2. Let λ ∈ Xalg (T × E) be an algebraic weight. Suppose we write

λ = (λτ )τ :F →E ,

λτ =

n−1 X

(aτi − 1)γi + d · δ = (bτ1 , . . . , bτn ).

i=1

If λ is dominant, integral, and pure, then: 1. Integrality: • • • •

aτi ∈ Z for all τ and all i; nd ∈ Z; and P rλτ ∈ Z, i.e., nd ≡ i(aτi − 1) (mod n); bτi ∈ Z for all τ and all i.

2. Dominance: • aτi ≥ 1 for all τ and all i; • bτ1 ≥ bτ2 ≥ · · · ≥ bτn . 3. Purity: • aτi = aτn−i and 2d ∈ Z. In particular, if n is odd, then d ∈ Z; • there exists w ∈ Z such that bτi + bτn−i+1 = w for all τ and all i. Indeed, w = 2d. Proof. For integrality and dominance see Sect. 2.2.5 and Sect. 2.2.6, respectively. Under purity, we need to show that 2d ∈ Z. Recall: by integrality rλτ ∈ Z and,

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30

CHAPTER 3

by definition, nd =

Pn−1 i=1

2nd = nd + nd

i(aτi − 1) + nrλτ . The purity condition implies =

n−1 X

i(aτi − 1) +

i=1

=

n

n−1 X

(n − i)(aτi − 1) + 2nrλτ

i=1

n−1 X

(aτi − 1) + 2nrλτ .

i=1

Hence we have 2d =

n−1 X

(aτi − 1) + 2rλτ .

(3.2)

i=1

This implies that 2d ∈ Z. Furthermore, if n is odd then d ∈ Z, since we already had nd ∈ Z by integrality. In terms of the standard basis, the condition aτi = aτn−i translates to bτi − bτi+1 + 1 = bτn−i − bτn−i+1 + 1, which is the same as bτi + bτn−i+1 = bτi+1 + bτn−i . In other words, bτi + bτn−i+1 is independent of i; say, bτi + bτn−i+1 = wτ . Then 2nd = nd + nd =

n X

bτi + bτn−i+1 = nwτ ,

i=1

or that wτ = 2d is independent of τ ; hence bτi + bτn−i+1 = 2d for all i and τ. Denote by X0∗ (T × E) the set of pure weights. For λ ∈ X0∗ (T × E), with the notations as earlier, we shall call the integer 2d the purity weight of λ; furthermore, if we write λ = λ(1) + dδ, then note that the dual weight of λ is given by λv = λ(1) −dδ; i.e., λv = λ−2dδ, which implies Mvλ = Mλ ⊗(det)−2d . 3.1.3

Motivic weight

Let λ ∈ X0∗ (T × E) be as earlier. The motivic weight of λ is defined to be the integer wλ := max{wλτ | τ : F → E}, where wλτ =

n−1 X

aτi .

i=1

From (3.2) we get the parity conditions wλτ ≡ 2d + n − 1

(mod 2), ∀τ ;

hence wλ ≡ 2d + n − 1

(mod 2). (3.3)

In particular, if n is odd then wλ ≡ 0 (mod 2). These parity conditions play an important role in the numerology concerning cuspidal parameters, Hodge pairs, critical points, etc.

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31

ANALYTIC TOOLS

3.1.4

Cuspidal parameters and the representation at infinity

Let λ ∈ X0∗ (T × C) with λ = (λν )ν:F →C . The infinitesimal character of an 0 0 irreducible admissible (g, K∞ )-module π∞ such that H • (g, K∞ ; π∞ ⊗ Mλ ) 6= 0, by Wigner’s Lemma, is uniquely determined by λ. Hence, up to isomorphism, there are only finitely many such π∞ (this is a consequence of the Langlands classification; the reader is referred to Wallach [71, Thm. 5.5.6]). We denote this 0 finite set by Coh∞ (G, λ). Among this finite set of possible (g, K∞ )-modules, exactly one, up to twisting by sign characters (see later), is generic, i.e., admits a Whittaker model. Only this representation can appear as the representation at infinity of a global cohomological cuspidal representation. Pn−1 Let λν = i=1 (aνi − 1)γi + d · δn . The cuspidal parameter of λ is defined as ` = (`ν )ν:F →C , where `ν = (`ν1 , . . . , `νn ) and `ν1 `ν2 `ν3

:= aν1 + aν2 + aν3 + · · · + aνn−1 := −aν1 + aν2 + aν3 + · · · + aνn−1 := −aν1 − aν2 + aν3 + · · · + aνn−1

`νn

:= −aν1 − aν2 − · · · − aνn−1 = −wλν

= aν1 + aν2 + aν3 + · · · + aνn−1 = wλν = aν2 + aν3 + · · · + aνn−2 = aν3 + · · · + aνn−3 .. . (3.4)

The integers `νj , are also called as the cuspidal parameters of λ. Observe that • ` depends only on the semisimple part λ(1) and not on the abelian part λab of λ; i.e., ` depends only on the aνi ’s, and not on the purity weight 2d. • `ν1 > `ν2 > · · · > `ν[n/2] > 0 and `νi = −`νn−i+1 . • `ν = (wλν , wλν − 2aν1 , wλν − 2aν1 − 2aν2 , . . . , −wλν ). It readily follows that `νi ≡ wλ ≡ 2d + n − 1

(mod 2);

(3.5)

i.e., every cuspidal parameter has the same parity as the motivic weight. Furthermore, if n is odd, then all the cuspidal parameters, the motivic weight and the purity weight are even; however, if n is even, then all the cuspidal parameters and the motivic weight have the same parity which is opposite to the parity of the purity weight. For any integer ` ≥ 1, let D` be the discrete series representation of GL2 (R) of lowest nonnegative K-type corresponding to ` + 1 and with central character R sgn`+1 . The Langlands parameter of D` is IndW C× (χ` ), where WR is the Weil × group of R, and χ` is the character of C sending z to (z/¯ z )`/2 , or χ` (reit ) = ei`t . The infinitesimal character of D` is (`/2, −`/2). For example, a holomorphic cuspidal modular form of weight k generates Dk−1 as the representation at infinity. (For more details, the reader is referred to [57, Sects. 3.1.2–3.1.5].)

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Let Rn stand for the standard parabolic subgroup of GLn of type (2, 2, . . . , 2) if n is even, and of type (2, 2, . . . , 2, 1) if n is odd. For λ = (λν )ν:F →C ∈ X0∗ (T × C), define Dλν as    GLn (R) −d −d  ν ⊗ | · | ν Ind (D ) ⊗ · · · ⊗ (D ⊗ | · | ) , if n is even; ` `  R R R (R) 1 n/2 n      −d −d IndGLn (R) (D ν ⊗ | · |−d ) ⊗ · · · ⊗ (D ν , if n is odd, `1 `(n−1)/2 ⊗ | · |R ) ⊗ | · |R R Rn (R) GL (R)

n where IndRn (R) denotes normalized parabolic induction. Identify the set S∞ of infinite places with the set ΣF ; say, v ∈ S∞ corresponds to νv ∈ ΣF . Define O Dλ = Dλνv . (3.6)

v∈S∞

We will now discuss twisting by sign characters. Identify the sign-character sgn : R× → {±1} with −1, and the trivial character of R× with +1. Let ε = (εv )v∈S∞ beQan r-tuple of signs; i.e., εv ∈ {±1}. Then ε gives a character of order 2 on v∈S∞ R× , and via the determinant map, ε gives of a character of G(R)/G(R)0 = π0 (G(R)). ∼ Dλ for every ε. However, if 1. If n is even then Dλ ⊗ ε = 2. n is odd, then for every nontrivial ε we have Dλ ⊗ ε 6∼ = Dλ . This may be seen by computing central characters. The central character ωDλνv ⊗εv of the v-th component of Dλ ⊗ ε is ωDλνv ⊗εv = sgn

(n−1) 2

· | |−nd · εv .

(We have used the fact that `νj is even when n is odd.) We collect some basic properties of Dλ in the following proposition. Proposition 3.3. Let λ ∈ X0∗ (T × C) and Dλ be as in (3.6). Let ε = (εv )v∈S∞ be any r-tuple of signs. Then 1. 2. 3. 4.

Dλ ⊗ ε is an irreducible essentially tempered representation. Dλ ⊗ ε admits a Whittaker model. 0 H • (g, K∞ ; (Dλ ⊗ ε) ⊗ Mλ ) 6= 0. Let π be an irreducible essentially tempered representation of G(R). Suppose 0 H • (g, K∞ ; π ⊗ Mλ ) 6= 0, then π = Dλ ⊗ ε for some ε.

Proof. These are well-known results for GLn (R); we refer the reader to the very useful survey articles of Knapp [47] and Mœglin [51] and to the references therein. (Irreduciblity follows from Speh’s results. A representation irreducibly Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:01 PM

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ANALYTIC TOOLS

induced from essentially discrete series representation is essentially tempered. Admitting a Whittaker model is a hereditary property, and an essentially discrete series representation of GL2 (R) has a Whittaker model. Nonvanishing of cohomology follows from Delorme’s Lemma; see later.)

3.1.5

The cohomology degrees

The relative Lie algebra cohomology group in Prop. 3.3 can be computed using Delorme’s Lemma; see, for example, Borel and Wallach [9, Thm. III.3.3]. We summarize what we need about these cohomology groups in the following proposition. Proposition 3.4. Let λ ∈ X0∗ (T × C), Dλ and ε be as earlier. Then 0 ˜F H • (g, K∞ ; (Dλ ⊗ ε) ⊗ Mλ ) 6= 0 ⇐⇒ bF n ≤ • ≤ tn ,

˜F where the bottom degree bF n and the top degree tn are defined as: 2 bQ n := bn /4c,

Q bF n := rF bn ,

Q tQ n := bn + dn/2e − 1,

Q tF n := rF tn ,

F t˜F n := tn + rF − 1,

where, recall that rF = [F : Q], and for any real number x, bxc denotes the greatest integer less than or equal to x, and dxe = −b−xc is the least integer greater than or equal to x. 0 Furthermore, the cohomology group H q (g, K∞ ; (Dλ ⊗ ε) ⊗ Mλ ), as a module 0 ˜F over K∞ /K∞ = π0 (K∞ ) = π0 (Gn (R)), in the extreme degrees q = bF n or q = tn is given by 1. If n is even, then Dλ ⊗ ε = Dλ for all ε, and 0 H q (g, K∞ ; Dλ ⊗ Mλ ) =

M

ε.

\ ε ∈ π0 (G n (R))

2. If n is odd then 0 H q (g, K∞ ; (Dλ ⊗ ε) ⊗ Mλ ) = (−1)rF (d+(n−1)/2) ε.

Proof. We just make a few comments, as all of this is well known. First of all, observe that g/k = g(1) /k(1) ⊕ z/s, and hence Λ• (g/k) = Λ• (g(1) /k(1) ) ⊗ Λ• (z/s). This implies: • 0 (Λ (g/k), (Dλ ⊗ ε) ⊗ Mλ ) HomK∞

= HomK (1) (Λ• (g(1) /k(1) ), (Dλ ⊗ ε) ⊗ Mλ ) ⊗ Hom(Λ• (z/s), C), ∞

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CHAPTER 3

since the Lie algebra z acts trivially on (Dλ ⊗ ε) ⊗ Mλ . The cohomology group (1) H • (g(1) , K∞ ; (Dλ ⊗ε)⊗Mλ ) is calculated in Clozel [14, Lem. 3.14]; in particular, F 0 one sees that it is nonvanishing if and only if bF n ≤ • ≤ tn . For (g, K∞ )F F ˜ cohomology we need to go up to tn = tn + rF − 1 since the dimension of z/s is rF − 1. If n is odd then O(n)/SO(n) is represented by {±1n }, where 1n is the identity n × n-matrix. So, in case (2), it simply boils down to computing the central character of (Dλ ⊗ ε) ⊗ Mλ .

3.2 3.2.1

SQUARE-INTEGRABLE COHOMOLOGY de Rham complex

For a level structure Kf ⊂ G(Af ), in what follows, we consider various spaces of functions on G(Q)\G(A)/Kf that are naturally G(R)-modules. Assume for the moment that Kf is a neat subgroup. Consider a pure weight λ ∈ X0∗ (T × C). fλ on S G , as constructed in Sect. 2.2.8, is a local system of CThe sheaf M Kf fλ ) is the cohomology of the de Rham vector spaces. The cohomology H • (S G , M Kf

G fλ ). We have the isomorphism between the de Rham complex complex Ω• (SK ,M f and the relative Lie algebra complex G fλ ) = HomK 0 (Λ• (g/k), C ∞ (G(Q)\G(A)/Kf , ω −1 |S(R)0 ) ⊗ Mλ ), Ω• (SK ,M λ f ∞

where C ∞ (G(Q)\G(A)/Kf , ωλ−1 |S(R)0 ) is the set all functions φ : G(A) → C such that φ is smooth (which means smooth in the usual sense at the infinite places and locally constant at the finite places) and φ(γ g k f a∞ ) = ωλ−1 (a∞ )φ(g),

∀g ∈ G(A), γ ∈ G(Q), k f ∈ Kf , a∞ ∈ S(R)0 .

−1 We will abbreviate ωλ−1 |S(R)0 = ω∞ . Differentiating the action of G(R) provides an action of the Lie algebra g and hence an action of the universal enveloping algebra U(g) that will be denoted by f 7→ U f for U ∈ U(g).

3.2.2

Definition of square-integrable cohomology ∞

−1 −1 Inside C (G(Q)\G(A)/Kf , ω∞ ) lies the subspace C2∞ (G(Q)\G(A)/Kf , ω∞ ) of smooth functions φ satisfying the square-integrability condition: Z |(U φ)(g)|2 |det(g)|2d F dg < ∞, S(R)0 G(Q)\G(A)

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ANALYTIC TOOLS

for all U ∈ U(g). The integrand is trivial on S(R)0 by (2.14). Now we define • G fλ ) as the submodule of H • (S G , M fλ ) consisting of those cohomolH(2) (SK ,M Kf f ogy classes that can be represented by square-integrable forms, i.e., by closed forms in • ∞ −1 0 (Λ (g/k), C HomK∞ 2 (G(Q)\G(A)/Kf , ω∞ ) ⊗ Mλ ).

3.2.3

A result of Borel and Garland

We know from the fundamental work of Langlands [50] that we have a decomposition into essentially unitary G(R)-modules: −1 L2 (G(Q)\G(A)/Kf , ω∞ ) −1 −1 = L2disc (G(Q)\G(A)/Kf , ω∞ ) ⊕ L2cont (G(Q)\G(A)/Kf , ω∞ ),

where the first summand in the right-hand side is the closure of the direct sum of irreducible subspaces. Hence, any square-integrable form • ∞ −1 0 (Λ (g/k), C ω ∈ HomK∞ 2 (G(Q)\G(A)/Kf , ω∞ ) ⊗ Mλ )

may be written as ω = ωdisc ⊕ ωcont . The two projections given by the above decomposition commute with the action of G(A); hence they commute with the action of U(g). This implies that both ωdisc and ωcont are smooth; i.e., −1 ωdisc , ωcont ∈ HomK∞ (Λ• (g/k), C2∞ (G(Q)\G(A)/Kf , ω∞ ) ⊗ Mλ ).

If ω is closed then dωdisc = dωcont = 0. Proposition 3.5. The cohomology class [ωcont ] = 0. Proof. Let q be the degree of ωcont . Since the C ∞ functions are dense in L2 , Lemma 5.5 in [7] implies that we can find a −1 ψ ∈ HomK∞ (Λq−1 (g/k), C2∞ (G(Q)\G(A)/Kf , ω∞ ) ⊗ Mλ )

such that the L2 -norm of ωcont − dψ becomes arbitrarily small (see also [30]). This fact can also be obtained by more general methods using the theory of distributions ([18, Thm. 24, 26]) or Hilbert space techniques ([1, Thm. C. 31]). We invoke Poincar´e duality ([26, Thm. 4.8.9]; the finiteness assumptions therein G fλ ) is zero if and are easily verified): The cohomology class [ωcont ] ∈ H p (SK ,M f fv ) only if the value of the cup product pairing with any class [ω2 ] ∈ H d−p (S G , M c

Kf

λ

is zero. But the absolute value |[ωcont ] ∪ [ω2 ]| of the cup product can be given by a scalar product integral. Therefore it can be estimated by the product of

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CHAPTER 3

the two L2 -norms |[ωcont ] ∪ [ω2 ]| = |[ωcont − dψ] ∪ [ω2 ]| ≤ ||ω − dψ||2 ||ω2 ||2 (Cauchy–Schwarz inequality) and hence must be zero. We define Coh(2) ∞ (G, λ) to be the finite set of isomorphism classes of essentially unitary G(R)-modules having nontrivial cohomology with Mλ -coefficients. For each π∞ ∈ Coh(2) ∞ (G, λ) choose a representative Vπ∞ , and let Vπ∞ be the resulting Harish-Chandra module of K∞ -finite vectors. Then, we put
−1 Wπ(2) = HomG(R) Vπ∞ , L2 (G(Q)\G(A)/Kf , ω∞ ) ∞
−1 = HomG(R) Vπ∞ , L2disc (G(Q)\G(A)/Kf , ω∞ ) . (2)

−1 It is clear that any Φ ∈ Wπ∞ sends Vπ∞ to C2∞ (G(Q)\G(A)/Kf , ω∞ ). It follows from Borel and Garland [8] that the induced map M 0 • G fλ ) Wπ(2) ⊗ H • (g, K∞ ; Vπ∞ ⊗ Mλ ) −→ H(2) (SK ,M (3.7) ∞ f (2)

π∞ ∈Coh∞ (G,λ)

is surjective. In [8] it is proved that only finitely many summands in the above decomposition contribute to cohomology. We can also take the action of the Hecke algebra into account; its action via convolution on the discrete spectrum −1 L2disc (G(Q)\G(A)/Kf , ω∞ ) is self-adjoint. For each isomorphism class π∞ ∈ (2) Coh∞ (G, λ), take Vπ∞ as earlier, and similarly, for each isomorphism class πf G of absolutely irreducible HK -module choose a representative Vπf for πf , and f define
(2) 2 −1 Wπ∞ ⊗πf = HomG(R)×HG V ⊗ V , L (G(Q)\G(A)/K , ω ) . π π f disc ∞ ∞ f K f

Define Coh(2) (G, λ, Kf ) to be the set of those πf for which there exists a π∞ (2) such that Wπ∞ ⊗πf 6= 0. Then we get the refined decomposition M

(2)

0 Wπ∞ ⊗πf ⊗ H • (g, K∞ ; Vπ∞ ⊗ Mλ ) ⊗ Vπf

π∞ ×πf

−→

M

• G fλ )(πf ), (3.8) H(2) (SK ,M f

πf ∈Coh(2) (G,λ,Kf ) • G fλ ), and the summand on the rightthat is a surjective map onto H(2) (SK ,M f hand side indexed by πf is in fact the πf -isotypic component. This implies that square-integrable cohomology is a semisimple module under the action of the Hecke algebra, hence so is inner cohomology. It is a delicate question to compute the kernel of the above map. Note that, by the multiplicity one theorem for the

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ANALYTIC TOOLS

discrete spectrum for GLn (which was proved in a special case by Jacquet [37], (2) and in general by Mœglin and Waldspurger [52]), we have dim(Wπ∞ ⊗πf ) = 1.

3.3 3.3.1

CUSPIDAL COHOMOLOGY The cohomological cuspidal spectrum

The space of square-integrable functions contains the space of smooth cusp forms ∞ −1 ∞ −1 Ccusp (G(Q)\G(A)/Kf , ω∞ ) ⊂ C(2) (G(Q)\G(A)/Kf , ω∞ ).

This inclusion induces an inclusion in cohomology (see Borel [6, Cor. 5.5]) and we define cuspidal cohomology by
• G fλ ) := H • g, K 0 ; C ∞ (G(Q)\G(A)/Kf , ω −1 ) ⊗ Mλ . Hcusp (SK ,M ∞ cusp ∞ f For π∞ ∈ Coh∞ (G, λ) and πf ∈ Coh(G, λ, Kf ), define Wπcusp := Hom(g,K∞ )⊗HG ∞ ⊗πf K

f


∞ −1 Vπ∞ ⊗ Vπf , Ccusp (G(Q)\G(A)/Kf , ω∞ ) .

Define Cohcusp (G, λ, Kf ) as the set of those πf ∈ Coh(G, λ, Kf ) for which we can find a π∞ ∈ Coh∞ (G, λ) such that Wπcusp 6= 0. We have a surjective map ∞ ⊗πf M

M

0 Wπcusp ⊗ H • (g, K∞ ; Vπ∞ ⊗ Mλ ) ⊗ Vπf ∞ ⊗πf

πf ∈Cohcusp (G,λ,Kf ) π∞ ∈Coh∞ (G,λ) • G −→ Hcusp (SK , Mλ ), f

which is in fact an isomorphism by a theorem of Borel, loc. cit.; this also implies • G G that Hcusp (SK , Mλ ) ⊂ H!• (SK , Mλ ). f f 3.3.2

Consequence of multiplicity one and strong multiplicity one

By multiplicity one for the cuspidal spectrum for G = RF/Q (GLn /F ) (Jacquet (cusp) [37] and Mœglin and Waldspurger [52]), if Wπ∞ ⊗πf is nonzero then it is of dimension 1. Furthermore, from strong multiplicity one for cuspidal representations due to Jacquet and Shalika [39], it follows that πf is determined by its G G restriction πfS to the central subalgebra HG,S = ⊗v∈S / HKp of HKf .

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38 3.3.3

CHAPTER 3

The character of the component group I

Since a cuspidal automorphic representation has a global Whittaker model, it follows that the representation at infinity π∞ is isomorphic to Dλ ⊗ ε for some sign character ε, because these are the only π∞ ∈ Coh∞ (G, λ) that have a local Whittaker model. Applying the isomorphism mentioned in Sect. 3.3.1, and the ˜F cohomology of Dλ ⊗ ε as described in Prop. 3.4 for extreme degrees q ∈ {bF n , tn }, we conclude the following decomposition of cuspidal cohomology into nonzero G absolutely irreducible π0 (G(R)) × HK -modules: f q G fλ ) Hcusp (SK ,M f L L  • G Hcusp (SK , Mλ )(πf × ε), if n is even,  f   π ∈Coh (G,λ,K ) \ cusp f  f ε ∈ π0 (Gn (R)) = L  • G   Hcusp (SK , Mλ )(πf × ε(πf )), if n is odd,  f πf ∈Cohcusp (G,λ,Kf )

(3.9) where, when n is odd the canonical character ε(πf ) that πf can pair with is given by ε(πf ) = (ε(πf )v )v∈S∞ ,

ε(πf )v (−1) = (−1)d ωπv (−1),

(3.10)

where ωπv is the central character of πv . This last assertion when n is odd may be seen as follows: Suppose we are given πf ∈ Cohcusp (G, λ, Kf ); then by strong multiplicity one, there is a unique π∞ that it can pair with to give a cuspidal automorphic representation π = π∞ ⊗ πf . Suppose, π∞ = ⊗v∈S∞ πv . By Prop. 3.4, (2), we also see that H q (g, Kv0 ; πv ⊗ Mλv ), as a Kv /Kv0 = O(n)/SO(n)-module, is the sign character whose value at −1 is ωπv (−1)(−1)d . To justify the notation ε(πf ), we note that ωπv is completely determined by ωπf by automorphy of the central character ωπ = ωπ∞ ⊗ ωπf . To parse it further, observe that ωπ = | |−nd ⊗ ωπ0 , where ωπ0 is a character of finite order. Fix v ∈ S∞ , and apply weak approximation to choose an a ∈ F × (which will depend on v) such that a < 0 as an element of Fv and a > 0 as an element of Fw for all w ∈ S∞ − {v}; then ωπv (−1) = ωπ0 f (a). Similarly, we also have (when n is odd): π∞ = Dλ ⊗ ε∞ (πf ); ε∞ (πf ) = (ε∞ (πf )v )v∈S∞ ; (n−1)/2

ε∞ (πf )v (−1) = (−1)

(3.11)

ωπv (−1).

The reader is also referred to the discussion in Gan and Raghuram [20, Sect. 3].

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3.3.4

A filtration

We now drop the assumption that we are working over C and go back to our coefficient system Mλ,E defined over E. An embedding ι : E → C gives a chain of subspaces • G fιλ,C ) ⊂ H • (S G , M fιλ,C ) Hcusp (SK ,M ! Kf f • G G fιλ,C ) ⊂ H • (SK fλ,E ) ⊗E,ι C. ⊂ H(2) (SK ,M ,M f f G fιλ,C ) = H • (S G , M fλ,E ) ⊗E,ι C is a subSince the Hecke module H!• (SK ,M Kf ! f module of a semisimple module it is also semisimple. But then already the G fλ,E ) is semisimple and if E is large enough we get an E-module H!• (SK ,M f isotypical decomposition M G G fλ,E ) ⊗E,ι C = fλ,E )(πf ) ⊗E,ι C, H!• (SK , M H!• (SK ,M f f πf ∈Coh! (G,λ,Kf )

where the πf are absolutely irreducible. We have to understand the discrepancies between these spaces, especially the difference between the cuspidal and the square-integrable cohomology. This issue may become very delicate for a general reductive group, but for GLn the situation is relatively simple: Take πf in Coh! (G, λ, Kf ); then for any ι : E → C, the module ι πf = πf ⊗E,ι C ∈ Coh(2) (G, λ, Kf ). By definition, there is a ι π∞ (2) such that Wι π∞ ⊗ι πf is one-dimensional. We have to find criteria to decide whether (2) Wιcusp or Wιcusp π∞ ⊗ι πf = 0 π∞ ⊗ι πf = Wι π∞ ⊗ι πf . We will show later (Thm. 4.7) that the isomorphism type ι πf ∈ Coh(2) (G, λ, Kf ) is cuspidal ⇐⇒ G G fλ,E )(πf ) ⊗E,ι C = H • (SK fλ,E ⊗E,ι C)(ι πf ). (3.12) H!• (SK ,M ,M (2) f f

This implies that if ι πf is cuspidal for one embedding ι0 , then it is cuspidal for every embedding ι, giving another proof of a result due to Clozel [14].

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Chapter Four Boundary Cohomology

We will discuss some relevant details of the cohomology of the boundary of G the Borel–Serre compactification of the locally symmetric space SK . (See [25, f Sect. 1.1] and [30] for more details and proofs.)

4.1

A SPECTRAL SEQUENCE CONVERGING TO BOUNDARY COHOMOLOGY

G Recall from Sect. 2.3.4 the Borel–Serre compactification of SK and the associf ated long exact sequence. There is a spectral sequence built out of the cohoG mology of the boundary strata ∂P SK that converges to the cohomology of the f boundary.

4.1.1

The spectral sequence at an arithmetic level

For 1 ≤ i ≤ n − 1, let P0,i be the standard maximal parabolic subgroup of G0 = GLn /F obtained by deleting the simple root αi . (See Sect. 2.2.4.) Let Pi = RF/Q (P0,i ). Then Pi is a standard maximal parabolic subgroup of G; note that Pi is not absolutely maximal unless F = Q. The standard parabolic subgroups P correspond to nonempty subsets I = ∆P ⊂ {1, 2, . . . , n − 1} where Pi ⊃ P ⇐⇒ i ∈ I. Define d(P ) := #I. If P ⊃ Q then ∆P ⊂ ∆Q and by the G G that gives a Borel–Serre construction we have an embedding ∂Q SK ,→ ∂P SK f f restriction map in cohomology G G fλ,E ) ' H • (∂P S G , M fλ,E ) −→ H • (∂Q SK fλ,E ). rP,Q : H • (∂P SK ,M ,M Kf f f G fλ,E ) whose E p,q From this we get a spectral sequence converging to H • (∂SK ,M 1 f term is M G fλ,E ). E p,q := H q (∂P SK ,M 1

f

d(P )=p+1

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The boundary map d : E1p,q → E1p+1,q is given by X d(ξP ) = (−1)σ(P,Q) rP,Q (ξP )

(4.1)

Q⊂P d(Q)=p+2 G fλ,E ) with d(P ) = p + 1, and σ(P, Q) is the place of for any ξP ∈ H q (∂P SK ,M f the vertex defining P in the ordered set of vertices defining Q.

4.1.2

The spectral sequence at a transcendental level

Take an ι : E → C and define
−1 0 ΩqP (Mιλ ) := HomK∞ Λq (g/k), C ∞ (P (Q)\G(A)/Kf , ω∞ ) ⊗ Mιλ . Also, define
−1 0 ΩqP (Mιλ )(0) := HomK∞ Λq (g/k), C ∞ (P (Q)UP (A)\G(A)/Kf , ω∞ ) ⊗ Mιλ . We have the inclusion ΩqP (Mιλ )(0) ⊂ ΩqP (Mιλ ), and taking the constant term F P along P gives a projection on to that subspace; recall that the constant −1 −1 term map F P : C ∞ (P (Q)\G(A)/Kf , ω∞ ) → C ∞ (P (Q)UP (A)\G(A)/Kf , ω∞ ) is defined as Z F P (φ)(g) = φ(ug) du. UP (Q)\UP (A)

Hence ΩqP (Mιλ )(0) is a direct summand of ΩqP (Mιλ ). Consider the doublecomplex Ω•• = Ω•• (Mιλ ): O

O

0

0

/

O

Ωn−2,0

/

O

Ω1,0

0

/

Ω0,0

/

O

Ωn−2,d

/

Ω1,1

O

/

Ω0,1

O

O

/

···

/

Ω1,d

O

/

···

/

Ω0,d

0

/0

. . .

O

O

0

···

. . .

O

/

/

0

Ωn−2,1

. . .

0

O

0

O

/0

O

/0

0

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CHAPTER 4

where Ωp,q =

M

ΩqP (Mιλ );

d(P )=p

the horizontal arrow Ωp,q → Ωp,q+1 is exterior differentiation; the number of columns d is the dimension of g/k; the vertical arrow Ωp,q → Ωp+1,q is defined exactly as in (4.1), i.e., as an alternating sum of maps ΩqP → ΩqQ for any Q ⊂ P −1 with d(Q) = d(P ) + 1, while using the canonical map C ∞ (P (Q)\G(A), ω∞ )→ ∞ −1 C (Q(Q)\G(A), ω∞ ). The associated simple complex made from this double G fιλ ). All of complex computes the cohomology of the boundary: H • (∂SK ,M f •• this remains true even if we work with the double subcomplex Ω (Mιλ )(0) . For the vertical arrows, one uses a partial constant term map: −1 −1 FPQ : C ∞ (P (Q)UP (A)\G(A), ω∞ ) → C ∞ (Q(Q)UQ (A)\G(A), ω∞ )

given by Z

FPQ (φ)(g) =

φ(u g) du. UQ (Q)UP (A)\UQ (A)

4.2

COHOMOLOGY OF ∂P S G AS AN INDUCED REPRESENTATION

It is clear from the E1p,q term of the spectral sequence in Sect. 4.1.1 that to understand the cohomology of the boundary, we need to understand the cohoG mology of a single stratum ∂P SK . In the following we do not need that P is f G maximal, i.e., that ∂P SKf is open. It is well known that G 0 fλ,E ) = H • (P (Q)\G(A)/K∞ fλ,E ). H • (∂P SK ,M Kf , M f 0 The space P (Q)\G(A)/K∞ Kf fibers over locally symmetric spaces of MP , as we now explain. Let ΞKf be a complete set of representatives for P (Af )\G(Af )/Kf . P 0 Let K∞ = K∞ ∩ P (R), and for ξf ∈ ΞKf , let KfP (ξf ) = P (Af ) ∩ ξf Kf ξf−1 . Then 0 P (Q)\G(A)/K∞ Kf =

a

P P (Q)\P (A)/K∞ KfP (ξf ).

ξf ∈ ΞKf MP P Let κP : P → P/UP = MP be the canonical map, and define K∞ = κP (K∞ ), MP P and for ξf ∈ ΞKf , let Kf (ξf ) = κP (Kf (ξf )). Define P S MM P

Kf

(ξf )

MP := MP (Q)\MP (A)/K∞ KfMP (ξf ).

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BOUNDARY COHOMOLOGY

MP The underline is to emphasize that we have divided out by K∞ which is not UP −1 connected; see Sect. 4.2.1. Let Kf (ξf ) = UP (Af ) ∩ ξf Kf ξf . We have the fibration: P P UP (Q)\UP (A)/KfUP (ξf ) ,→ P (Q)\P (A)/K∞ KfP (ξf )  S MM P Kf

(ξf )

.

The corresponding Leray–Serre spectral sequence degenerates at E2 -level. See, for example, Schwermer’s articles [60, Sect. 7] and [59, Thm. 2.7]. The cohomology of the total space is given in terms of the cohomology of the base with coefficients in the cohomology of the fiber. For the cohomology of the fiber, one uses a classical theorem due to van Est: if uP is the Lie algebra of UP then the cohomology of the fiber is the same as the unipotent Lie algebra cohomology group H • (uP , Mλ,E ), which is naturally an algebraic representation of MP ; the P associated sheaf on S MM is denoted by putting a tilde on top. Putting all P Kf

(ξf )

this together, we get H

•

G fλ,E ) (∂P SK ,M f

M

=

H

•



ξf ∈ ΞKf



P S MM , H • (u^ P , Mλ,E ) Kf P (ξf )

.

(4.2)

It is convenient to pass to the limit over all open-compact subgroups Kf and define fλ,E ) := lim H • (∂P S G , M fλ,E ). H • (∂P S G , M Kf −→ Kf

MP Let S MP := MP (Q)\MP (A)/K∞ . Now we can rewrite (4.2) as

M

fλ,E )Kf = H • (∂P S G , M

 KfMP (ξf ) H • S MP , H • (u^ , M ) . P λ,E

(4.3)

ξf ∈ ΞKf

Let’s recall an exercise in basic Mackey theory for taking invariants in an induced representation and stated in a context immediately relevant to us. Proposition 4.1. Let V be a module for MP (Af ) that is inflated up to a module G(A ) for P (Af ) via the canonical projection P (Af ) → MP (Af ). Let a IndP (Aff ) (V ) stand for the algebraic (that is un-normalized) induction from P (Af ) to G(Af ) of V . Let Kf ⊂ G(Af ) be an open-compact subgroup. Then 

a

G(A )

IndP (Aff ) (V )

Kf

=

M

MP

V Kf

(ξf )

.

ξf ∈ ΞKf

It is clear from Mackey theory as earlier that the right-hand side of (4.3) is the Kf -invariants of an algebraically induced representation. Passing to the limit over all Kf , we get the following important result. Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:01 PM

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Proposition 4.2. The cohomology of ∂P S G is given by   fλ,E ) = a Indπ0 (G(R))×G(Af ) H • (S MP , H • (u^ H • (∂P S G , M P , Mλ,E )) . π0 (P (R))×P (Af ) The notation a Ind stands for algebraic, or un-normalized, induction. The process of induction from π0 (P (R)) to π0 (G(R)) is important and needs some explanation.

4.2.1

Induction from π0 (P (R)) to π0 (G(R))

The parabolic subgroup P is of the form P = RF/Q (P0 ) for a parabolic subgroup P0 /F of G0 = GLn /F . Since F is totally real, we Q identify the set of infinite places S with Σ = Hom(F, R), and G(R) = ∞ F τ ∈ΣF GLn (R) and P (R) = Q τ τ P (R), where P = P × R. Furthermore, suppose, P0 corresponds 0 F,τ 0 0 τ ∈ΣF to the partition n = n1 + · · · + nk with k ≥ 2 and nj ≥ 1, then MP0 = GLn1 × · · · × GLnk /F. We have MP K∞

=

0 κP (P (R) ∩ K∞ )

! =

κP

0

P (R) ∩ (S(R) ·

Y

SO(n))

τ

=

κP

! Y S(R)0 · (P0τ (R) ∩ SO(n)) τ

! =

κP

0

S(R) ·

Y

S(O(n1 ) × · · · × O(nk ))

τ

'

S(R)0 ·

Y

S(O(n1 ) × · · · × O(nk )).

τ MP Note that π0 (P (R)) has order 2rk and π0 (K∞ ) has order 2r(k−1) . (Recall, r = [F : Q].) Inclusion of components induces a canonical surjective map π0 (P (R)) → π0 (G(R)) giving a short exact sequence: MP 1 −→ π0 (K∞ ) −→ π0 (P (R)) −→ π0 (G(R)) −→ 1.

(4.4)

MP In the definition of S MP we have divided out by K∞ . For S MP we divide only MP ,0 MP by K∞ , the connected component of the identity in K∞ . Hence, for any f coming from an algebraic representation M of MP /Q we have sheaf M M

f = H • (S MP , M) f π0 (K∞P ) . H • (S MP , M) This means that H • (S MP , H • (u^ P , Mλ,E )) is a module for π0 (P (R)) × P (Af ) on MP which π0 (K∞ ) acts trivially and so is naturally a module for π0 (G(R))×P (Af ), Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:01 PM

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BOUNDARY COHOMOLOGY

which is then induced up to get a module for π0 (G(R)) × G(Af ). See also Rem. 4.4.

4.2.2

Kostant’s theorem on unipotent cohomology

The structure of the unipotent cohomology group H • (uP , Mλ,E ) is well known by results of Kostant [46], and we briefly describe these results in our situation. The calculation of the unipotent cohomology groupQis over the field E. Recall that we are dealing with the split group G × E = τ :F →E Gτ0 , where Gτ0 = G0 ×F,τ E. Let ∆G0 stand for the set of roots of G0 and ∆+ G0 the subset of positive roots (for choice of Borel subgroup being the upper triangular subgroup). Let ΠG0 τ be the set of simple roots. The notations ∆Gτ0 , ∆+ Gτ0 , and ΠG0 are clear. Let P = RF/Q (P0 ) be a parabolic subgroup of G as earlier, and we let P0τ := P0 ×τ E. Q The Weyl group factors as W = τ :F →E W0τ , with each W0τ isomorphic to the permutation group Sn on n-letters. Let W P be the set of Kostant representatives in the Weyl group W of G corresponding to the parabolic subgroup P which is defined as τ

W P = {w = (wτ ) : wτ ∈ W0τ P0 }, where

τ

W0τ P0 := {wτ ∈ W0τ : (wτ )−1 α > 0, ∀α ∈ ΠMP τ }. 0

(Here ΠMP τ ⊂ ΠGτ0 denotes the set of simple roots in the Levi quotient MP0τ 0 of P0τ .) For w ∈ W , and in particular for w ∈ W P , and for λ ∈ X ∗ (T ), by w · λ we mean the twisted action of w on the weight λ, i.e., w · λ = (wτ · λτ )τ :F →E ,

wτ · λτ = wτ (λτ + ρn ) − ρn .

Since w ∈ W P , the weight w·λ is dominant and integral as a weight for MP × E and we consider the associated irreducible finite-dimensional representation Mw·λ of MP × E. Kostant’s theorem asserts that as MP × E-modules, one has a multiplicity-free decomposition: M H q (uP , Mλ,E ) ' Mw·λ,E . (4.5) w∈W P l(w)=q

The reader should bear in mind that the above result of Kostant can be

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parsed further over the set of embeddings τ : F → E. We have M O H q (uP , Mλ,E ) = H qτ (uP0τ , Mλτ ,E ) (K¨ unneth theorem) P

qτ =q τ :F →E

M

=

P

P

Mwτ ·λτ ,E

(Kostant for each uP0τ )

M

O

Mwτ ·λτ ,E

(⊗ commutes ⊕)

qτ =q wτ ∈W τ P0τ τ :F →E 0 l(wτ )=qτ

M

=

M

qτ =q τ :F →E wτ ∈W τ P0τ 0 l(wτ )=qτ

M

=

O

Mw·λ,E

(since w = (wτ ) and l(w) =

P

τ

l(wτ )).

w∈W P l(w)=q

Applying (4.5) to the boundary cohomology as in Prop. 4.2, while using the description of the action of π0 (P (R)) in Sect. 4.2.1, we get the following proposition. Proposition 4.3. The cohomology of ∂P S G is given by fλ,E ) H q (∂P S G , M M =

a

  M π (G(R))×G(A ) fw·λ,E )π0 (K∞P ) . Indπ00 (P (R))×P (Aff ) H q−l(w) (S MP , M

w∈W P

Remark 4.4. Suppose we have an irreducible MP (Af )-module πf such that for some λ ∈ X ∗ (T ) and some w ∈ W P , we have πf ∈ Coh(MP , w · λ). Let ε be a character of π0 (MP (R)) = π0 (P (R)). Then the induction to π0 (G(R)) × fλ,E ) if and only if ε is trivial on G(Af ) of ε × πf will contribute to H • (∂P S G , M MP π0 (K∞ ). In other words, not all of the cohomology of the Levi contributes to boundary cohomology. This is a subtle point that will be relevant later in the monograph; see Sect. 5.3.5.

4.3

CUSPIDAL SPECTRUM VERSUS RESIDUAL SPECTRUM

We describe the contribution of the discrete but noncuspidal spectrum to cohomology. This means we formulate the consequences of the description of the discrete spectrum in Mœglin–Waldspurger for the square integrable cohomology, in a sense we make their results more explicit. We work at a transcendental level: our coefficient systems are C-vector spaces. Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:01 PM

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BOUNDARY COHOMOLOGY

We go back to the global situation. Take a parabolic (not necessarily maximal) subgroup P ⊂ G, and let M be its reductive quotient. Let d(P ) be the parabolic rank of P and let γ P = {. . . , γPi , . . . } be the array of dominant weight characters γPi on the maximal parabolic subgroups Pi ⊃ P. We consider an iso• M typical σf in the cuspidal cohomology Hcusp (SK M , Dµ ⊗ Mw·λ ). Then we have f

the induced representation a

where |γ|z =

G(A)

IndP (A) Dµ ⊗ Vσf ⊗ |γ|z ⊂ C∞ (P (Q)U (A)\G(A))

Q

|γiu |zi . We get the Eisenstein intertwining operator (Sect. 6.3.4)

Eis(z, Dµ ⊗ σf ) :

a

G(A)

IndP (A) Dµ ⊗ Vσf ⊗ |γ|z ,→ C∞ (G(Q)\G(A)).

We know that this series defines a holomorphic intertwining operator as long as if 1, a unitary cuspidal representation σ of RF/Q (GLa /F ) such that ι πf ⊗| |d is the finite part of the unique irreducible quotient of the representation of G(A) parabolically induced from σ[(b−1)/2]×σ[(b−3)/2]×· · ·×σ[−(b−1)/2]. (Here, σ[s] := σ ⊗ |det|s .) It’s now an easy exercise to show that inequalities in (2) are violated.

5.1.3

Cuspidal cohomology for GLn

G fλ ), where ? = {c, !, empty, ∂}. Define Coh(H • (S G , M fλ )) Consider H?• (SK ,M Kf ? f G to be the set of isomorphism classes of absolutely irreducible HKf -modules that

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THE STRONGLY INNER SPECTRUM AND APPLICATIONS

53

occur in a Jordan–H¨ older series; this definition makes sense even if the module (S) is not semisimple. For an isomorphism class πf let πf be the restriction to the (S)

central subalgebra HKf . The various criteria for πf to be cuspidal are collected together in the following theorem. G Theorem 5.2. Let λ ∈ X0∗ (T ) be a pure weight and πf ∈ Coh(H!• (SK , Mλ,E )) = f Coh! (G, λ, Kf ). The following are equivalent:

1. There exists ι : E → C such that ι πf is cuspidal. 2. For all ι : E → C, ι πf is cuspidal. (S)

3. The isomorphism type πf

G fλ,E )). does not occur in Coh(H • (∂SK ,M f

G fλ,E )(πf ) 6= (0) if and only if bF ˜F 4. We have H q (SK ,M n ≤ q ≤ tn . f

5. There exists an ι : E → C such that (Dιλ ⊗ ε∞ (ιπf )) ⊗ ι πf is automorphic; see (3.11). ¯ → C extending ι, the Satake parameters satisfy 6. For ι : E → C and any ¯ι : E the estimates d− 1 d+ 1 qp 2 < |¯ι(ϑi,p )| < qp 2 . The above theorem has already been proved by all that was discussed before stating it; however, for the reader’s benefit let’s add a series of comments concerning the proof of the above theorem: • • • • •

(1) ⇐⇒ (3): see Thm. 4.7 and Thm. 4.8. (1) =⇒ (5): see first paragraph of Sect. 3.1.4 and Prop. 3.3. (5) =⇒ (4): standard computation using Delorme’s lemma; see Prop. 3.4. (4) =⇒ (1): see Thm. 4.7. (1) ⇐⇒ (2): follows from (1) ⇐⇒ (4). (This equivalence is a theorem of Clozel [14].) • (2) ⇐⇒ (6): see Prop. 5.1.

5.1.4

An arithmetic definition of Eisenstein cohomology

Let V be a finite-dimensional E-vector space that is also a module for a commutative Q-algebra A. Let us assume that all absolutely irreducible subquotients are one-dimensional (i.e., all eigenvalues lie in E). Define SpecA (V ) to be the set of isomorphism classes of absolutely irreducible A-modules over E that appear as irreducible subquotients of V. Let W be an A-stable E-subspace of V such that SpecA (W ) ∩ SpecA (V /W ) = ∅. Then there is an A-equivariant projection πW : V → W ; i.e., we have a splitting V ' W ⊕ V /W of A-modules. We say that the summands have disjoint supBrought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:02 PM

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port with respect to A, or simply that they have disjoint support with A being understood from the context at hand. We apply this to the global cohomology and for A we take the central subal(S) G fλ,E ). Then Thm. 4.8 gives that strongly inner gebra HKf acting on H • (SK ,M f cohomology splits off within global cohomology; i.e., we have a canonical decomposition with disjoint supports: G fλ,E ) = H • (S G , M fλ,E ) ⊕ H • (S G , M fλ,E ), H • (SK ,M !! Kf Eis Kf f

(5.3)

• G fλ,E ). Such a dewhich defines the Eisenstein cohomology denoted HEis (SK ,M f composition, and a finer one at that, is also given by Franke and Schwermer [19]; but their decomposition is given only at a transcendental level.

5.2

DEFINITION OF THE RELATIVE PERIODS

In this section we will consider the group G = RF/Q (GLn /F ) only when n is an even positive integer. The purpose of this section then is to define and analyze certain relative periods Ωε (ιπf ) ∈ C× , where λ = (λτ )τ :F →E is a pure weight, πf ∈ Coh!! (G, Kf , λ) for some level structure Kf , ι : E → C, and ε = (εv )v∈S∞ is a character of π0 (G(R)).

5.2.1

The arithmetic identification

Consider inner cohomology in the bottom degree bF n , and take the field E large enough so that we have the decomposition in (3.9). Fix an E-linear isomorphism G of HK -modules f bF

bF

ε G fλ,E )(πf × ε) −→ H n (S G , M fλ,E )(πf × −ε), Tarith (λ, πf ) : H! n (SK ,M Kf ! f

which we call an arithmetic identification between these absolutely irreducible ε modules. The choice of Tarith (λ, πf ) is well defined up to E(πf )× -multiples. We may and will ask that these isomorphisms be compatibly chosen; i.e., for any ι : E → E 0 we ask for the commutativity of F

b G fλ,E )(πf × ε) H! n (SK ,M f

F



ε Tarith (λ, πf )

fλ,E )(πf × −ε) / H bn (S G , M Kf ! F

ι•

b G fιλ, E 0 )(ι πf × ε) H! n (SK ,M f

ε Tarith (ιλ, ι πf )



ι•

fιλ, E 0 ) )(ι πf × −ε). / H bn (S G , M Kf ! F

(5.4) −ε ε We also want Tarith (λ, πf )−1 = Tarith (λ, πf ).

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THE STRONGLY INNER SPECTRUM AND APPLICATIONS

5.2.2

The transcendental identification

We start from πf ∈ Coh!! (G, Kf , λ) and our aim is to construct—using the transcendental description of the cohomology—a family of isomorphisms: bF

bF

ε G G n n Ttrans (ιλ, ιπf ) : Hcusp (SK , Mιλ,C )(ι πf × ε) −→ Hcusp (SK , Mιλ,C )(ι πf × −ε). f f F

b G fλ,E )(πf × ε0 ). We Fix an ε0 : π0 (G(R)) → {±1} and put Vπf = H! n (SK ,M f do this simply because we want to realize the isomorphism type by an explicitly given module. It follows from Thm. 5.2 that for any ι : E → C there is a unique irreducible admissible (g, K∞ )-module Dιλ so that there is a nonzero intertwining operator
∞ −1 Φ ∈ Hom(g,K∞ )⊗HG Dιλ ⊗ Vπf ⊗E,ι E 0 ), Ccusp (G(Q)\G(A)/Kf , ω∞ ) . K f

(See Sect. 3.3.1.) Appealing to the multiplicity one result for cusp forms for GLn /F we see that Φ is unique up to a scalar. This induces a map still called Φ on the relative Lie algebra complexes: HomK∞ (Λ• (g/k), Dιλ ⊗ Mιλ,C ⊗ Vπf ⊗E,ι E 0 )
∞ −1 → HomK∞ (Λ• (g/k), Ccusp G(Q)\G(A)/Kf , ω∞ ⊗ Mιλ,C ) and hence a map G Φ• : H • (g, K∞ , Dιλ ⊗ Mιλ,C ) ⊗ Vπf ⊗E,ι C → H!• (SK , Mιλ,C ). f

This map is an injection and in degree bF n it yields an isomorphism: ∼

F

bF

G H bn (g, K∞ , Dιλ ⊗ Mιλ,C )) ⊗ Vπf ⊗E,ι C −→ H! n (SK , Mιλ,C )(ι πf ). f

(5.5)

We decompose the factor on the left into eigenspaces: F

H bn (g, K∞ , Dιλ ⊗ Mιλ,C ) =

M

F

H bn (g, K∞ , Dιλ ⊗ Mιλ,C )(ε).

ε:π0 (G(R))→{±1} F

The spaces H bn (g, K∞ , Dιλ ⊗ Mιλ,C )(ε) are one-dimensional. Our next step will be the construction of an entangled system of basis elements {wε (ιλ)}ε for these one-dimensional spaces, and this system will be well defined up to a scalar in C× . The construction of an entangled system {wε (ιλ)}ε. We will now discuss the choice of basis wε (ιλ). Let ιλ = (ιλτ )τ :F →E = (λν )ν:F →C ; here ν = ι ◦ τ. Since n is even, the restriction to GLn (R)0 of Dλν breaks up as Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:02 PM

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− Dλν = D+ λν ⊕ Dλν , and the nontrivial element of π0 (GLn (R)) switches these two summands. Implicit in the proof of Prop. 3.4 is the fact that cohomology in the + bottom degree H bn (gln , SO(n)R× + ; Dλν ⊗ Mλν ) is one-dimensional, and we fix + ν a basis for this space, say w (λ ); i.e., + + ν H bn (gln , SO(n)R× + ; Dλν ⊗ Mλν ) = C w (λ ).

Apply the K¨ unneth theorem, to get that the element w++ (ι λ) := ⊗ν w+ (λν ) F

++ 0 generates the one-dimensional space H bn (g, K∞ ; D++ = ι λ ⊗ Mι λ ), where Dι λ + ⊗ν Dλν . For any character ε of π0 (Gn (R)), let’s define

wε (ι λ) =

X

ε(a)a · w++ (ι λ).

(5.6)

a∈π0 (Gn (R))

Then, clearly,
F 0 H bn g, K∞ ; Dιλ ⊗ Mιλ,C (ε) = C wε (ι λ). If we change w++ (ι λ) by a scalar µ, then each wε (ι λ) is multiplied by the same scalar, and hence the entangled system {wε (ιλ)}ε is well defined up to scalars. Starting from this entangled system we get isomorphisms ∼

bF

G Ψ(wε (ιλ)) : Vπf ⊗E,ι C −→ H! n (SK , Mιλ,C )(ι πf × ε), f

which are given by the composition v 7→ wε (ιλ) ⊗ v 7→ Φ• (wε (ιλ) ⊗ v). Then we define ε Ttrans (ιλ, ιπf , w) := irn/2 Ψ(w−ε (ιλ)) ◦ Ψ(wε (ιλ))−1 . (5.7) ε It’s clear that these operators Ttrans do not depend on the choices for Φ, {wε (ιλ)}, or w+ (λν ). (The scaling factor of irn/2 ensures that the map induced 0 by the standard intertwining operator in (g, K∞ )-cohomology is defined over Q; this will be relevant later in Chap. 8.)

We may orient our thoughts a bit differently for the transcendental identification. Consider the linear map of one-dimensional spaces: T ε (ι λ) : Cwε (ι λ) −→ Cw−ε (ι λ)

(5.8)

defined by T ε (ι λ)(wε (ι λ)) = irn/2 w−ε (ι λ). ε ι

(5.9) ++ ι

It is clear that T ( λ) is independent of the choice of basis w ( λ) because, if we change w++ (ι λ) by a nonzero scalar, then from (5.6), the same scalar appears

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in both w±ε (ι λ). Then the transcendental identification may be parsed as

V πf

/ H bn (S G , Mιλ,C )(ι πf × ε) Kf ! F

Vπf ⊗E,ι Cwε (ι λ) nn6 nnn n n n nnn nnn 1⊗T ε (ι λ) ⊗E,ι C PPP PPP PPP PPP P(  Vπf ⊗E,ι Cw

ε Ttrans (ιλ,ιπf )

 F / H bn (S G , Mιλ,C )(ι πf × −ε), Kf !

−ε ι

( λ)

ε and we may deduce, as before, that Ttrans (ιλ, ιπf ) is independent of any choice of basis elements.

5.2.3

The relative periods

Definition 5.3. There exists Ωε (ιλ, ιπf ) ∈ C× such that ε ε Ωε (ιλ, ιπf ) Ttrans (ιλ, ιπf ) = Tarith (λ, πf ) ⊗E,ι 1. ε ε If we change the choice of Tarith (λ, πf ) to α Tarith (λ, πf ) for an α ∈ E(πf )× then ε ι ι ε ι ι Ω ( λ, πf ) changes to ι(α) Ω ( λ, πf ); i.e., we have an array of nonzero complex numbers {. . . , Ωε (ιλ, ιπf ), . . . }ι:E→C

well defined up to multiplication by E(πf )× . In other words, we have defined a period Ωε (λ, πf ) ∈ (E ⊗ C)× /E(πf )× . Sometimes, we suppress the λ in the notation, and will just write Ωε (πf ). The reader is referred to Raghuram [54] to see how the relative periods Ωε (ιπf ) are related to some other periods attached to ιπf that have played an important role in certain recent results on the special values of L-functions. To get these relations we have to replace the above choice of a “model space” Vπf by its Whittaker realization Wπf , which is also an E-vector space. If WDι λ is the Whittaker model of Dι λ then we have the canonical inclusion ∞ −1 WDι λ ⊗ Wπf ⊂ Ccusp (G(Q)\G(A)/Kf , ω∞ )).

and therefore we get equalities F

H bn (g, K∞ , WDι λ ⊗ Mιλ,C )(ε) ⊗ (Wπf ⊗E,ι C) F

G = H bn (SK , Mλ,E )(πf × ) ⊗E,ι ⊗C. (5.10) f

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The comparison of the two E-structures on both sides of this equality yields the aforementioned periods c(ι πf , ε). They are related to our relative periods as: Ωε (ιλ, ιπf ) =

c(ι πf , ε) , c(ι πf , −ε)

with equality to be construed as elements of (E ⊗ C)× /E × .

5.2.4

Period relations under Tate twists

The periods Ωε (ιπf ) have a simple behavior under Tate twists. For the moment, let n be any positive integer. Let λ = (λτ )τ :F →E be a pure weight, where, as Pn−1 before, λτ = i=1 (aτi − 1)γi + d · δn = (bτ1 , . . . , bτn ). For m ∈ Z, define a pure weight λ + mδn := (λτ + mδn )τ :F →E , where λτ + mδn =

n−1 X

(aτi − 1)γi + (d + m) · δn .

i=1

Also, let mδn stand for the weight (mδn )τ :F →E . We have the fundamental class G eδn ∈ H 0 (SK , Q[δn ]) f (see [30]) and the cup product by this class yields an isomorphism ∪eδm

n • G G TTate (m) : H?• (SK , Mλ,E ) −→ H?• (SK , Mλ+mδn ,E ) f f

called the m-th Tate twist. If πf ∈ Coh? (G, λ, Kf ) with ? ∈ {!, !!} then it is clear that • TTate (m)(πf × ε) = πf (−m) × (−1)m ε, (5.11) where πf (−m) := πf ⊗ | |−m stands for g f 7→ |det(g f )|−m πf (g f ), and ε is a character of π0 (G(R)). ε For later use, we introduce the notation TTate (λ, πf , m) for the m-th Tate twist on the module πf × ε appearing in Coh!! (G, Kf , λ).

Proposition 5.4. For any integer m, if πf ∈ Coh!! (G, Kf , λ) then πf (m) ∈ Coh!! (G, Kf , λ − mδn ). Assume now that n is even; then, for any ε we have m

Ωε (ιπf (m)) = Ω(−1)

ε ι

( πf ).

Proof. Follows from Thm. 5.2, Def. 5.3, and (5.11). We will say that the isomorphism types πf and πf (m) are conformally equivalent and we will denote by {πf } = {. . . , πf (m), . . . }m∈Z the conformal Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:02 PM

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59

equivalence class of isomorphism classes. We say that an isomorphism type πf ∈ Coh!! (G, Kf , λ) is realized if we have picked—by a certain rule—an irreG ducible submodule H(πf ) ⊂ H!!• (SK , Mλ,E ); such a rule may be given by the f G choice of a character ε, i.e., we may choose H(πf ) = H!!• (SK , Mλ,E )(πf × ε). f 5.2.5

Period relations under twisting by Dirichlet characters

Let us record the behavior of the relative periods under twisting πf by a finite× order character of the id`ele class group of F. Suppose χ : F × \A× is a F → E character of finite order. We let εχ = (χv (−1))v∈S∞ be the signature of χ. We have the following proposition. Proposition 5.5. If πf ∈ Coh!! (G, Kf , λ) then πf ⊗ χf ∈ Coh!! (G, Kf , λ). Assume now that n is even; then, for any ε we have Ωε (ιπf ⊗ ιχf ) = Ωεεχ (ιπf ). Proof. We will only briefly sketch the proof and leave the details to the reader. The character χ gives a degree zero class θχ = χf ⊗ εχ for GL1 with constant coefficients. Pulling this back via the determinant δ : GLn → GL1 , we get a G class δ ∗ θχ in H 0 (SK , E). We may cup the class for σf with δ ∗ θχ , or, using an f embedding ι : E → C, we may wedge the differential for ισf with ι δ ∗ θχ = δ ∗ θιχ , and the key point is that both of these have the same effect. Furthermore, if we have two characters χ and χ0 , both of these operations are associative, i.e., cupping by θχχ0 is the same as first cupping with θχ followed by θχ0 , etc.; i.e., we can define Ttrans and Tarith in a compatible manner for the family of twists πf ⊗ χf . From here on it is a routine exercise to see the rest of the proof of the period relation. (The reader should compare with the proof of the main theorem of [56].) We may derive the following corollary to Prop. 5.4 and Prop. 5.5: Suppose n is even, and let πf be as above, and χ be an algebraic Hecke character of F with coefficients in E, then χ = | |m χ◦ for m ∈ Z and χ◦ is a character of finite order with values in E. Let εχ = (−1)m εχ◦ be the signature of χ. Then Ωε (ιπf ⊗ ιχf ) = Ωεεχ (ιπf ).

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5.3

THE STRONGLY INNER COHOMOLOGY OF THE BOUNDARY

5.3.1

The restriction to the maximal strata

We have the restriction from the cohomology of the boundary to the cohomology of the maximal boundary strata M G G fλ,E ) −→ fλ,E ) H • (∂SK ,M H • (∂P SK ,M (5.12) f f P : maximal G and for the cohomology of each boundary strata ∂P SK , after passing to the f limit over all Kf , from Prop. 4.3 we have

M

fλ,E ) = H • (∂P S G , M

a

π (G(R))×G(A ) fw·λ,E ). Indπ00 (P (R))×P (Aff ) H •−l(w) (S MP , M

w∈W P

Our previous result (5.3) applied to the quotients MP yields a decomposition •−l(w)

fw·λ,E ) = H H •−l(w) (S MP , M !!

•−l(w)

fw·λ,E ) ⊕ H (S MP , M Eis

fw·λ,E ) (S MP , M

and therefore we get homomorphisms between Hecke modules: fλ,E ) −→ rmax : H • (∂S G , M

M M P

w∈W P



π (G(R))×G(A ) a Indπ00 (P (R))×P (Aff ) a



•−l(w)

H!!

 fw·λ,E ) ⊕ (S MP , M

  π (G(R))×G(A ) •−l(w) fw·λ,E ) . (5.13) Indπ00 (P (R))×P (Aff ) HEis (S MP , M

On the right-hand side we project to the first set of summands to get fλ,E ) rmax,! : H • (∂S G , M M M −→ P

a

π (G(R))×G(A )

•−l(w)

Indπ00 (P (R))×P (Aff ) H!!

fw·λ,E ). (S MP , M

w∈W P

This homomorphism is now surjective because the right-hand side is in the kernel of all differentials in the spectral sequence. Take Kf -invariants and if S is the finite set of places outside of which Kf is the full maximal compact, then G we restrict the action of the Hecke algebra HK to the commutative subalgef G,S G bra H = ⊗p∈S / HKp . Then, as in the proof of Thm. 4.8, once again applying

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Jacquet and Shalika [39, Thm. 4.4], we have

SpecHG,S

M M  P

π (G(R))×G(A ) a Indπ00 (P (R))×P (Aff )

•−l(w) fw·λ,E ) H!! (S MP , M

!

w∈W P

∩ SpecHG,S ker(rmax,! )Kf

5.3.2

Kf




= ∅. (5.14)

An interlude on Kostant’s representatives

Let the notations be as in Sect. 4.2.2, but take P0 to be a maximal parabolic subgroup of G0 . Let ΠMP0 = ΠG0 − {αP0 }. Let wP0 be the unique element of W0 = WG0 such that wP0 (ΠMP0 ) ⊂ ΠG0 and wP0 (αP0 ) < 0; it is the longest Kostant representative for W P0 . Let Q0 be the parabolic subgroup associated to P0 ; we have (i) wP0 (ΠMP0 ) = ΠMQ0 ; (ii) wP0 (∆UP0 ) = −∆UQ0 . (Here ∆UP0 is the set of those positive roots whose root spaces are in UP0 ; similarly, ∆UQ0 .) For (ii), observe that if α ∈ ∆UP0 then in the expression for α in terms of simple roots, the root αP0 has to appear with a positive integral coefficient, and since wP0 (αP0 ) < 0, there is a negative coefficient in the expression for wP0 (α); but all coefficients have the same sign and hence wP0 (α) < 0. Lemma 5.6. With notations as above, we have: 1. The map w 7→ w0 := wP w gives a bijection W P → W Q . If w = (wτ )τ :F →E , then by definition, wP w = (wP0 wτ )τ :F →E . 2. This bijection has the property that l(wτ ) + l(w0τ ) = dim (UP0τ ). Hence l(w) + l(w0 ) = dim(UP ). Proof. The proof is the same for every component indexed by τ. We will suppress τ from the notation and just work over the group G0 /F and its subgroups. Let w0 ∈ W0 . (In our simplified notation, this w0 is any of the wτ for the original w in (1) above.) To prove the first statement of the lemma: wP0 w0 ∈ W Q0

⇐⇒

w0−1 wP−1 (β) > 0, ∀β ∈ ΠMQ0 (put β = wP0 α; see (i)) 0

⇐⇒ ⇐⇒

w0−1 (α) > 0, ∀α ∈ ΠMP w0 ∈ W P 0 .

To prove l(w0 ) + l(w00 ) = dim (UP0 ), partition ∆UP0 as a disjoint union

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CHAPTER 5

∆+ U

−1 P0 ,w0

∪ ∆− U

−1 P0 ,w0

, where

∆+ U

:= {α ∈ ∆UP0 : w0−1 (α) > 0}

∆− U

:= {α ∈ ∆UP0 : w0−1 (α) < 0}.

−1 P0 ,w0

and −1 P0 ,w0

Since w0 ∈ W P0 , it follows that −1 −1 − {α ∈ ∆+ G0 : w0 α < 0} = {α ∈ ∆UP0 : w0 α < 0} = ∆U

−1 P0 ,w0

Hence, l(w0 ) = l(w0−1 ) = |∆− U gives a bijection

∆+ UP0 ,w0−1

−1 P ,w0

→

.

|. Next, observe that the map α 7→ −wP0 α

∆− UQ0 ,w00−1

since

w00−1 (−wP0 (α)) = −w00−1 wP0 (α) = −w0−1 α < 0. We have dim (UP0 ) = |∆UP0 | = |∆− U

| + |∆+ U

= |∆− U

| + |∆− U

−1 P0 ,w0 −1 P0 ,w0

−1 P0 ,w0

|

0−1 Q0 ,w0

|

= l(w0−1 ) + l(w00−1 ) = l(w0 ) + l(w00 ).

Similarly, we have the following self-bijection of W P : Lemma 5.7. Let the notations be as in Lem. 5.6. Let wG be the element of longest length in the Weyl group WG of G, and similarly, let wMP be the element of longest length in the Weyl group WMP of the Levi quotient MP of P . Then: 1. The map w 7→ wv := wMP w wG gives a bijection W P → W P . 2. This bijection has the property that l(w) + l(wv ) = dim(UP ). Proof. Similar to the proof of Lem. 5.6. If Mλ is a highest weight module and Mλv is its dual, then we have a nondegenerate pairing H q (uP , Mλ,E ) × H dU −q (uP , Mλv ,E ) −→ E.

(5.15)

If we decompose the two cohomology modules according to (4.5) then the pairing becomes a direct sum over w ∈ W P of nondegenerate pairings Mw·λ,E × Mwv ·λv ,E

−→ E.

(5.16)

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5.3.3

63

Further decomposition into isotypical pieces

We define fλ,E ) H!!• (∂S G , M M =

M

a

  π (G(R))×G(A ) •−l(w) fw·λ,E ) , Indπ00 (P (R))×P (Aff ) H!! (S MP , M

P : maximal w∈W P

where each term decomposes further into isotypic components: M •−l(w) •−l(w) fw·λ,E ) = fw·λ,E )(σ ). H!! (S MP , M H!! (S MP , M f σ f ∈Coh!! (MP ,w·λ)

Given two maximal parabolic subgroups P and Q, with reductive quotients MP and MQ , respectively, we have to understand under what conditions we have nontrivial intertwining operators between the induced modules

a

•−l(w)

π (G(R))×G(A )

Indπ00 (P (R))×P (Aff ) H!! a

fw·λ,E )(σ ) and (S MP , M f

π (G(R))×G(A )

•−l(w1 )

Indπ00 (Q(R))×Q(Aff ) H!!

fw ·λ,E )(σ ), (5.17) (S MQ , M 1 f,1

where w ∈ W P , w1 ∈ W Q , σ f ∈ Coh!! (MP , w · λ), and σ f,1 ∈ Coh!! (MQ , w1 · λ). It is again the theorem of Jacquet–Shalika that tells us that this will almost never be the case unless we are in a special situation which is summarized in the following proposition. Proposition 5.8. We have a nontrivial intertwining between two such modules as in (5.17) if and only if 1. P = Q or P and Q are associate, and furthermore, if this condition is fulfilled, then 2. under the bijection w 7→ w0 from W P → W Q (Lem. 5.6) which gives a bijection σ f → σ ∗f between Coh!! (MP , w · λ) and Coh!! (MQ , w0 · λ), we have w1 = w0 and σ f,1 = σ ∗f . Therefore, strongly inner cohomology of the boundary has an isotypical decomposition M

fλ,E ) = H!!• (∂S G , M

M

M

P : maximal w∈W P σ f ∈Coh!! (MP ,w·λ)

a

π (G(R))×G(A )

•−l(w)

Indπ00 (P (R))×P (Aff ) [H!! a

π (G(R))×G(A )

fw·λ,E )(σ ) ⊕ (S MP , M f •−l(w0 )

Indπ00 (Q(R))×Q(Aff ) H!!

 fw0 ·λ,E )(σ ∗ ) . (5.18) (S MQ , M f

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  Here isotypical means that two different summands of the form . . . have disjoint supports (see Sect. 5.1.4). Observe that in the above decomposition the cohomology in the two summands is possibly living in different degrees because of the shifts by l(w) and l(w0 ). This motivates the following definition. Definition 5.9. An w = (wτ )τ :F →E ∈ W is called a balanced Kostant representative for P if w ∈ W P and l(wτ ) = dim (UP0 )/2,

∀τ : F → E.

It follows from Lem. 5.6 that if w is a balanced Kostant representative for P then w0 is also balanced as a Kostant representative for Q. An obvious necessary condition for the existence of balanced elements in W P is that dim (UP0 ) = nn0 is even. In this monograph we are concerned only with the contribution coming from such balanced Kostant representatives.

5.3.4

Interlude on induced representations

Let σ f ∈ Coh!! (MP , w · λ) be as earlier. Write MP = Gn × Gn0 , σ f = σf ⊗ σf0 with σf ∈ Coh!! (Gn , µ) and σf0 ∈ Coh!! (Gn0 , µ0 ) with pure weights µ and µ0 such that w · λ = µ + µ0 . We will henceforth assume that µ and µ0 are such that there is a balanced w ∈ W P for which w−1 · (µ + µ0 ) is a dominant weight. This condition on the weights µ and µ0 has many interesting and crucial consequences that are captured by the combinatorial lemma; see Sect. 7.2.3. Consider now the associate parabolic subgroup Q with reductive quotient MQ = Gn0 × Gn . The Kostant representative w0 ∈ W Q given as in Lem. 5.6 is also balanced, and furthermore it is easy to see that w0 · λ = w0 · (w−1 · (µ + µ0 )) = (µ0 − nδn0 ) + (µ + n0 δn ). If σf ∈ Coh!! (Gn , µ) then σf (−n0 ) ∈ Coh!! (Gn , µ + n0 δn ). Similarly, σf0 ∈ Coh!! (Gn0 , µ0 ) implies σf0 (n) ∈ Coh!! (Gn0 , µ0 − nδn0 ). Here the module σ ∗f is nothing but σf0 (n) × σf (−n0 ). Consider the corresponding algebraically induced representations appearing as in (5.18): a

G(A )

IndP (Aff ) σf ⊗ σf0




and

a


G(A ) IndQ(Aff ) σf0 (n) ⊗ σf (−n0 ) .

(5.19)

By Jacquet and Shalika [39, (4.3)], they are equivalent almost everywhere. Take any open-compact subgroup Kf , and an associated finite set of finite places S such that Kf is unramified outside S, and consider the Kf -invariants as an HG,S -module. There is only one isomorphism type of simple HG,S -module in the Kf -invariants of the induced representations in (5.19); denote this particular isomorphism type as I S (σf , σf0 ). Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:02 PM

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65

Taking contragredients we have: a

G(A )

IndP (Aff ) σf ⊗ σf0


v

∼

−→

a


G(A ) IndP (Aff ) σf v (n0 ) ⊗ (σf0 )v (−n) ,

(5.20)

with σfv (n0 ) ∈ Coh!! (Gn , µv − n0 δn ) and σf0v (−n) ∈ Coh!! (Gn0 , µ0v + nδn0 ). As a notational artifice, given weights µ ∈ X ∗ (Tn ) and µ0 ∈ X ∗ (Tn0 ), we will denote the weight µ + µ0 ∈ X ∗ (TN ) also as   µ 0 µ+µ = . µ0 Note that the inducing data σfv (n0 ) ⊗ σf0v (−n) for the contragredient representation considered earlier has cohomology with respect to the weight  v  µ − n0 . µ0v + n

Lemma 5.10. Let µ ∈ X0∗ (Tn ) be a pure weight, and similarly, µ0 ∈ X0∗ (Tn ). Assume that there exists a balanced Kostant representative w ∈ W P such that λ := w−1 · (µ + µ0 ) is dominant. Let wv ∈ W P be the Kostant representative associated to w given by Lem. 5.7. Then wv is also balanced and furthermore  v  µ − n0 w v · λv = . µ0v + n Proof. That wv is balanced follows from Lem. 5.7. The rest of the proof may be parsed over the embeddings τ : F → E and for each τ it is the same calculation, and so we just suppress τ from notation. Next, as a notational artifice, let wN be the element of longest length in the Weyl group of G0 := GLN /F . As a τ permutation matrix, we have wN (i, j) = δi,N −j+1 . For τ : F → E, let wN be the same element but now thought of as the element of longest length in G0 ×F,τ E. τ Then, wG = (wN )τ :F →E . Similarly, representing MP0 as block diagonal matrices permits us to think of wMP as an array indexed by τ with each entry being the block diagonal matrix: wMP0 = ( wn wn0 ) , where wn (i, j) = δi,n−j+1 , etc. Note that −wN λ = λv , −wn µ = µv and −wn0 µ0 = µ0v . We have:

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v

w ·λ

v

 =



wn



wwN · (−wN λ)   wn w · (−λ − 2ρN ) wn0      wn −µ · − 2ρ N wn0 −µ0  v    µ wn − ρN − ρN µ0v wn0  v  µ − n0 . µ0v + n wn0

 = = = =

Again, there is only one isomorphism type of simple HG,S -module in the Kf -invariants of the induced representation in (5.20).

5.3.5

The character of the component group II

The assumption of wanting a balanced Kostant representative necessitates nn0 to be even. Without loss of generality, we take n to be even and let n0 be any positive integer. Given σf ∈ Coh!! (Gn , µ) and σf0 ∈ Coh!! (Gn0 , µ0 ) we begin by taking a character ε0 of π0 (Gn0 (R)) as  ε(σf0 ) if n0 is odd, and ε0 = (5.21) any character of π0 (Gn0 (R)) if n0 is even, where ε(σf0 ) is as in (3.10). Note that ε(σf0 (n)) = ε(σf0 ) since n is an even integer. Now take any character ε of π0 (Gn (R)). From the short exact sequence of the group of connected components in (4.4) it follows that MP the character ε × ε0 of π0 (P (R)) is trivial on π0 (K∞ ) ⇐⇒ ε = ε0 . M

fµ+µ0 ,E )π0 (K∞P ) . The character ε0 ×ε0 Hence (ε0 ×σf )×(ε0 ×σf0 ) ,→ H!!• (S MP , M of π0 (P (R)) maps to ε0 on π0 (G(R)). From Sect. 4.2.1 it follows that a



π (G(R))×G(A ) G(A ) Indπ00 (P (R))×P (Aff ) (ε0 ⊗ σf ) ⊗ (ε0 ⊗ σf0 ) = ε0 ⊗ a IndP (Aff ) σf ⊗ σf0 .

Proposition 5.11. We assume that N is odd. Then the Hecke module σ f = M bF +bF fw·λ,E )π0 (K∞P ) . σf ⊗ σ 0 occurs with multiplicity at most one in H n n0 (S MP , M f

!!

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THE STRONGLY INNER SPECTRUM AND APPLICATIONS

Proof. Assume n is even and n0 is odd. Take pure weights µ and µ0 , and write µ + µ0 =: µ. We abbreviate bF +bF n0

HPb (σ f ) := H!!n

M

fµ,E )π0 (K∞P ) (σ ). (S MP , M f

(5.22)

Then we get a decomposition into irreducible Hecke modules bF +bF n0

H!!n

fµ,E )(σ ) (S MP , M f M bF bF = H!!n (S Gn , Mµ )(σf × ε) ⊗ H!!n0 (S Gn0 , Mµ0 )(σf0 ) ε

and the submodule HPb (σ f ) is the (irreducible) summand with ε = ε(σf0 ). (See Sect. 3.3.3.) Hence we see that HPb (σ f ) realizes σ f .

5.3.6

Interlude on arithmetic identifications

Following Sect. 5.2.4, we call two isomorphism types σ f and σ1 f to be conformally equivalent if there is a δ = dδn + d0 δn0 such that σ f ⊗ |δf |−1 = σ1 f . For any such δ we have the canonical generator eδ ∈ H 0 (S MP , Q[δ]), and the cup product yields an isomorphism bF +bF n0

• TTate (δ) = ∪eδ : H!!n

fµ,E )(σ ) (S MP , M f bF +bF n0

∼

−→ H!!n

fµ+δ,E )(σ ⊗ |δf |−1 ). (S MP , M f

We may use these identifications and form the direct limit bF +bF n0

H!!n

F

F

fµ+,E )({σ }) := lim H bn +bn0 (S MP , M fµ+δ,E )(σ ⊗ |δf |−1 }). (S MP , M f f −→ !! δ

This is a vector space that has a distinguished isomorphism to any member of the family bF +bF fµ+δ,E )(σ ⊗ |δf |−1 )}δ . {H!!n n0 (S MP , M f • Now we notice that TTate (δ) gives an isomorphism from bF

bF

HPn (S Gn , Mµ )(σf × ) ⊗ HPn0 (S Gn0 , Mµ0 )(σf0 ) to bF

HPn (S Gn , Mµ+dδn )(σf ⊗ |δn,f |−d × (−1)d ) bF

0

⊗ HPn0 (S Gn0 , Mµ0 +d0 δn0 )(σf0 ⊗ |δn0 ,f |−d ). Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:02 PM

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CHAPTER 5 0

0

Since (σf0 ⊗ |δn0 ,f |−d ) = (σf0 )(−1)d we get ∼

• TTate (δ) : HPb (σ f ) −→ HPb (σ f ⊗|δf |−1 ) ⇐⇒ d(δ) = d−d0 ≡ 0

mod 2. (5.23)

This suggests that we divide the family of weights {µ+δ} into two classes {+, −} according to the value of d(δ) mod 2. We can define the two limits HPb+ ({σ f }) :=

lim −→

HPb (σ f ⊗ |δf |−1 ),

lim −→

HPb (σ f ⊗ |δf |−1 ).

µ+δ:d(δ)≡0(2)

HPb− ({σ f }) :=

µ+δ:d(δ)≡1(2)

Finally, we can construct an isomorphism ∼

Tarith ({σ f }, ±) : HPb+ ({σ f }) −→ HPb− ({σ f }).

(5.24)

To do so we go back to our σ f = σf ⊗ σf0 and the description bF

bF

HPb+ ({σ f }) = HPn (S Gn , Mµ )(σf × (σf0 )) ⊗ HPn0 (S Gn0 , Mµ0 )(σf0 ). In Sect. 5.2.1 we have chosen an isomorphism between irreducible Hecke modules bF

∼

bF

ε Tarith (µ, σf ) : HPn (S Gn , Mµ )(σf × (σf0 )) −→ HPn (S Gn , Mµ )(σf × −(σf0 )), (5.25) which is unique up to an element of E × . We take δ1 = δn0 and then ∼

ε • Tarith (λ, σf ) ⊗ TTate (δ1 ) : HPb (σ f ) −→ HPb (σ f ⊗ |δ1,f |−1 ),

(5.26)

and this induces our searched for Tarith ({σ f }, ±). If we are in the case that n, n0 are both even then M

bF

bF

H!!n (S Gn , Mµ )(σf × ε) ⊗ H!!n0 (S Gn0 , Mµ0 )(σf0 × ε)

ε

= HPb (σ f ) =

M

HPb (σ f × ε).

ε • It will be important that TTate (δQ ) maps HPb (σ 0 f × ε) to HPb (σ 0 f ⊗ |δQ |f × ε).

5.3.7

A summary of our notation with some simplifications

The ambient group will be G = RF/Q (GLN /F ). Let n and n0 be positive integers so that N = n+n0 . Let G0 = GLN /F and let P0 be the standard (i.e., containing the standard Borel subgroup) maximal parabolic subgroup of G0 of type (n, n0 ), Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:02 PM

THE STRONGLY INNER SPECTRUM AND APPLICATIONS

69

UP0 its unipotent radical, and MP0 = P0 /UP0 = GLn × GLn0 /F is the Levi quotient. Let Q0 be the standard maximal parabolic subgroup of G0 of type (n0 , n), and UQ0 and MQ0 are similarly defined. Let G, P , UP , MP , Q, UQ , and MQ be the restriction of scalars from F to Q of G0 , P0 , UP0 , MP0 , Q0 , UQ0 , and MQ0 , respectively. If n = n0 then Q0 = P0 , Q = P , etc.; we will call this the self-associate case. We identify MP and MQ as in the introduction. Let µ ∈ X0∗ (Tn ) be a pure weight, and similarly, µ0 ∈ X0∗ (Tn0 ). Assume there exists a balanced Kostant representative w ∈ W P such that λ := w−1 · (µ + µ0 ) is dominant. So nn0 is even, and without loss of generality, we take n to be even, and n0 may be even or odd. (See Sect. 7.4.1 for the case nn0 is odd.) Let wv ∈ W P correspond to w as in Lem. 5.7. Let w0 (resp., wv0 ) be in W Q corresponding to w (resp., wv ) via Lem. 5.6. All the elements w, w0 , wv , and wv0 are balanced; i.e., their lengths are dim(UP )/2 = dim(UQ )/2. In the selfassociate case (n = n0 , P = Q), we note that w0 6= w and wv 6= wv0 . Write µ = µ(1) + µab with µab = dδn , and similarly, µ = µ0(1) + µ0ab with µ0ab = d0 δn0 . Write µ = µ + µ0 , and put a(µ) = a(µ, µ0 ) = d − d0 . If d(µ) = (nd + n0 d0 )/N then note that µ = µ(1) + µ0(1) + a(µ)γP + d(µ)δN (see Sect. 6.3.1 for γP ); the quantity a(µ) plays an important role for much that follows; however, d(µ) is uninteresting. Let σf ∈ Coh!! (Gn , µ) and σf0 ∈ Coh!! (Gn0 , µ0 ). Take a large enough finite set of finite places S containing all the places where either σf or σf0 is ramified. Take Q Kf = p Kp an open-compact subgroup of G(Af ) such that wP Kf wP−1 = Kf and Kp = GLN (Op ) for and p|p with p ∈ / S. By HG,S we mean the product of local spherical Hecke algebras outside of S. Take ε0 as in (5.21), and for brevity, we also denote ε˜0 for the character ε0 ⊗ε0 of π0 (MP (R)) or of π0 (MQ (R)). In the bottom degree, we observe F bF n + bn0 +

1 2

dim(UP ) = bF N,

and write IbS (σf , σf0 , ε0 )P,w for a

 F F K f M b +b π (G(R))×G(A ) fw·λ,E )π0 (K∞P ) (˜ Indπ00 (P (R))×P (Aff ) H!!n n0 (S MP , M ε0 ⊗ (σf ⊗ σf0 )) .

Note that the inducing data is HPb (σ f ) as in (5.22) when n0 is odd; however, if n0 is even, we have a choice of ε0 ; nevertheless, the reader should bear in mind that the explicit model for ε0 ⊗ σf or ε0 ⊗ σf0 is as it appears (with multiplicity one) in strongly inner cohomology in the bottom degree. Similarly, IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0 denotes a

 F F Kf MQ b +b 0 π (G(R))×G(A ) fw0 ·λ,E )π0 (K∞ ) (˜ Indπ00 (Q(R))×Q(Aff ) H!!n n (S MQ , M ε0 ⊗ (σf0 (n) ⊗ σf (−n0 )) .

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CHAPTER 5

In the top degree we work with the contragredient modules. The module σfv (n0 ) ⊗ σf0v (−n) is strongly inner for MP with respect to the weight wv · λv . Likewise, σfv 0 ⊗ σfv is strongly inner for MQ for the weight wv0 · λv . We also have ˜F t˜F n + tn0 +

1 2

dim(UP ) = t˜F N − 1.

In this case ItS (σfv (n0 ), σf0v (−n), ε0 )P,wv denotes a

 F F K f M t˜ +t˜ 0 π (G(R))×G(A ) fwv ·λv ,E )π0 (K∞P ) (˜ Indπ00 (P (R))×P (Aff ) H!!n n (S MP , M ε0 ⊗ (σfv (n0 ) ⊗ σf0v (−n))) ,

and similarly ItS (σf0v , σfv , ε0 )Q,wv 0 . 5.3.8

A strong form of the Manin–Drinfeld principle

Our main theorem on boundary cohomology as a Hecke module is the following theorem, which is the culmination of the entire discussion in this section. Theorem 5.12. Let the notations be as in Sect. 5.3.7. Then: 1. The π0 (G(R))×HG,S -modules IbS (σf , σf0 , ε0 )P,w and IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0 and similarly, the modules ItS (σfv (n0 ), σf0v (−n), ε0 )P,wv and ItS (σf0v , σfv , ε0 )Q,wv 0 are finite-dimensional E-vector spaces, all of which have the same dimension, say, denoted k. 2. The sum IbS (σf , σf0 , ε0 )P,w ⊕ IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0 F fλ,E )Kf . is a 2k-dimensional E-vector space that is isotypic in H bN (∂S G , M 0 (When P = Q, note that w 6= w.) Furthermore, there is a π0 (G(R)) × HG,S equivariant projection:

F fλ,E )Kf Rbσf ,σ0 ,ε0 : H bN (∂S G , M f

−→ IbS (σf , σf0 , ε0 )P,w ⊕ IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0 . 3. The sum ItS (σfv (n0 ), σf0v (−n), ε0 )P,wv ⊕ ItS (σf0v , σfv , ε0 )Q,wv 0 F fλv ,E )Kf . is a 2k-dimensional E-vector space that is isotypic in H t˜N −1 (∂S G , M v0 v (When P = Q, note that w 6= w .) Furthermore, there is a π0 (G(R))×HG,S equivariant projection:

˜F

fλv ,E )Kf Rtσf ,σ0 ,ε0 : H tN −1 (∂S G , M f

−→ ItS (σfv (n0 ), σf0v (−n), ε0 )P,wv ⊕ ItS (σf0v , σfv , ε0 )Q,wv 0 .

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Chapter Six Eisenstein Cohomology

G Recall from Sect. 2.3.4 the long exact sequence of π0 (G(R)) × HK -modules f •

•

•

i r d G G G fλ,E ) −→ fλ,E ) −→ fλ,E ) −→ · · · −→ Hc• (SK ,M H • (S¯K ,M H • (∂SK ,M ··· f f f

Eisenstein cohomology is defined as:   r• • G G • G fλ,E ) := Image H • (S¯K fλ,E ) −→ f HEis (∂SK , M , M H (∂S , M ) . (6.1) λ,E Kf f f We may pass to the limit over all Kf and also look at π0 (G(R))×G(Af )-modules   r• • fλ,E ) := Image H • (S¯G , M fλ,E ) −→ fλ,E ) . HEis (∂S G , M H • (∂S G , M

(6.2)

• fλ,E ) and The reader should note the subtle difference between HEis (∂S G , M • fλ,E ). The latter, we recall, is a complement to strongly inner cohoHEis (S G , M mology in global cohomology. Since strongly inner cohomology does not intertwine with boundary cohomology, we get a surjective map: • fλ,E ) HEis (S G , M

6.1

6.1.1

fλ,E ). / / H • (∂S G , M Eis

´ DUALITY AND MAXIMAL ISOTROPIC POINCARE SUBSPACE OF BOUNDARY COHOMOLOGY Poincar´ e duality

G Let the notations be as in Sect. 5.3.7. Let d := dF N = dim(SKf ). We have the G following Poincar´e duality pairing for sheaf cohomology on SK : f G G fλ,E ) × Hcd−• (SK fλv ,E ) −→ E. H • (SK ,M ,M f f

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(6.3)

72

CHAPTER 6

G G The boundary ∂SK is a compact manifold with corners with dim(∂SK ) = d−1. f f G We also have Poincar´e duality on ∂SKf : G G fλ,E ) × H d−1−• (∂SK fλv ,E ) −→ E. H • (∂SK ,M ,M f f

(6.4)

We will denote either of the two dualities simply by ( , ).

6.1.2

Compatibility of duality isomorphisms with the connecting homomorphism

Consider the following diagram: G fλ,E ) H • (SK ,M f



×

G fλv ,E ) Hcd−• (SK ,M f O

r∗

G fλ,E ) H • (∂SK ,M f

−→

E

−→

E

d∗

×

G fλv ,E ) H d−1−• (∂SK ,M f

(6.5) where the horizontal arrows are the Poincar´e duality pairings (6.3) and (6.4), and the vertical arrows r∗ and d∗ are as in the long exact sequence. For any G fλ,E ) and any ς ∈ H d−1−• (∂S G , M fλv ,E ) we have class ξ ∈ H • (SK ,M Kf f (r∗ (ξ), ς) = (ξ, d∗ (ς)).

6.1.3

(6.6)

Maximal isotropic subspaces

The following proposition asserts that Eisenstein cohomology is a maximal isotropic subspace of boundary cohomology under Poincar´e duality. Proposition 6.1. Under the duality pairing (6.4) for boundary cohomology, we have: • G G fλ,E ) = H d−1−• (∂SK fλv ,E )⊥ . HEis (∂SK ,M ,M Eis f f • G fλ,E ). Then Proof. This is an exercise in using (6.6). Let r∗ (ξ) ∈ HEis (∂SK ,M f

(r∗ (ξ), r∗ (ς 0 )) = (ξ, d∗ r∗ (ς 0 )) = (ξ, 0) = 0,

d−1−• G fλv ,E ). ∀r∗ (ς 0 ) ∈ HEis (∂SK ,M f

Hence the left-hand side is contained in the right-hand side. For the reverse inG fλ,E ) is orthogonal to H d−1−• (∂S G , M fλv ,E ), clusion, suppose ξ 0 ∈ H • (∂SK ,M Kf Eis f then G fλv ,E ). 0 = (ξ 0 , r∗ (ς 0 )) = (d∗ (ξ 0 ), ς 0 ), ∀ς 0 ∈ H d−1−• (SK ,M f Nondegeneracy of the duality pairing (6.3) in degree d − 1 − • implies ξ 0 ∈ Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:04 PM

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EISENSTEIN COHOMOLOGY

• G fλ,E ). Ker(d∗ ) = Im(r∗ ). Hence, ξ 0 ∈ HEis (∂SK ,M f

6.2

THE MAIN RESULT ON RANK-ONE EISENSTEIN COHOMOLOGY

Notations are as in Thm. 5.12. Consider the following maps starting from F fλ,E )Kf and ending with an isotypic component global cohomology H bN (S G , M in boundary cohomology: F

fλ,E )Kf H bN (S G , M

(6.7)

r∗

 F fλ,E )Kf H bN (∂S G , M Rbσ

0 0 f ,σf ,ε

IbS (σf , σf0 , ε0 )P,w

 ⊕ IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0

Recall, from Thm. 5.12, that IbS (σf , σf0 , ε0 )P,w ⊕ IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0 is a E-vector space of dimension 2k. In the self-associate case just change the Q to P . Our main result on Eisenstein cohomology (see Thm. 6.2) says that the bF fλ,E )Kf = Im(r∗ ) under Rb 0 0 is a middle-dimensional image of H N (∂S G , M Eis

σf ,σf ,ε

(i.e., k-dimensional) subspace of this 2k-dimensional space. (It helps to have a mental picture of when k = 1, i.e., of a line in an ambient two-dimensional space; we will see later that the slope of this line contains arithmetic information about L-values.) The proof of this main result also needs the analogue of (6.7) in the top degree. We now state and prove this main result on Eisenstein cohomology.

6.2.1

The image of Eisenstein cohomology under R•σf ,σ0 ,ε0 f

Theorem 6.2. Let the notations be as in 5.3.7. Furthermore, for brevity, let Ib (σf , σf0 , ε0 ) t

I

(σf , σf0 , ε0 )v

F

:=

bN fλ,E )Kf ), Rbσf ,σ0 ,ε0 (HEis (∂S G , M

:=

t˜F N −1 fλv ,E )Kf ). Rtσf ,σ0 ,ε0 (HEis (∂S G , M f

f

1. In the non-self-associate cases (n 6= n0 ) we have:

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CHAPTER 6

(a) Ib (σf , σf0 , ε0 ) is a k-dimensional E-subspace of IbS (σf , σf0 , ε0 )P,w ⊕ IbS (σf0 (n), σf (−n0 ), ε0 )Q,w0 . (b) It (σf , σf0 , ε0 )v is a k-dimensional E-subspace of ItS (σfv (n0 ), σf0v (−n), ε0 )P,wv ⊕ ItS (σf0v , σfv , ε0 )Q,wv 0 . 2. In the self-associate case (n = n0 ) the same assertions hold by putting Q = P.

6.2.2

Proof of Thm. 6.2

The proof of (2) is almost identical to the proof of (1), and so we give the details only for (1). The proof of (1) involves two steps: (i) The first step is to show that both Ib (σf , σf0 , ε0 ) and It (σf , σf0 , ε0 )v are at least k-dimensional; this is achieved by going to a transcendental level and appealing to Langlands’s constant term theorem and producing enough cohomology classes in the image. (ii) The second step, after invoking properties of the Poincar´e duality pairing, is to show that both Ib (σf , σf0 , ε0 ) and It (σf , σf0 , ε0 )v have dimension exactly k. We take up these two arguments in the Sects. 6.2.2.1 and 6.2.2.2. 6.2.2.1

The cohomological meaning of the constant term theorem of Langlands

Take an embedding ι : E → C and pass to a transcendental level via ι. To show that Ib (σf , σf0 , ε0 ) or It (σf , σf0 , ε0 )v has dimension at least k as an E-vector space, it suffices to show that their base change to C via ι has dimension at least k as a C-vector space; i.e., we would like to show:   bF N fιλ )Kf ) ≥ k and dimC Rbισf ,ισ0 ,ε0 (HEis (∂S G , M f   t˜F N fιλ )Kf ) ≥ k. (6.8) dimC Rtισf ,ισ0 ,ε0 (HEis (∂S G , M f

In Sect. 6.3 we briefly introduce the L-functions at hand, recall the celebrated theorem of Langlands on the constant term of an Eisenstein series, and then we will come back to the proof of this part in Sect. 6.3.7. 6.2.2.2

Application of Poincar´e duality

The proofs of (1)(a) and (1)(b), assuming that we have proved (6.8), are an exercise involving properties of Poincar´e duality especially that it is nondegenerate, Hecke-equivariant, and that the Eisenstein part is maximal isotropic. Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:04 PM

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EISENSTEIN COHOMOLOGY

The bare-bones linear algebra looks like: Suppose we have decompositions V = VP ⊕ VQ and W = WP ⊕ WQ , where VP , VQ , WP , and WQ are all kdimensional vector spaces over E; suppose also that we have a nondegenerate pairing ( , ) : V × W → E such that (VP , WQ ) = (VQ , WP ) = 0 and the pairing is nondegenerate on VP × WP and VQ × WQ ; furthermore, suppose we are given subspaces I ⊂ V and J ⊂ W such that dimE (I) ≥ k, dimE (J) ≥ k, and (I, J) = 0. Then it is easy to see that dimE (I) = k = dimE (J). This concludes the proof of Thm. 6.2 under the assumption that we have proved (6.8).

6.3

A THEOREM OF LANGLANDS: THE CONSTANT TERM OF AN EISENSTEIN SERIES

We will need some details from the Langlands–Shahidi method in our context. The reader is referred to Kim [45] and Shahidi [65] for details, proofs, and further references.

6.3.1

Notations for certain characters attached to P

Let the notations be as in Sect. 5.3.7. In particular, P0 = MP0 UP0 is the standard (n, n0 ) parabolic subgroup of GLN /F, and P = RF/Q (P0 ), etc. We write     h 0 0 0 0 MP0 = m = diag(h, h ) = : h ∈ GLn , h ∈ GLn , 0 h0 and let AP0 := ZMP0 =

  t1n a= 0

0 t0 1n0



 : t, t0 ∈ GL1 .

Fix an identification X ∗ (AP0 ) = Z2 , by letting (k, k 0 ) ∈ Z2 correspond to 0 the character that sends a to tk t0k . We have M M X ∗ (AP × E) = X ∗ (AP0 ×τ E) = Z2 . τ :F →E

τ :F →E

Similarly, fix X ∗ (MP0 ) = Z2 , by letting (k, k 0 ) ∈ Z2 correspond to the character 0 that sends diag(h, h0 ) to det(h)k det(h0 )k . Restriction from MP0 to AP0 gives ∗ ∗ an inclusion X (MP0 ) ,→ X (AP0 ) that is given by (k, k 0 ) 7→ (nk, n0 k 0 ). Clearly, X ∗ (MP0 ) ⊗ Q = X(AP0 ) ⊗ Q, which fixes an identification X ∗ (MP0 ) ⊗ Q = Q2 via X(AP0 ) ⊗ Q = Q2 . Similarly, X ∗ (MP0 ) ⊗ R = X ∗ (AP0 ) ⊗ R and fix X ∗ (MP0 ) ⊗ R = R2 . This fixes X ∗ (MP ) ⊗ R = ⊕τ :F →E R2 . Let ρP0 be half the sum of positive roots whose root spaces appear in UP0 . Brought to you by | Ludwig-Maximilians-Universität München Universitätsbibliothek (LMU) Authenticated Download Date | 10/20/19 2:04 PM

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CHAPTER 6

The restriction of ρP0 to AP0 is in X ∗ (AP0 ) ⊗ Q ,→ a∗P0 := X(AP0 ) ⊗ R = X ∗ (MP0 ) ⊗ R and under the above identification of the latter with R2 , one has ρP0 = (n0 /2, −n/2), or using the notations of 2.2.4, we have ρP0 =

1 2

X

ei − ej =

1≤i≤n

n0 n (e1 + · · · + en ) − (en+1 + · · · + en+n0 ). 2 2

n+1≤j≤N

And ρP = (ρP0τ )τ :F →E , with each ρP0τ given as earlier. Let αP0 = αn = en − en+1 be the unique simple root of G0 that is not among the roots of MP0 . Consider the corresponding fundamental weight γP0 = γn = hρP0 , αP0 i−1 ρP0 . Identify X ∗ (TN,0 ) ⊗ R with RN using the ei ’s, and let ( , ) be the usual euclidean inner product on RN . It is easy to see that hρP0 , αP0 i =

2(ρP0 ,αP0 ) (αP0 ,αP0 )

= N/2, and hence γn = γP0 =

2 ρ . N P0

We also have αP = (αP0τ )τ :F →E and γP = (γP0τ )τ :F →E , with N γP0τ = 2 ρP0τ . Let |δP0 | be the modular character of MP0 (k), where k is any local field (such as k = R or k = Fv any p-adic completion of F ), which is defined as |δP0 |(m) = |det(AduP (m))| for m ∈ MP0 (k). If m = diag(h, h0 ) with h ∈ GLn (k) and h0 ∈ GLn0 (k) then 0

|δP0 |(diag(h, h0 )) = |det(h)|n |det(h0 )|−n . Note that |2ρP0 |(·) = |δP0 |(·). Also, |δP |, the modular character of MP (k), for k = R or k = Qp , is defined via Q the various completions of F over that place; for example, |δP | on MP (Qp ) is p|p |δP0 |p , where by |δP0 |p we mean the character as above on MP0 (Fp ). For any character γ : M0 → Gm and m ∈ M0 (Fp ) we can ord (γ(m)) write γ(m) = $p p × unit, and then −ordp (γ(m))

|γ|(m) = qp

6.3.2

.

Induced representations

Let σ (resp., σ 0 ) be a cuspidal automorphic representation of Gn (A) (resp., Gn0 (A)). The relation with our previous arithmetic notation is that given σf ∈ Coh!! (Gn , µ) and given ι : E → C, think of ι σf to be the finite part of a cuspidal automorphic representation ι σ, etc. The ι is fixed, and we suppress it until otherwise mentioned. Consider the induced representation IPG (s, σ ⊗ σ 0 )

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EISENSTEIN COHOMOLOGY

consisting of all smooth functions f : G(A) → Vσ ⊗ Vσ0 such that 1

s

f (mug) = |δP |(m) 2 |δP |(m) N (σ ⊗ σ 0 )(m) f (g)

(6.9)

for all m ∈ MP (A), u ∈ UP (A), and g ∈ G(A), where Vσ (resp., Vσ0 ) is the subspace inside the space of cusp forms on Gn (A) (resp., Gn0 (A)) realizing the representation σ (resp., σ 0 ). In other words, n0

G(A)

IPG (s, σ ⊗ σ 0 ) = IndP (A) ((σ ⊗ | | N s ) ⊗ (σ 0 ⊗ | |

−n N s

)),

where IndG P denotes the normalized parabolic induction. In terms of algebraic or un-normalized induction, we have G(A)

n0

IPG (s, σ ⊗ σ 0 ) = a IndP (A) ((σ ⊗ | | N s+

6.3.3

n0 2

) ⊗ (σ 0 ⊗ | |

−n n N s− 2

)).

(6.10)

Standard intertwining operators

There is an element wP ∈ WG , the Weyl group of G, which is uniquely determined by the property wP (ΠG − {αP }) ⊂ ΠG and wP (αP ) < 0. This element looks like wP = (wPτ 0 )τ :F →E , where for each τ , as a permutation matrix in GLN , we have   1n wPτ 0 = . 1 n0 The parabolic subgroup Q, which is associate to P , corresponds to wP (ΠG − {αP }). Since wPτ 0 −1 diag(h, h0 )wPτ 0 = diag(h0 , h) for all diag(h, h0 ) ∈ MP0τ , we get wP (σ ⊗ σ 0 ) = σ 0 ⊗ σ as a representation of MQ (A). The global standard intertwining operator: G TstP Q (s, σ ⊗ σ 0 ) : IPG (s, σ ⊗ σ 0 ) −→ IQ (−s, σ 0 ⊗ σ)

is given by the integral (TstP Q (s, σ ⊗ σ 0 )f )(g) =

Z UQ (A)

f (wP−1 ug) du. 0

(6.11)

Often, we will abbreviate TstP Q (s, σ ⊗ σ 0 ) as Tst (s, σ ⊗ σ 0 ). The global standard intertwining operator factorizes as a product of local standard intertwining operators: Tst (s, σ ⊗ σ 0 ) = ⊗v Tst (s, σv ⊗ σv0 ), where the local operator G Tst (s, σv ⊗ σv0 ) : IPG (s, σv ⊗ σv0 ) −→ IQ (−s, σv0 ⊗ σv )

is given by a similar local integral.

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(6.12)

78 6.3.4

CHAPTER 6

Eisenstein series

Let f ∈ IPG (s, σ × σ 0 ); for g ∈ G(A) the value f (g) is a cusp form on MP (A). By the defining equivariance property of f , the complex number f (g)(m) determines and is determined by f (mg)(1) for any m ∈ MP (A). Henceforth, we identify f ∈ IPG (s, σ × σ 0 ) with the complex valued function g 7→ f (g)(1); i.e., we have embedded:

−1 −1 IPG (s, σ × σ 0 ) ,→ C ∞ UP (A)MP (Q)\G(A), ω∞ ⊂ C ∞ P (Q)\G(A), ω∞ , −1 where ω∞ is a simplified notation for the central character of σ ⊗ σ 0 restricted ◦ to S(R) . If σf ∈ Coh(Gn , µ), σf0 ∈ Coh(Gn0 , µ0 ), and ι : E → C then ω∞ is the product of the central characters ωMιµ ωMιµ0 restricted to S(R).

Given f ∈ IPG (s, σ × σ 0 ), thought of as a function on P (Q)\G(A),
define the −1 corresponding Eisenstein series EisP (s, f ) ∈ C ∞ G(Q)\G(A), ω∞ by averaging over P (Q)\G(Q): X f (γg); (6.13) EisP (s, f )(g) := γ∈P (Q)\G(Q)

this is convergent if