Classification of Pseudo-reductive Groups (AM-191) 9781400874026

Table of contents :
Cover
Title
Copyright
Contents
1 Introduction
1.1 Motivation
1.2 Root systems and new results
1.3 Exotic groups and degenerate quadratic forms
1.4 Tame central extensions
1.5 Generalized standard groups
1.6 Minimal type and general structure theorem
1.7 Galois-twisted forms and Tits classification
1.8 Background, notation, and acknowledgments
2 Preliminary notions
2.1 Standard groups, Levi subgroups, and root systems
2.2 The basic exotic construction
2.3 Minimal type
3 Field-theoretic and linear-algebraic invariants
3.1 A non-standard rank-1 construction
3.2 Minimal field of definition for Ru(Gk̅)
3.3 Root field and applications
3.4 Application to classification results
4 Central extensions and groups locally of minimal type
4.1 Central quotients
4.2 Beyond the quadratic case
4.3 Groups locally of minimal type
5 Universal smooth k-tame central extension
5.1 Construction of central extensions
5.2 A universal construction
5.3 Properties and applications of G̃
6 Automorphisms, isomorphisms, and Tits classification
6.1 Isomorphism Theorem
6.2 Automorphism schemes
6.3 Tits-style classification
7 Constructions with regular degenerate quadratic forms
7.1 Regular degenerate quadratic forms
7.2 Conformal isometries
7.3 Severi–Brauer varieties
8 Constructions when Φ has a double bond
8.1 Additional constructions for type B
8.2 Constructions for type C
8.3 Exceptional construction for rank 2
8.4 Generalized exotic groups
8.5 Structure of ZG and ZG,C
9 Generalization of the standard construction
9.1 Generalized standard groups
9.2 Structure theorem
A Pseudo-isogenies
A.1 Main result
A.2 Proof of Pseudo-Isogeny Theorem
A.3 Relation with the semisimple case
B Clifford constructions
B.1 Type B
B.2 Type C
B.3 Cases with [k : k^2] ≤ 8
B.4 Type BC
C Pseudo-split and quasi-split forms
C.1 General characteristic
C.2 Quasi-split forms
C.3 Rank-1 cases
C.4 Higher-rank and non-reduced cases
D Basic exotic groups of type F4 of relative rank 2
D.1 General preparations
D.2 Forms of k-rank 2
Bibliography
Index

Citation preview

Annals of Mathematics Studies Number 191

Classification of Pseudo-reductive Groups

Brian Conrad Gopal Prasad

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2016

c 2016 by Princeton University Press Copyright 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 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Conrad, Brian, 1970Classification of Pseudo-reductive Groups / Brian Conrad, Gopal Prasad. pages cm. – (Annals of mathematics studies; number 191) Includes bibliographical references and index. ISBN 978-0-691-16792-3 (hardcover : alk. paper) – ISBN 978-0-691-16793-0 (pbk. : alk. paper) 1. Linear algebraic groups. 2. Group theory. 3. Geometry, Algebraic. I. Prasad, Gopal. II. Title. QA179.C665 2016 512’.55–dc23 2015023803 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX using MathTime fonts. The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. 1 Printed in the United States of America 1

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Contents

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Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . 1.2 Root systems and new results . . . . . . . . . 1.3 Exotic groups and degenerate quadratic forms 1.4 Tame central extensions . . . . . . . . . . . . 1.5 Generalized standard groups . . . . . . . . . 1.6 Minimal type and general structure theorem . 1.7 Galois-twisted forms and Tits classification . 1.8 Background, notation, and acknowledgments

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Preliminary notions 2.1 Standard groups, Levi subgroups, and root systems . . . . . . 2.2 The basic exotic construction . . . . . . . . . . . . . . . . . . 2.3 Minimal type . . . . . . . . . . . . . . . . . . . . . . . . . .

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Field-theoretic and linear-algebraic invariants 3.1 A non-standard rank-1 construction . . . . 3.2 Minimal field of definition for Ru .Gk / . . 3.3 Root field and applications . . . . . . . . . 3.4 Application to classification results . . . . .

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Central extensions and groups locally of minimal type 4.1 Central quotients . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Beyond the quadratic case . . . . . . . . . . . . . . . . . . . 4.3 Groups locally of minimal type . . . . . . . . . . . . . . . . .

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Universal smooth k-tame central extension 5.1 Construction of central extensions . . . . . . . . . . . . . . . 5.2 A universal construction . . . . . . . . . . . . . . . . . . . .

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CONTENTS

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Automorphisms, isomorphisms, and Tits classification 6.1 Isomorphism Theorem . . . . . . . . . . . . . . . . . . . . . 6.2 Automorphism schemes . . . . . . . . . . . . . . . . . . . . . 6.3 Tits-style classification . . . . . . . . . . . . . . . . . . . . .

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Constructions with regular degenerate quadratic forms 7.1 Regular degenerate quadratic forms . . . . . . . . . . . . . . 7.2 Conformal isometries . . . . . . . . . . . . . . . . . . . . . . 7.3 Severi–Brauer varieties . . . . . . . . . . . . . . . . . . . . .

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Constructions when ˆ has a double bond 8.1 Additional constructions for type B . . 8.2 Constructions for type C . . . . . . . 8.3 Exceptional construction for rank 2 . . 8.4 Generalized exotic groups . . . . . . 8.5 Structure of ZG and ZG;C . . . . . .

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Generalization of the standard construction 9.1 Generalized standard groups . . . . . . . . . . . . . . . . . . 9.2 Structure theorem . . . . . . . . . . . . . . . . . . . . . . . .

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A Pseudo-isogenies A.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Proof of Pseudo-Isogeny Theorem . . . . . . . . . . . . . . . A.3 Relation with the semisimple case . . . . . . . . . . . . . . .

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B Clifford constructions B.1 Type B . . . . . . . . . B.2 Type C . . . . . . . . . B.3 Cases with Œk W k 2  6 8 B.4 Type BC . . . . . . . .

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C Pseudo-split and quasi-split forms C.1 General characteristic . . . . . . . . C.2 Quasi-split forms . . . . . . . . . . C.3 Rank-1 cases . . . . . . . . . . . . C.4 Higher-rank and non-reduced cases .

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D Basic exotic groups of type F4 of relative rank 2 D.1 General preparations . . . . . . . . . . . . . . . . . . . . . . D.2 Forms of k-rank 2 . . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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Classification of Pseudo-reductive Groups

1 Introduction

1.1 Motivation Algebraic and arithmetic geometry in positive characteristic provide important examples of imperfect fields, such as (i) Laurent-series fields over finite fields and (ii) function fields of positive-dimensional varieties (even over an algebraically closed field of constants). Generic fibers of positive-dimensional algebraic families naturally lie over a ground field as in (ii). For a smooth connected affine group G over a field k, the unipotent radical Ru .Gk /  Gk may not arise from a k-subgroup of G when k is imperfect. (Examples of this phenomenon will be given shortly.) Thus, for the maximal smooth connected unipotent normal k-subgroup Ru;k .G/  G (the k-unipotent radical), the quotient G=Ru;k .G/ may not be reductive when k is imperfect. A pseudo-reductive group over a field k is a smooth connected affine kgroup G such that Ru;k .G/ is trivial. For any smooth connected affine k-group G, the quotient G=Ru;k .G/ is pseudo-reductive. A pseudo-reductive k-group G that is perfect (i.e., G equals its derived group D.G/) is called pseudosemisimple. If k is perfect then pseudo-reductive k-groups are connected reductive k-groups by another name. For imperfect k the situation is completely different: Example 1.1.1. Weil restrictions G D Rk 0 =k .G 0 / for finite extensions k 0 =k and connected reductive k 0 -groups G 0 are pseudo-reductive [CGP, Prop. 1.1.10]. If G 0 is nontrivial and k 0 =k is not separable then such G are never reductive [CGP, Ex. 1.6.1]. A solvable pseudo-reductive group is necessarily commutative [CGP, Prop. 1.2.3], but the structure of commutative pseudo-reductive groups appears to be intractable (see [T]). The quotient of a pseudo-reductive k-group by a smooth connected normal k-subgroup or by a central closed k-subgroup scheme can fail to be pseudo-reductive, and a smooth connected normal k-subgroup of

2

Introduction

a pseudo-semisimple k-group can fail to be perfect; see [CGP, Ex. 1.3.5, 1.6.4] for such examples over any imperfect field k. A typical situation where the structure theory of pseudo-reductive groups is useful is in the study of smooth affine k-groups about which one has limited information but for which one wishes to prove a general theorem (e.g., cohomological finiteness); examples include the Zariski closure in GLn of a subgroup of GLn .k/, and the maximal smooth k-subgroup of a schematic stabilizer (as in local-global problems). For questions not amenable to study over k when k is imperfect, this structure theory makes possible what had previously seemed out of reach over such k: to reduce problems for general smooth affine k-groups to the reductive and commutative cases (over finite extensions of k). Such procedures are essential to prove finiteness results for degree-1 Tate-Shafarevich sets of arbitrary affine group schemes of finite type over global function fields, even in the general smooth affine case; see [C1, §1] for this and other applications. A detailed study of pseudo-reductive groups was initiated by Tits; he constructed several instructive examples and his ultimate goal was a classification. The general theory developed in [CGP] by characteristic-free methods includes the open cell, root systems, rational conjugacy theorems, the Bruhat decomposition for rational points, and a structure theory “modulo the commutative case” (summarized in [C1, §2] and [R]). The lack of a concrete description of commutative pseudo-reductive groups is not an obstacle in applications (see [C1]). In general, if G is a smooth connected affine k-group then Ru;k .G/K  Ru;K .GK / for any extension field K=k, and this inclusion is an equality when K is separable over k [CGP, Prop. 1.1.9] but generally not otherwise (e.g., equality fails with K D k for any imperfect k and non-reductive pseudo-reductive G). Taking K D ks shows that G is pseudo-reductive if and only if Gks is pseudoreductive (and also shows that if k is perfect then pseudo-reductive k-groups are precisely connected reductive k-groups). Hence, any smooth connected normal k-subgroup of a pseudo-reductive k-group is pseudo-reductive. Every smooth connected affine k-group G is generated by D.G/ and a single Cartan k-subgroup. Since D.G/ is pseudo-semisimple when G is pseudoreductive [CGP, Prop. 1.2.6], and Cartan k-subgroups of pseudo-reductive kgroups are commutative and pseudo-reductive, the main work in describing pseudo-reductive groups lies in the pseudo-semisimple case. A smooth affine k-group G is pseudo-simple (over k) if it is pseudo-semisimple, nontrivial, and has no nontrivial smooth connected proper normal k-subgroup; it is absolutely pseudo-simple if Gks is pseudo-simple. (See [CGP, Def. 3.1.1, Lemma 3.1.2] for equivalent formulations.) A pseudo-reductive k-group G is pseudo-split if it

1.1 Motivation

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contains a split maximal k-torus T , in which case any two such tori are conjugate by an element of G.k/ [CGP, Thm. C.2.3] Remark 1.1.2. If G is a pseudo-semisimple k-group then the set fGi g of its pseudo-simple normal k-subgroups is finite, the Gi ’s pairwise commute and generate G, and every perfect smooth connected normal k-subgroup of G is generated by the Gi ’s that it contains (see [CGP, Prop. 3.1.8]). The core of the study of pseudo-reductive groups G is the absolutely pseudo-simple case. Although [CGP] gives general structural foundations for the study and application of pseudo-reductive groups over any imperfect field k, there are natural topics not addressed in [CGP] whose development requires new ideas, such as: (i) Are there versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups? (ii) The standard construction (see §2.1) is exhaustive when p WD char.k/ ¤ 2; 3. Incorporating constructions resting on exceptional isogenies [CGP, Ch. 7–8] and birational group laws [CGP, §9.6–§9.8] gives an analogous result when p D 2; 3 provided that Œk W k 2  D 2 if p D 2; see [CGP, Thm. 10.2.1, Prop. 10.1.4]. More examples exist if p D 2 and Œk W k 2  > 2 (see §1.3); can we generalize the standard construction for such k? (iii) Is the automorphism functor of a pseudo-semisimple group representable? (Representability fails in the commutative pseudo-reductive case.) If so, what can be said about the structure of the identity component and component group of its maximal smooth closed subgroup Autsm G=k (thereby 0 defining a notion of “pseudo-inner” ks =k-form via .Autsm G=k / )? (iv) What can be said about existence and uniqueness of pseudo-split ks =kforms, and of quasi-split pseudo-inner ks =k-forms? (“Quasi-split” means the existence of a solvable pseudo-parabolic k-subgroup.) (v) Is there a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case? (Many ingredients in the semisimple case break down for pseudo-semisimple G; e.g., G may have no pseudo-split ks =k-form, and the quotient G=ZG of G modulo the 0 scheme-theoretic center ZG can be a proper k-subgroup of .Autsm G=k / .) The special challenges of characteristic 2 are reviewed in §1.3–§1.4 and §4.2. Recent work of Gabber on compactification theorems for arbitrary linear algebraic groups uses the structure theory of pseudo-reductive groups over general (imperfect) fields. That work encounters additional complications in characteristic 2 which are overcome via the description of pseudo-reductive groups as

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Introduction

central extensions of groups obtained by the “generalized standard” construction given in Chapter 9 of this monograph (see the Structure Theorem in §1.6).

1.2 Root systems and new results A maximal k-torus T in a pseudo-reductive k-group G is an almost direct product of the maximal central k-torus Z in G and the maximal k-torus T 0 WD T \ D.G/ in D.G/ [CGP, Lemma 1.2.5]. Suppose T is split, so the set ˆ WD ˆ.G; T / of nontrivial T -weights on Lie.G/ injects into X.T 0 / via restriction. The pair .ˆ; X.T 0 /Q / is always a root system (coinciding with ˆ.D.G/; T 0 / since G=D.G/ is commutative) [CGP, Thm. 2.3.10], and can be canonically enhanced to a root datum [CGP, §3.2]. In particular, to every pseudo-semisimple ks -group we may attach a Dynkin diagram. However, .ˆ; X.T 0 /Q / can be nonreduced when k is imperfect of characteristic 2 (the non-multipliable roots are the roots of the maximal geometric reductive quotient G red ). A pseudo-split k pseudo-semisimple group is (absolutely) pseudo-simple precisely when its root system is irreducible [CGP, Prop. 3.1.6]. This monograph builds on earlier work [CGP] via new techniques and constructions to answer the questions (i)–(v) raised in §1.1. In so doing, we also simplify the proofs of some results in [CGP]. (For instance, the standardness of all pseudo-reductive k-groups if char.k/ ¤ 2; 3 is recovered here by another method in Theorem 3.4.2.) Among the new results in this monograph are: (i) pseudo-reductive versions of the Existence, Isomorphism, and Isogeny Theorems (see Theorems 3.4.1, 6.1.1, and A.1.2), (ii) a structure theorem over arbitrary imperfect fields k (see §1.5–§1.6), (iii) existence of the automorphism scheme AutG=k for pseudo-semisimple G, and properties of the identity component and component group of its maximal smooth closed k-subgroup Autsm G=k (see Chapter 6), (iv) uniqueness and optimal existence results for pseudo-split and “quasi-split” ks =k-forms for imperfect k, including examples (in every positive characteristic) where existence fails (see §1.7), (v) a Tits-style classification of pseudo-semisimple k-groups G in terms of both the Dynkin diagram of Gks with -action of Gal.ks =k/ on it and the k-isomorphism class of the embedded anisotropic kernel (see §1.7). We illustrate (v) in Appendix D by using anisotropic quadratic forms over k to construct and classify absolutely pseudo-simple groups of type F4 with k-rank 2 (which never exist in the semisimple case).

1.3 Exotic groups and degenerate quadratic forms

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1.3 Exotic groups and degenerate quadratic forms If p D 2 and Œk W k 2  > 2 then there exist families of non-standard absolutely pseudo-simple k-groups of types Bn , Cn , and BCn (for every n > 1) with no analogue when Œk W k 2  D 2. Their existence is explained by a construction with certain degenerate quadratic spaces over k that exist only if Œk W k 2  > 2: Example 1.3.1. Let .V; q/ be a quadratic space over a field k with char.k/ D 2, d WD dim V > 3, and q ¤ 0. Let Bq W .v; w/ 7! q.v C w/ q.v/ q.w/ be the associated symmetric bilinear form and V ? the defect space consisting of v 2 V such that the linear form Bq .v;
/ on V vanishes. The restriction qjV ? is 2-linear (i.e., additive and q.cv/ D c 2 q.v/ for v 2 V , c 2 k) and dim.V =V ? / D 2n for some n > 0 since Bq induces a non-degenerate symplectic form on V =V ? . Assume 0 < dim V ? < dim V . Now q is non-degenerate (i.e., the projective hypersurface .q D 0/  P.V  / is k-smooth) if and only if dim V ? D 1, which is to say d D 2n C 1. It is well-known that in such cases SO.q/ is an absolutely simple group of type Bn with O.q/ D 2  SO.q/, so SO.q/ is the maximal smooth closed k-subgroup of O.q/ since char.k/ D 2. Assume also that .V; q/ is regular; i.e., ker.qjV ? / D 0. Regularity is preserved by any separable extension on k (Lemma 7.1.1). For such (possibly degenerate) q, define SO.q/ to be the maximal smooth closed k-subgroup of the k-group scheme O.q/; i.e., SO.q/ is the k-descent of the Zariski closure of O.q/.ks / in O.q/ks . In §7.1–§7.3 we prove: SO.q/ is absolutely pseudo-simple with root system Bn over ks where 2n D dim.V =V ? /, the dimension of a root group of SO.q/ks is 1 for long roots and dim V ? for short roots, and the minimal field of definition over k for the geometric unipotent radical of SO.q/ is the k-finite subextension K  k 1=2 generated over k by the square roots .q.v 0 /=q.v//1=2 for nonzero v; v 0 2 V ? . For any nonzero v0 2 V ? , the map v 7! .q.v/=q.v0 //1=2 is a k-linear injection of V ? into k 1=2 with image V containing 1 and generating K as a k-algebra. If we replace v0 with a nonzero v1 2 V ? then the associated k-subspace of K is V where  D .q.v0 /=q.v1 //1=2 2 K  . In particular, the case K ¤ k occurs if and only if dim V ? > 2, which is precisely when the regular q is degenerate, and always Œk W k 2  D Œk 1=2 W k > dim V ? . If V ? D K, as happens whenever Œk W k 2  D 2, then SO.q/ is the quotient of a “basic exotic” k-group [CGP, §7.2] modulo its center. The SO.q/’s with V ? ¤ K (so Œk W k 2  > 2) are a new class of absolutely pseudo-simple k-groups of type Bn (with trivial center); for n D 1 and isotropic q these are the type-A1 groups PHV ? ;K=k built in §3.1. In §7.2–§7.3 we show that every k-isomorphism SO.q 0 / ' SO.q/ arises from a conformal isometry q 0 ' q and use this to construct more absolutely

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Introduction

pseudo-simple k-groups of type B with trivial center via geometrically integral non-smooth quadrics in Severi–Brauer varieties associated to certain elements of order 2 in the Brauer group Br.k/. Remarkably, this accounts for all nonreductive pseudo-reductive groups whose Cartan subgroups are tori (see Proposition 7.3.7), and when combined with the exceptional isogeny Sp2n ! SO2nC1 in characteristic 2 via a fiber product construction it yields (in §8.2) new absolutely pseudo-simple groups of type Cn when n > 2 and Œk W k 2  > 2 (with short root groups over ks of dimension ŒK W k and long root groups over ks of dimension dim V ? ). A generalization in §8.3 gives even more such k-groups for n D 2 if Œk W k 2  > 8 (using that B2 D C2 ). In §1.5–§1.6 we provide a context for this zoo of constructions.

1.4 Tame central extensions A new ingredient in this monograph is a generalization of the “standard construction” (from §2.1) that is better-suited to the peculiar demands of characteristic 2. Before we address that, it is instructive to recall the principle underlying the ubiquity of standardness away from the case char.k/ D 2 with Œk W k 2  > 2, via splitting results for certain classes of central extensions. We now review the most basic instance of such splitting, to see why it breaks down completely (and hence new methods are required) when char.k/ D 2 with Œk W k 2  > 2 (see 1.4.2). 1.4.1. Let G be an absolutely pseudo-simple k-group with minimal field of ss definition K=k for Ru .Gk /  Gk , and let G 0 WD GK be the maximal semisimple e0 ! G 0 and  WD quotient of GK . For the simply connected central cover q W G ker q  ZG e0 , there is (as in [CGP, Def. 5.3.5]) a canonical k-homomorphism e0 /=RK=k ./ G W G ! D.RK=k .G 0 // D RK=k .G

(1.4.1.1)

induced by the natural map iG W G ! RK=k .G 0 /. The map G makes sense for any pseudo-reductive G but (as in [CGP]) it is of interest only for absolutely pseudo-simple G. By Proposition 2.3.4, ker G is central if char.k/ ¤ 2. The key to the proof that G is standard if char.k/ ¤ 2; 3 is the surjectivity of e0 / G for such k, as then (1.4.1.1) pulls back to a central extension E of RK=k .G by ker G . This central extension is split due to a general fact: if k 0 =k is an arbitrary finite extension of fields and G 0 a connected semisimple k 0 -group that is simply connected then for any commutative affine k-group scheme Z of finite type with no nontrivial smooth connected k-subgroup (e.g., Z D ker G as

1.4 Tame central extensions

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above), every central extension of k-group schemes 1 ! Z ! E ! Rk 0 =k .G 0 / ! 1

(1.4.1.2)

is (uniquely) split over k (see [CGP, Ex. 5.1.4]). In contrast, for many imperfect k and k-finite k 0  k 1=p for p D char.k/, the k-group Rk 0 =k .SLn / seems to admit non-split central extensions by Ga when n > 2 [CGP, Rem. 5.1.5].

1.4.2. For an absolutely pseudo-simple k-group G, two substantial difficulties arise if G is not surjective (so char.k/ D 2; 3) or if Gks has a non-reduced root system (which can occur only if the field k is imperfect and of characteristic 2): (i) Assume Gks has a reduced root system (so ker G is central in G, by Proposition 2.3.4) but that G is not surjective. The possibilities for G .G/ force us e0 of G ss and conto go beyond the simply connected semisimple central cover G K sider a wider class of absolutely pseudo-simple groups over finite extensions of k, called generalized basic exotic and basic exceptional, building on §1.3; see Chapter 8. (The maximal geometric semisimple quotient of these new groups is simply connected, and the basic exceptional case – which occurs over k if and only if char.k/ D 2 with Œk W k 2  > 8 – rests on the equality B2 D C2 .) (ii) Assume k is imperfect with char.k/ D 2. If Œk W k 2  D 2 then every pseudoQ reductive k-group uniquely has the form H  Hi where Hks has a reduced root system and each Hi is absolutely pseudo-simple over k with a non-reduced root system over ks [CGP, Prop. 10.1.4, Prop. 10.1.6] (and each Hi is pseudosplit, has trivial center, and Autk .Hi / D Hi .k/ [CGP, Thm. 9.9.3]). In contrast, when Œk W k 2  > 2 it is generally impossible to split off (as a direct factor) the contribution from non-reduced irreducible components of the root system over ks ; see Example 6.1.5. Moreover, as is explained in [CGP, §9.8–§9.9], the classification of pseudo-split absolutely pseudo-simple k-groups with a non-reduced root system over ks rests on invariants of linear algebraic nature that do not arise (in nontrivial ways) when Œk W k 2  D 2.

1.4.3. To classify absolutely pseudo-semisimple G over any k whatsoever, we shall use the following new construction. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. For example, if k 0 =k is an extension of finite degree and 0 is a k 0 group scheme of multiplicative type then Rk 0 =k .0 / is k-tame. If G is a perfect

8

Introduction

smooth connected affine group over a field k then a central extension 1!Z!E !G!1 with affine E of finite type is called k-tame if Z is k-tame. In Theorem 5.1.3 we show that for any such G, if K=k is the minimal field of definition for Ru .Gk /  Gk then the category of k-tame central extensions E of G that are smooth, connected, and perfect is equivalent to the category of connected semisimple ss central extensions of G 0 WD GK over K via E E 0 WD EK =Ru;K .EK /. e of G for which The perfect smooth connected k-tame central extension G 0 the associated connected semisimple central extension of G is simply connected is called the universal smooth k-tame central extension of G (it is initial among smooth k-tame central extensions of G). It is elementary that if G is pseudoe and that if G WD D.Rk 0 =k .G 0 // for a finite extension k 0 =k reductive then so is G e D Rk 0 =k .G e0 / for the simply and connected semisimple k 0 -group G 0 then G 0 0 e ! G . In proofs of general theorems it is often connected central cover G e (which has better properties), and by Theorem possible to pass from G to G e is described in terms of a known 9.2.1 (and Proposition 5.3.3) the k-group G e list of constructions when G is of “minimal type” in a sense discussed in §1.6.

1.5 Generalized standard groups In [CGP, Ch. 7–8], we constructed a class of pseudo-semisimple groups over any imperfect field k of characteristic p 2 f2; 3g by using certain non-standard absolutely pseudo-simple groups G 0 – called “basic exotic” – over finite extensions k 0 =k such that .G 0 0 /ss is simply connected and the irreducible root system k ˆ of Gk0 0 is reduced with an edge of multiplicity p: such ˆ can be F4 , Bn , or s Cn in characteristic 2 (with any n > 2) and G2 in characteristic 3. Letting K 0 =k 0 be the minimal field of definition for Ru .G 0 0 /  G 0 0 , we have k 0 ( K 0  k 0 1=p k k and over ks0 the long root groups are 1-dimensional whereas short root groups have dimension ŒK 0 W k 0  > 1 (short root groups are isomorphic to RKs0 =ks0 .Ga /). Going beyond these constructions, in [CGP, Ch. 9] pseudo-split absolutely pseudo-simple groups G 0 with root system BCn (for any n > 1) are constructed over any imperfect field k 0 with characteristic 2. If Œk 0 W k 0 2  D 2 then by [CGP, Thm. 9.9.3(1)] the k 0 -group G 0 is classified up to k 0 -isomorphism by the rank n > 1 and the minimal field of definition K 0 =k 0 for Ru .G 0 0 /  G 0 0 ; here, n can k k be arbitrary and K 0 =k 0 can be any nontrivial purely inseparable finite extension. For imperfect k of characteristic 2 or 3, with Œk W k 2  D 2 in the BCn -cases, Weil restrictions to k of groups G 0 as above over finite extensions k 0 =k are

1.5 Generalized standard groups

9

perfect and satisfy the splitting result for central extensions as in (1.4.1.2); see [CGP, Prop. 8.1.2, Thm. 9.9.3(3)]. (That splitting result fails in some BCn -cases with k 0 D k and n > 1 whenever Œk W k 2  > 2; see Examples B.4.1 and B.4.3.) If k is imperfect and either char.k/ D 3 or char.k/ D 2 with Œk W k 2  D 2 then the preceding constructions capture all deviations from standardness over k (see [CGP, Thm. 10.2.1]). However, over any field k of characteristic 2 with Œk W k 2  > 2 there exist many other pseudo-reductive groups, starting with: Example 1.5.1. Consider imperfect k of characteristic 2 and a pseudo-split absolutely pseudo-simple k-group G with a reduced root system such that G ss ' SL2 . k If Œk W k 2  D 2 then G ' RK=k .SL2 / for a purely inseparable finite extension K=k (see [CGP, Prop. 9.2.4]). In contrast, as we review in §3.1, if Œk W k 2  > 2 then many more possibilities for G occur: in addition to the field invariant K=k, there are linear algebra invariants (such as certain K  -homothety classes of nonzero kK 2 -subspaces V of K, with the case V ¤ K occurring if Œk W k 2  > 2). The groups in Example 1.5.1 can be used to “shrink” short root groups (for type B) or “fatten” long root groups (for type C) in pseudo-split basic exotic k-groups with rank n > 2. When Œk W k 2  > 2, this relates the new classes of absolutely pseudo-simple groups G mentioned in 1.4.2(i) to the basic exotic cases. For these additional constructions (and the derived groups of their Weil restrictions through finite extensions of the ground field) we prove a splitting result for central extensions as in (1.4.1.2) when Œk W k 2  6 8 (see Proposition B.3.4), but this splitting result fails whenever Œk W k 2  > 8 (see §B.1–§B.2). For any imperfect field k of characteristic 2, the data classifying pseudo-split absolutely pseudo-simple k-groups G with root system BCn (n > 1) is much more intricate when Œk W k 2  > 2 than when Œk W k 2  D 2, and one encounters new behavior when Œk W k 2  > 2 that never occurs when Œk W k 2  D 2. For instance, if K=k is the minimal field of definition for Ru .Gk /  Gk and Œk W k 2  > 2 then there can be proper subfields F  K over k such that the non-reductive maximal pseudo-reductive quotient GF =Ru;F .GF / of GF has a reduced root system (see [CGP, Ex. 9.1.8]); this never happens if Œk W k 2  D 2. Weil restrictions to k of generalized basic exotic groups, basic exceptional groups, and the constructions in [CGP, Ch. 9] with non-reduced root systems are used to define a generalized standard construction over any field k in Definitions 9.1.5 and 9.1.7. This construction satisfies many nice properties (see §9.1). The information required to make a “generalized standard presentation” of a pseudoreductive k-group G (if it admits such a presentation at all!) consists of data that is uniquely functorial with respect to isomorphisms in the pair .G; T / where T is a maximal k-torus of G (see Proposition 9.1.12); any T may be used.

10

Introduction

1.6 Minimal type and general structure theorem For a pseudo-reductive group G over any field k, the important notion of G being of minimal type was introduced in [CGP, Def. 9.4.4] and is reviewed in §2.3. Every pseudo-reductive k-group G admits a canonical pseudo-reductive central quotient G of minimal type with the same root datum as G (over ks ) [CGP, Prop. 9.4.2(iii)], and the central quotient G=ZG is always pseudo-reductive and of minimal type (see Proposition 4.1.3). Many of our results for general G rest on a classification and structure theorem for pseudo-split absolutely pseudo-simple groups of minimal type (over any field) given in Theorem 3.4.1 in the spirit of the Existence and Isomorphism Theorems for split connected semisimple groups. This classification in the pseudo-split minimal-type case supplements the root datum with additional field-theoretic data, as well as linear-algebraic data in characteristic 2. To prove theorems about general pseudo-reductive groups, it is often harmless and genuinely useful to pass to the minimal type case (e.g., see the proofs of Proposition 3.4.4, Proposition 6.1.4, Proposition B.3.1, and Theorem C.2.10). Generalized basic exotic and basic exceptional k-groups from 1.4.2(i) are of minimal type and admit an intrinsic characterization via this condition; see Theorem 8.4.5 (and Definition 8.4.1). However, a pseudo-reductive central quotient of a pseudo-reductive group of minimal type is generally not of minimal type; absolutely pseudo-simple counterexamples that are standard exist over every imperfect field (see Example 2.3.5). Hence, a general structure theorem for pseudo-reductive groups must go beyond the minimal type case. There is a weaker condition on a pseudo-reductive k-group G that we call locally of minimal type: for a maximal k-torus T  G, this is the property that for all non-divisible roots a of .Gks ; Tks /, the pseudo-simple ks -group .Gks /a of rank 1 generated by the ˙a-root groups admits a pseudo-simple central extension of minimal type. This notion might appear to be ad hoc, but it is not because it admits an elegant global characterization in the pseudo-semisimple case: such a k-group G is locally of minimal type if and only if its universal e is of minimal type (Proposition 5.3.3). In smooth k-tame central extension G particular, for every pseudo-semisimple G, the universal smooth k-tame central extension of G=ZG is always of minimal type; this is convenient in general proofs. By design, if G is pseudo-reductive and locally of minimal type then so is any pseudo-reductive central quotient of G. It is also easy to check that all generalized standard pseudo-reductive groups are locally of minimal type. Example 1.6.1. Rank-1 absolutely pseudo-simple ks -groups are classified in

1.7 Galois-twisted forms and Tits classification

11

Proposition 3.1.9 if char.k/ ¤ 2 and in [CGP, Prop. 9.2.4, Thm. 9.9.3(1)] if char.k/ D 2 with Œk W k 2  6 2. This classification implies that over such k every pseudo-reductive k-group is locally of minimal type. Example 1.6.1 is optimal in the k-aspect: if char.k/ D 2 with Œk W k 2  > 2 then for any n > 1 there are pseudo-split pseudo-simple k-groups with root system BCn that are not locally of minimal type (see §B.4), and likewise (see 4.2.2 and §B.1–§B.2) for pseudo-split pseudo-simple k-groups with root systems Bn and Cn for any n > 1 when Œk W k 2  > 8 (optimal by Proposition B.3.1); here B1 and C1 mean A1 . These examples suggest that there is no analogue of the “standard construction” beyond the locally minimal type class. Since “locally of minimal type” is more robust than “minimal type”, we seek to describe all pseudo-reductive groups locally of minimal type. One of our main results (Theorem 9.2.1) is a converse to the elementary fact that generalized standard pseudo-reductive groups (see §1.5) are locally of minimal type: Structure Theorem. A pseudo-reductive group locally of minimal type is generalized standard. In particular, if G is an arbitrary pseudo-reductive group then G=ZG is generalized standard. The novelty is that when char.k/ D 2 there is no restriction on Œk W k 2 . The cases char.k/ ¤ 2 or char.k/ D 2 with Œk W k 2  6 2 are part of [CGP, Thm. 10.2.1]; that earlier result is reproved in a new way in this monograph (using certain inputs from [CGP]) in the course of proving the above more general theorem.

1.7 Galois-twisted forms and Tits classification A ks =k-form of a pseudo-reductive group G over a field k is a pseudo-reductive k-group H such that Hks ' Gks . The theory of Chevalley groups ensures that if G is reductive then it admits a unique split ks =k-form. Uniqueness of pseudosplit ks =k-forms holds in the pseudo-reductive case (Proposition C.1.1). Existence of a pseudo-split ks =k-form seems to be intractable for commutative pseudo-reductive G, so now consider pseudo-semisimple G. Over many imperfect k (with arbitrary characteristic p > 0) there are pseudosemisimple G without a pseudo-split ks =k-form, due to a field-theoretic obstruction that cannot arise if G is absolutely pseudo-simple or if Œk W k p  D p; see Example C.1.2. Additional examples allowing Œk W k p  D p are given in Example C.1.6, but those are also not absolutely pseudo-simple. Existence result: For any absolutely pseudo-simple G, a pseudo-split ks =kform exists if char.k/ ¤ 2 and also if char.k/ D 2 with Œk W k 2  6 4 except

12

Introduction

possibly (for the latter case) if G is standard of type D2n with n > 2 and k admits a quadratic Galois extension (or a cubic Galois extension when n D 2); see Proposition C.1.3 and Corollary C.2.12. The same conclusion holds in the standard absolutely pseudo-simple case when char.k/ D 2 without restriction on Œk W k 2  (subject to the same exceptions for type D2n ). Avoidance of type D2n (n > 2) is necessary because for all n > 2 and imperfect k of characteristic 2 that admits a quadratic (or cubic when n D 2) Galois extension there exists a standard absolutely pseudo-simple k-group G of type D2n with no pseudo-split ks =k-form; see Proposition C.1.4 and Remark C.1.5. More counterexamples: What about the non-standard case if char.k/ D 2 and Œk W k 2  > 4? If Œk W k 2  > 4 and k has sufficiently rich Galois theory then in Example C.3.1 we make (non-standard) absolutely pseudo-simple groups of type A1 over k without a pseudo-split ks =k-form. These are used in §C.4 to make many more non-standard absolutely pseudo-simple k-groups without a pseudo-split ks =k-form: generalized basic exotic k-groups whose root system over ks is Bn or Cn for any n > 2, and absolutely pseudo-simple k-groups of minimal type whose root system over ks is BCn for any n > 1. Going beyond the study of pseudo-split ks =k-forms, it is natural to seek a pseudo-reductive analogue of the existence and uniqueness of quasi-split inner forms for connected reductive groups. Recall that for connected reductive G the notion of inner form involves Galois-twisting against the action of the identity component G=ZG of the automorphism scheme of G. Due to the mysterious nature of commutative pseudo-reductive groups, for an analogous result in the pseudo-reductive case we shall restrict attention to pseudo-semisimple G. The analogue of “inner form” for pseudo-semisimple groups G is not defined via the action of G=ZG , but rather via the identity component of the maximal smooth closed k-subgroup of the automorphism scheme of G. To be precise, in §6.2 we prove for pseudo-semisimple G that the automorphism functor of G on the category of k-algebras is represented by an affine k-group scheme AutG=k of finite type (this functor is often not representable for commutative G; see Example 6.2.1). In general AutG=k is not k-smooth (Example 6.2.3), but its maximal smooth closed k-subgroup Autsm G=k has structure analogous to 0 the semisimple case (see Propositions 6.2.4 and 6.3.10): .Autsm G=k / is pseudoreductive and its derived group is G=ZG . (Absolutely pseudo-simple G with 0 G=ZG ¤ .Autsm G=k / arise over every imperfect field; see Remark 6.2.5.) Inspired by the semisimple case, for pseudo-semisimple G we prove that 0 .Autsm G=k /.ks / is a subgroup of the automorphism group of the based root datum over ks (Remark 6.3.6). For absolutely pseudo-simple G we show that

1.8 Background, notation, and acknowledgments

13

this subgroup inclusion is often an equality. Counterexamples to equality in the absolutely pseudo-simple case exist precisely for type D2n (n > 2) with k imperfect of characteristic 2 (for the same reason that such cases may not have pseudo-split ks =k-form); see Proposition 6.3.10. In §6.3 we use our study of the structure of Autsm G=k (including its behavior under passage to pseudo-reductive central quotients of G) to prove a Tits-style classification theorem in the general pseudo-semisimple case (no minimal-type hypothesis!), recovering the well-known result due to Tits in the semisimple case. As an illustration of the method, in Appendix D we show that if k is imperfect of characteristic 2 then absolutely pseudo-simple k-groups of type F4 that are not pseudo-split cannot have k-rank 3 whereas they can have k-rank 2 (in contrast with the semisimple case!); all instances of the latter are described via anisotropic quadratic forms over k (with examples given over specific k). In §C.2 the notion of pseudo-inner form of a pseudo-reductive k-group G sm 0 0 is defined in terms of .Autsm D.G/=k / . We use the structure of .AutD.G/=k / to prove uniqueness of pseudo-inner ks =k-forms that are quasi-split (i.e., admit a solvable pseudo-parabolic k-subgroup). The existence of such ks =k-forms is proved assuming when char.k/ D 2 that Œk W k 2  6 4 or G is standard (Theorem C.2.10). This is optimal because if char.k/ D 2, Œk W k 2  > 4, and k has sufficiently rich Galois theory (more precisely, k admits a quadratic Galois extension k 0 such that ker.Br.k/ ! Br.k 0 // ¤ 1) then for every n > 1 there exist non-standard absolutely pseudo-simple k-groups of types Bn , Cn , and BCn over ks without a quasi-split ks =k-form: examples without a pseudo-split ks =k-form (see §C.3–§C.4) do the job, by Lemma C.2.2.

1.8 Background, notation, and acknowledgments In this monograph we use many constructions and results from [CGP]. Familiarity with Chapters 1–5, §7.1–§7.2, §8.1, Chapter 9, parts of Appendix A (A.5, A.7, A.8), Theorem B.3.4, and Appendix C.2 (especially Theorem C.2.29) of [CGP] is sufficient for understanding our main techniques. Chapter 9 in the first edition of [CGP] has been completely rewritten in the second edition, incorporating significant improvements that are used throughout this monograph, and some results outside Chapter 9 of [CGP] are improved in the second edition and used here. We provide many cross-references to aid the reader. (All numerical labeling in the first edition of [CGP] is unchanged in the second edition except in Chapter 9, apart from an equation label in [CGP, Ex. 1.6.4].) For a scheme X of finite type over a field k and a closed subscheme Z of

14

Introduction

XK for an extension field K=k, the intersection of all subfields k 0  K over k such that Z descends (necessarily uniquely) to a closed subscheme of Xk 0 is also such a subfield, called the minimal field of definition for Z  XK relative to k; see [EGA, IV2 , §4.8] for a detailed discussion of the existence of such a field. The behavior of K=k with respect to extension of k is addressed in [CGP, Lemma 1.1.8]. (In [CGP] the phrase “field of definition” is understood to require minimality, but in this monograph we keep “minimality” in the terminology.) For an automorphism  of a field k and a k-scheme X, X denotes the kscheme k ˝;k X. For a map f W X ! Y of k-schemes, f denotes the induced map X ! Y over k. The maximal smooth closed k-subgroup of a k-group scheme H of finite type is denoted H sm (see [CGP, Rem. C.4.2]). For a smooth connected affine k-group G and closed k-subgroup scheme H  G that is either smooth or of multiplicative type, the scheme-theoretic centralizer ZG .H / for the H -action on G via conjugation is defined as in [CGP, A.1.9ff., A.8.10]; the scheme-theoretic center is ZG WD ZG .G/ [CGP, A.1.10]. The k-unipotent radical Ru;k .G/ is the maximal unipotent smooth connected normal k-subgroup of G. The k-radical Rk .G/ is the maximal solvable smooth connected normal k-subgroup of G. If K=k denotes the minimal field of definition for Ru .Gk /  Gk then we dered fine GK to be the quotient GK =Ru;K .GK / that is a K-descent of the maximal reductive quotient G red of Gk . Taking K=k instead to be the minimal field of k ss definition for R.Gk /  Gk yields the quotient GK WD GK =RK .GK / of GK as a K-descent of the maximal semisimple quotient of Gk . A Levi k-subgroup of G is a smooth closed k-subgroup L  G such that Lk ! G red is an isomorphism. k The Weyl group of a root system ˆ is is denoted W .ˆ/. If ˆ is irreducible and not simply laced then for any basis  of ˆ we denote by > and < the respective subsets of longer and shorter roots in . We thank Ofer Gabber for his very illuminating advice and suggestions, K˛esˇ tutis Cesnaviˇ cius and the referees for helpful comments, and Stella Gastineau for typesetting assistance. We thank Indu Prasad for her encouragement, hospitality, and support over the years. G.P. thanks the Institute for Advanced Study (Princeton), and his host Peter Sarnak, for hospitality and support during 2012–13. He also thanks the Mathematics Research Center at Stanford University for support during a visit in the summer of 2013 and RIMS (Kyoto) for its hospitality during July 2014. B.C. is grateful to IAS for its hospitality during several visits. We thank Vickie Kearn, Betsy Blumenthal, Nathan Carr, and Glenda Krupa for their editorial work. B.C. was supported by NSF grant DMS-1100784 and G.P. was supported by NSF grants DMS-1001748 and DMS-1401380.

2 Preliminary notions

2.1 Standard groups, Levi subgroups, and root systems 2.1.1. Let k be a field. The “standard construction” of pseudo-reductive kgroups, as reviewed below, provides both a general structure theorem for pseudoreductive k-groups when char.k/ ¤ 2 as well as a guide for the main results we shall prove in this monograph. As motivation, consider a smooth connected affine k-group G with minimal field of definition K=k for its geometric unipotent radical. The extension K=k is purely inseparable of finite degree and Ru .Gk / descends to the maximal smooth connected unipotent normal K-subgroup Ru;K .GK /  GK . Consider the maximal reductive quotient G 0 WD GK =Ru;K .GK / over K. The natural map iG W G ! RK=k .G 0 /

(2.1.1)

is a first attempt to relate G to a canonically associated Weil restriction of a connected reductive group. The image has good properties, given in (ii) below: Proposition 2.1.2. Let G, K=k, and G 0 be as above with G pseudo-reductive. (i) Let L  G be a Levi k-subgroup. Any smooth connected k-subgroup H  G containing L is pseudo-reductive, L is a Levi k-subgroup of H , and Hk \ Ru .Gk / D Ru .Hk /. (ii) The image G WD iG .G/  RK=k .G 0 / is pseudo-reductive, K=k is the minimal field of definition for its geometric unipotent radical, and the inclusion of G into RK=k .G 0 / is iG . Part (i) is [CGP, Lemma 7.2.4], and we give its proof for the convenience of the reader. If G is pseudo-split then L as in (i) exists by [CGP, Thm. 3.4.6].

16

Preliminary notions

Proof. We first prove (i). Since Lk ! Gk =Ru .Gk / is an isomorphism, Hk D Lk n .Hk \ Ru .Gk // as schemes. Thus, the smoothness and connectedness of Hk imply that the unipotent normal subgroup scheme Hk \ Ru .Gk / in Hk is smooth and connected, with quotient by this isomorphic to Lk . Hence, Ru .Hk / D Hk \ Ru .Gk /, so Ru;k .H /  Ru;k .G/ D 1 and hence H is pseudo-reductive. This proves (i). Since an equality among purely inseparable extensions of k can be checked after scalar extension to ks , to prove (ii) we may and do assume k D ks . Now G is pseudo-split, so by [CGP, Thm. 3.4.6] it contains a Levi k-subgroup L. By definition of a Levi subgroup, LK ! G 0 is an isomorphism. The natural map RK=k .G 0 /K ! G 0 is a K-descent of the maximal geometric reductive quotient (by [CGP, Prop. A.5.11(1),(2)]), so the homomorphism L ! RK=k .G 0 / induced by iG is a Levi k-subgroup inclusion. Thus, we may apply (i) to the ambient group RK=k .G 0 / and its k-subgroup G D iG .G/ to deduce that G is pseudoreductive with geometric unipotent radical defined over K and with L as a Levi k-subgroup, so the evident map GK ! G 0 is a K-descent of the maximal geometric reductive quotient. Hence, the minimal field of definition F=k for the geometric unipotent radical of G is a subextension of K=k and it remains to show that the inclusion F  K over k is an equality. By the definition of K=k, it is sufficient to prove that the geometric unipotent radical of G is defined over F . That is, it suffices to construct a smooth connected unipotent normal F -subgroup U of GF such that GF =U is reductive. Consider the composite F -homomorphism red

q W GF  G F  G F : It is sufficient to prove that the kernel of this map is smooth, connected, and unipotent. For this purpose we may extend scalars to K, but the K-descent of the maximal geometric reductive quotient of G has been identified with the map GK ! G 0 corresponding to the given inclusion j W G ,! RK=k .G 0 /. Hence, qK is the map GK ! G 0 corresponding to the composition of G  G with j , which is to say that qK corresponds to iG W G ! RK=k .G 0 /. By the definition of iG in terms of the universal property of Weil restriction, this implies that qK is the canonical quotient map GK ! G 0 whose kernel is a K-descent of Ru .Gk /.  For pseudo-reductive G, the map iG is problematic for two reasons: (i) if G is non-commutative (hence non-solvable) then the field K is a coarser

17

2.1 Standard groups, Levi subgroups, and root systems

invariant than the collection of minimal fields of definition Kj =k for the kernels of projections of Gk onto the simple factors Hj of the adjoint central quotient G ad ¤ 1 of the maximal reductive quotient G red , k k (ii) iG might have nontrivial kernel. The following construction bypasses both of these problems. Let k 0 be a nonzero finite reduced k-algebra. Let G 0 ! Spec.k 0 / be a smooth Q affine group scheme with connected reductive fibers. Write k 0 D ki0 for fields Q ki0 and let Gi0 denote the ki0 -fiber of G 0 , so Rk 0 =k .G 0 / D Rki0 =k .Gi0 /. Let T 0 0

be a maximal k 0 -torus in G 0 , and define T WD T 0 =ZG 0 to be the associated 0 maximal k 0 -torus in G WD G 0 =ZG 0 , where ZG 0 denotes the scheme-theoretic center of G 0 . Suppose there is given a commutative pseudo-reductive k-group C and a factorization in k-homomorphisms 

0

Rk 0 =k .T 0 / ! C ! Rk 0 =k .T /

(2.1.2.1) 0

of the Weil restriction to k of the canonical quotient map q W T 0 ! T over k 0 . (Beware that Rk 0 =k .q/ may not be surjective when k 0 is not k-étale, and we do not require  to be surjective.) 0 0 The natural G -action on G 0 over k 0 defines a natural Rk 0 =k .G /-action on Rk 0 =k .G 0 / over k, and hence a natural action of C on Rk 0 =k .G 0 / via composition 0 0 with C ! Rk 0 =k .T / ! Rk 0 =k .G /. The anti-diagonal map Rk 0 =k .T 0 / ! Rk 0 =k .G 0 / o C is an inclusion with central image, and the associated central quotient G D .Rk 0 =k .G 0 / o C /=Rk 0 =k .T 0 /

(2.1.2.2)

is always pseudo-reductive [CGP, Prop. 1.4.3]. Informally, G is obtained from Rk 0 =k .G 0 / by replacing the Cartan k-subgroup Rk 0 =k .T 0 / with the commutative pseudo-reductive k-group C (whose structure we treat as a black box); G is a pushout of Rk 0 =k .G 0 / along . Note that C is a Cartan k-subgroup of G, since Rk 0 =k .T 0 /  C is a Cartan k-subgroup of Rk 0 =k .G 0 / o C . Definition 2.1.3. A pseudo-reductive k-group is standard if it is k-isomorphic to a k-group as on the right side of (2.1.2.2). Every commutative pseudo-reductive k-group G is standard, by using k 0 D k, T 0 D G 0 D 1, and C D G. By [CGP, Thm. 4.1.1], any non-commutative standard pseudo-reductive k-group G admits a standard presentation: a description

18

Preliminary notions

as in (2.1.2.2) using a 4-tuple .G 0 ; k 0 =k; T 0 ; C / as above (including a specified factorization (2.1.2.1) that we suppress from the notation) for which the fibers of G 0 ! Spec.k 0 / are semisimple, absolutely simple, and simply connected. If G is a non-commutative standard pseudo-reductive group and the map j W Rk 0 =k .G 0 / ! G arises from a standard presentation of G then the triple .G 0 ; k 0 =k; j / is uniquely determined by G up to unique isomorphism [CGP, Prop. 4.2.4] and there is a natural bijection between the sets of maximal k 0 tori T 0 of G 0 and Cartan k-subgroups C of G via the relation j.Rk 0 =k .T 0 //  C [CGP, Prop. 4.1.4(2),(3)]. In this sense, a standard presentation of a noncommutative standard pseudo-reductive k-group G amounts to a choice of a Cartan k-subgroup of G (or equivalently, a choice of maximal k-torus of G). Let G be a pseudo-split pseudo-reductive k-group and T a split maximal k-torus in G, so S WD T \ D.G/ is a split maximal k-torus of D.G/ and T is an almost direct product of S and the maximal central k-torus Z of G [CGP, Lemma 1.2.5]. Consider the T -action and S -action on Lie.G/. As is explained in the proof of [CGP, Thm. 2.3.10], the weight spaces in Lie.G/ for nontrivial T -weights are supported inside Lie.D.G// and coincide with the weight spaces for nontrivial S -weights. These weight spaces can have very high dimension, but nonetheless the pair .X.T /; ˆ.G; T // can be naturally enhanced to a root datum R.G; T / such that the Q-span of ˆ.G; T / maps isomorphically onto the quotient X.S /Q of X.T /Q [CGP, §3.2]. The root system ˆ.G; T / is reduced whenever char.k/ ¤ 2 and also when char.k/ D 2 provided that G red has no connected semisimple normal subgroup k that is simple and simply connected of type C (where C1 D A1 ). In general the set of non-multipliable roots in ˆ.G; T / is equal to ˆ.G red ; Tk / and this maps k isomorphically onto a reduced root system in X.S/Q [CGP, Thm. 2.3.10]. Lemma 2.1.4. Let H  H 0 be an inclusion of pseudo-reductive groups over a field k, and assume that their maximal tori have the same dimension. Let T be a maximal k-torus of H , and assume that ˆ.Hks ; Tks / D ˆ.Hk0 s ; Tks /. A connected reductive k-subgroup L  H is a Levi k-subgroup of H if and only if it is a Levi k-subgroup of H 0 . Proof. We may and do assume k D ks . By Proposition 2.1.2(i), if L is a Levi k-subgroup of H 0 then it is a Levi k-subgroup of H (without needing to assume the equality of root systems, which could fail when ˆ.H 0 ; T / is non-reduced). Assume instead that L is a Levi k-subgroup of H , and let ˆ denote the common root system for .H; T / and .H 0 ; T /. Fix a positive system of roots ˆC and let ˆ0 denote the set of non-multipliable roots in ˆ, so ˆ0 D ˆ.L; T /. Then ˆ0 C WD ˆC \ ˆ0 is a positive system of roots in ˆ0 ; let 0 be the basis of

2.2 The basic exotic construction

19

simple roots in ˆ0 C . By [CGP, Thm. 3.4.6], Levi k-subgroups of H 0 containing T are uniquely determined by their root groups Ea0 for a 2 0 , and each Ea0 may be chosen arbitrarily among the 1-dimensional T -stable smooth connected k-subgroups of the a-root group of .H 0 ; T / (defined as in [CGP, Def. 2.3.13]: its Lie algebra is the a-weight space since a is not multipliable). Hence, there is a unique Levi k-subgroup L0 of H 0 containing T such that its a-root group Ea0 coincides with the a-root group Ea of .L; T / for all a 2 0 . Our task is to prove L0 D L as k-subgroups of H 0 . It suffices to prove E 0 a D E a for all a 2 0 because any connected reductive group equipped with a chosen split maximal torus is generated by that maximal torus and its root groups for the simple positive and negative roots relative to a choice of positive system of roots in the root system. Let U 0 a be the a-root group for .H 0 ; T /, and let U a be the a-root group for .H; T /. By the dynamic construction of such root groups, U 0 a \ H D U a . Choose a nontrivial element x 2 Ea .k/ D Ea0 .k/. By [CGP, Prop. 3.4.2], there are unique elements u0 ; v 0 2 U 0 a .k/ f1g such that u0 xv 0 2 NH 0 .T /.k/ and unique u; v 2 U a .k/ f1g such that uxv 2 NH .T /.k/. From the uniqueness we conclude that u0 D u and v 0 D v, so u0 ; v 0 2 U a .k/ f1g. The same reasoning applies to .L; T / and .L0 ; T /, so in fact u0 ; v 0 2 E 0 a .k/ \ E a .k/. The T -orbits of these nontrivial elements under conjugation exhaust E 0 a f1g and E a f1g, so E 0 a D E a as desired. 

2.2 The basic exotic construction The only nontrivial multiplicities that occur for the edges of Dynkin diagrams of reduced irreducible root systems are 2 and 3. This underlies the existence of exceptional isogenies that only arise in characteristics 2 and 3, and such isogenies are used to build non-standard absolutely pseudo-simple groups called “basic exotic” (see Definition 2.2.2). One of our main tasks (see Chapters 7–8) is to generalize the “basic exotic” construction so that we can prove a structure theorem over any imperfect k of characteristic 2, without restriction on Œk W k 2 . 2.2.1. Let k be an imperfect field with p WD char.k/ 2 f2; 3g, and let k 0 =k be a nontrivial finite extension satisfying k 0 p  k. Let G 0 be a connected semisimple k 0 -group that is absolutely simple and simply connected with an edge of multiplicity p in its Dynkin diagram (so G 0 has type G2 if p D 3 and type F4 or Bn or Cn with n > 2 if p D 2). By [CGP, Lemma 7.1.2], there is a unique minimal non-central normal k 0 -subgroup scheme N 0  G 0 whose relative Frobenius mor-

20

Preliminary notions 0

phism is trivial; we call  0 W G 0 ! G WD G 0 =N 0 the very special isogeny of G 0 . 0

Let T 0  G 0 be a maximal k-torus, and T D .T 0 /. By [CGP, Prop. 7.1.5],  0 carries long root groups for .G 0 ; T 0 / isomorphically onto short root groups 0 0 for .G ; T / and carries short root groups for .G 0 ; T 0 / onto long root groups 0 0 0 of .G ; T / via a Frobenius isogeny. Moreover, G is simply connected with root system dual to that of G 0 , and the factorization of the Frobenius isogeny 0 FG 0 =k 0 W G 0 ! G 0 .p/ through  0 is via an isogeny  0 W G ! G 0 .p/ that is the very 0 special isogeny of G . For types F4 in characteristic 2 and G2 in characteristic 3, this provides the unique nontrivial factorization of FG 0 =k 0 [CGP, Lemma 7.1.2]. 0 Consider the Weil restriction f D Rk 0 =k . 0 / W Rk 0 =k .G 0 / ! Rk 0 =k .G /; this is not surjective. By definition of Levi k-subgroups, for any Levi k-subgroup | W 0 0 G ,! Rk 0 =k .G / (if one exists) the associated map G k 0 ! G is an isomorphism. 0 (For a link between such Levi k-subgroups and k-descents of G , see [CGP, Lemma 7.2.1].) By [CGP, Prop. 7.3.1] the following are equivalent: (i) G lies inside the image of f , (ii) f 1 .G/ is smooth, (ii) f 1 .G/ks contains a Levi ks -subgroup of Rk 0 =k .G 0 /ks . When these equivalent conditions hold, by [CGP, 7.2.6–7.2.7] the fiber product G WD f 1 .G/ in the diagram G

j





G

/ Rk 0 =k .G 0 /

|

f

(2.2.1)

/ Rk 0 =k .G 0 /

is absolutely pseudo-simple, k 0 =k is the minimal field of definition for its geometric radical, and the given inclusion G ,! Rk 0 =k .G 0 / corresponds to the maximal reductive quotient Gk 0  G 0 . Also, G is not standard [CGP, Prop. 8.1.1]. Definition 2.2.2. A pseudo-reductive k-group G as in (2.2.1) is basic exotic. By [CGP, Prop. 7.2.7(3)], if E=k is a separable extension of fields then a pseudo-reductive k-group H is basic exotic if and only if HE is basic exotic. Q If K is a nonzero finite reduced k-algebra, so K D Ki for finite extension ` fields Ki =k, and if Gi is a basic exotic Ki -group for each i , then for G WD Gi Q over Spec.K/ the pseudo-reductive k-group RK=k .G / D RKi =k .Gi / is perfect by [CGP, Prop. 8.1.2]. This k-group uniquely determines the pair .G ; K=k/ up

2.3 Minimal type

21

to unique k-isomorphism in the following sense: if .H ; L=k/ is another such pair then any k-isomorphism RK=k .G / ' RL=k .H / arises from a uniquely determined pair .; '/ consisting of a k-algebra isomorphism  W K ' L and group isomorphism ' W G ' H over  . The existence and uniqueness of .; '/ rests on intrinsically characterizing .G ; K=k/ in terms of the k-group RK=k .G /; this is carried out as part of [CGP, Prop. 8.2.4], but an alternative approach is given in Corollary 3.3.9 as an application of ideas developed in §2.3 and §3.3. Definition 2.2.3. A k-group of the form RK=k .G / for a pair .G ; K=k/ as above is called an exotic pseudo-reductive k-group.

2.3 Minimal type 2.3.1. Let G be a pseudo-reductive group over a field k, and recall the map iG as in (2.1.1). It is not clear how to describe the possibilities for ker iG in general; the most favorable situation is when .ker iG /T D 1 for a maximal k-torus T of G, as then we see using [CGP, Prop. 2.1.12(2)] that ker iG is a connected group scheme and we can try to understand its structure in terms of the intersection of .ker iG /ks with root groups of .Gks ; Tks /. Since .ker iG /T D ker iG \ ZG .T /, we are led to consider the intersection of the unipotent k-group scheme ker iG with a Cartan k-subgroup C D ZG .T / of G. The intersection .ker iG / \ C has remarkable properties (proved in [CGP, Prop. 9.4.2, Cor. 9.4.3]): it is independent of C (so we denote it as CG ), it is central in G, the central quotient G=CG is pseudo-reductive with the same root datum as G over ks and the same minimal field of definition over k for its geometric unipotent radical, and iG is the composition of the central quotient map G  G=CG and iG=CG . Thus, .ker iG /=CG D ker iG=CG , so CG=CG D 1. The kgroup CG has no nontrivial smooth connected k-subgroup since ker iG contains no such k-subgroup (as .ker iG /k  Ru .Gk / and G is pseudo-reductive). Definition 2.3.2. A pseudo-reductive k-group G is of minimal type if CG is trivial. Example 2.3.3. For any basic exotic pseudo-simple group G over an imperfect field of characteristic 2 or 3, ker iG is trivial by [CGP, Prop. 7.2.7(1),(2)] and so such G are of minimal type. Proposition 2.3.4. Let G be a pseudo-reductive k-group whose root system over ks is reduced. Let K be the minimal field of definition over k for the geometric unipotent radical Ru .Gk / of G, and let G 0 D GK =Ru;K .GK /. The kernel of the homomorphism iG W G ! RK=k .G 0 / is central .so ker iG is trivial if G is

22

Preliminary notions

of minimal type/ and it does not contain any nontrivial smooth connected ksubgroup. If G 0 is semisimple and simply connected and iG is surjective then iG is an isomorphism, so in such cases G is of minimal type and standard. The reducedness hypothesis on the root system cannot be dropped: if k is imperfect of characteristic 2 then there exist absolutely pseudo-simple k-groups G of minimal type whose root system over ks is non-reduced, and for any such G the kernel of iG is connected and commutative but not central (and such G exist for which iG is surjective); see [CGP, Thm. 9.4.7, Thm. 9.8.1(4)]. Proof. To prove the proposition we may and do replace k with ks so that k is separably closed. Let T be a maximal k-torus in G and let ˆ WD ˆ.G; T / be the root system of G with respect to T . We view T 0 WD TK as a maximal K-torus of G 0 . As ˆ has been assumed to be reduced, ˆ.G 0 ; T 0 / D ˆ. Let  W GL1 ! T be a 1-parameter k-subgroup such that ha; i ¤ 0 for every a 2 ˆ, so ZG ./ D ZG .T /. For a 2 ˆ, let Ua be the corresponding root group in G. According to [CGP, Prop. 2.3.11], Ua is a k-vector group admitting a unique linear structure with respect to which T acts linearly, and Lie.Ua / is the a-root space for T in Lie.G/. Since a 2 ˆ.G 0 ; T 0 /, the restriction of iG to Ua is a T -equivariant homomorphism between a-root groups. The 1-dimensional aK root group of G 0 has a unique K-linear structure with respect to which T 0 acts linearly, and the map induced by iG between the a-root groups respects the linear structures since GL1 -compatibility follows from the nontriviality of a. Thus, ker.iG jUa / is also a vector group, so it is a smooth connected k-subgroup of G whose geometric fiber lies in the unipotent radical. By pseudo-reductivity of G it follows that ker.iG jUa / D 1, so the a-weight space in Lie.ker iG / vanishes. We conclude that the k-subgroups Uker iG .˙/ .D UG .˙/ \ ker iG / are trivial. Now using [CGP, Prop. 2.1.12 (2)] for H WD .ker iG /k contained in the connected unipotent group Ru .Gk /, it follows that ker iG D Zker iG ./ D ker iG \ ZG ./  ZG .T /. As T was an arbitrary maximal k-torus of G and the Cartan subgroup C D ZG .T / is commutative, we infer that ker iG commutes with C and with the k-subgroup G t of G generated by the k-tori of G. Since D.G/ is perfect, it is contained in G t and thus G D C
G t . Hence, ker iG is central. Since .ker iG /k  Ru .Gk /, by applying [CGP, Lemma 1.2.1] we see that ker iG does not contain any nontrivial smooth connected k-subgroup. Now if G 0 is semisimple and simply connected and iG W G ! RK=k .G 0 / is surjective then 1 ! ker iG ! G ! RK=k .G 0 / ! 1

2.3 Minimal type

23

is a central extension of RK=k .G 0 / by the k-group scheme ker iG which does not contain a nontrivial smooth connected k-subgroup. By [CGP, Prop. 5.1.3, Ex. 5.1.4] such a central extension must be trivial. But G is smooth and connected, so it follows that ker iG D 1 and hence iG W G ! RK=k .G 0 / is an isomorphism. Thus, G is of minimal type.  As noted above, if G is a pseudo-reductive k-group then CG=CG D 1. Thus, the pseudo-reductive k-group G=CG is of minimal type. If the common root system of Gks and .G=CG /ks is reduced then ker iG D CG and ker iG=CG D 1 by Proposition 2.3.4 (applied to G=CG ), so in such cases the image iG .G/ D G=CG (which is pseudo-reductive with the same minimal field of definition over k for its geometric unipotent radical as G, by Proposition 2.1.2(ii)) is of minimal type. Over any imperfect field k of characteristic p > 0 there exist standard absolutely pseudo-simple groups not of minimal type: 2

Example 2.3.5. Let k be imperfect of characteristic p > 0, and let k 0 D k.a1=p / and k D k.a1=p / D k 0 \ k 1=p with a 2 k k p . Consider the k-group G D Rk 0 =k .SLp /=Rk=k .p /

(2.3.5)

of rank p 1 (so rank 1 when p D 2). By [CGP, Ex. 5.3.7] G is a standard pseudo-simple k-group and dim ker iG D .p 1/2 > 0. Nontriviality of ker iG implies that G is not of minimal type since its root system is reduced. For pseudo-reductive k-groups G1 and G2 it is clear that CG1  CG2 D CG1 G2 inside G1  G2 , so G1  G2 is of minimal type if and only if G1 and G2 are of minimal type. As the following lemma shows, the formation of CG (and hence of the minimal type property) behaves well with respect to separable extension of the ground field (such as scalar extension to ks , or from a global function field to its completion at some place): Lemma 2.3.6. Let G be a pseudo-reductive group over a field k. For any separable extension k 0 =k, .CG /k 0 D CGk0 inside Gk 0 . In particular, G is of minimal type if and only if Gk 0 is of minimal type. Proof. Let K=k be the minimal field of definition for the geometric unipotent radical of G. Clearly K is purely inseparable of finite degree over k and K 0 WD k 0 ˝k K is a field separable over K. By [CGP, Prop. 1.1.9(2)], K 0 =k 0 is the minimal field of definition for the geometric unipotent radical of Gk 0 . red red Since .GK / ˝K K 0 D .Gk 0 /K 0 , it follows that .iG /k 0 D iGk 0 . For any Cartan k-subgroup C of G, Ck 0 is a Cartan k 0 -subgroup of Gk 0 and .CG /k 0 D .ker iG \ C /k 0 D ker iGk0 \ Ck 0 D CGk0 . 

24

Preliminary notions

The following proposition provides an alternative description of the central unipotent k-subgroup scheme CG . Proposition 2.3.7. Let G be a pseudo-reductive k-group. The central unipotent k-subgroup scheme CG of G contains every central unipotent k-subgroup scheme of G. Proof. Let G D G=CG . Recall that G is a pseudo-reductive k-group of minimal type. It suffices to show that any central unipotent k-subgroup scheme of G (such as the image of a central unipotent k-subgroup scheme of G) is trivial. Let T be a maximal k-torus of G and let U be a central unipotent ksubgroup scheme of G , so U is contained in the Cartan subgroup ZG .T / of G . Let K=k be the minimal field of definition for the geometric unipotent radical of G , and define G 0 D GKred . As G is of minimal type, the restriction of iG W G ! RK=k .G 0 / to the Cartan subgroup ZG .T / has trivial kernel. If T 0 is the image of TK in G 0 then iG carries the Cartan k-subgroup ZG .T /, and hence also U , isomorphically onto its image in the Cartan k-subgroup RK=k .T 0 / of RK=k .G 0 /. Thus, it suffices to show that every unipotent k-subgroup scheme U of RK=k .T 0 / is trivial. The inclusion j W U ,! RK=k .T 0 / corresponds to a Khomomorphism UK ! T 0 , and this latter homomorphism is trivial since UK is unipotent and T 0 is a K-torus. Hence, j is trivial, so U is trivial as desired.  Remark 2.3.8. Let f W G ! G be a quotient homomorphism between pseudoreductive groups. The preceding proposition implies that f .CG /  CG . Hence, G=CG uniquely dominates every minimal type pseudo-reductive quotient of G (i.e., it is the unique maximal such quotient) and f induces a homomorphism between the maximal pseudo-reductive minimal type quotients G=CG and G=CG of G and G respectively. Example 2.3.9. If F is a nonzero finite reduced k-algebra and G is a smooth affine F -group with pseudo-reductive fibers of minimal type over the factor fields of F then the pseudo-reductive k-group G WD RF =k .G / is also of minimal type. To see this, we note that if Z is the center of G , then by [CGP, Prop. A.5.15(1)] the center of G is RF =k .Z /. Now if G is of minimal type then its center Z does not contain a nontrivial unipotent F -subgroup scheme. This implies that the center RF =k .Z / of G does not contain any nontrivial unipotent k-subgroup scheme (see the argument used towards the end of the proof of Proposition 2.3.7), so G is of minimal type as claimed. Thus, by Example 2.3.3, if k is imperfect of characteristic 2 or 3 then every exotic pseudo-semisimple k-group is of minimal type.

2.3 Minimal type

25

Later arguments with torus centralizers and their derived groups will require good interaction of such constructions with the minimal-type property: Lemma 2.3.10. Let G be a pseudo-reductive k-group of minimal type. Any smooth connected normal k-subgroup N of G is of minimal type, as is ZG ./0 for any closed k-subgroup scheme  of a k-torus in G. In particular, the pseudosemisimple k-group D.ZG ./0 / is of minimal type. Pseudo-reductivity of ZG ./0 is a special case of [CGP, Prop. A.8.14(2)]. Proof. We may assume k D ks . Now all k-tori are split, so by [CGP, Prop. 9.4.5] both N and ZG ./0 are of minimal type. Since D.ZG ./0 / is normal in ZG ./0 , it is of minimal type since the same holds for ZG ./0 .  2.3.11. We finish §2.3 by analyzing maximal pseudo-reductive quotients of certain k-groups. As motivation, for a finite purely inseparable extension K=k and pseudo-reductive K-group G 0 , note that the natural map RK=k .G 0 /K ! G 0 has smooth connected unipotent kernel [CGP, Prop. A.5.11(1),(2)], so it is the maximal pseudo-reductive quotient over K. Thus, given the k-group RK=k .G 0 / and the extension K of k we can canonically reconstruct the K-group G 0 . We seek a refinement in which G 0 is pseudo-semisimple and the Weil restriction RK=k .G 0 / (which is generally not perfect) is replaced with its derived group. That is, is the pseudo-reductive quotient D.RK=k .G 0 //K D D.RK=k .G 0 /K /  G 0 also maximal? If k is imperfect of characteristic 2 and either ŒkI k 2  > 16 0 or GK has a non-reduced root system then the answer may in general be “no” s (in all other cases the answer is “yes”, by Proposition B.3.5), so we shall consider a finer condition than pseudo-semisimplicity in order to recover G 0 from D.RK=k .G 0 // and K=k in general. A suitable refinement involves incorporating the “minimal type” property. We shall present this in a wider setting, as follows. Let H be a smooth connected affine group over a field k (such as H D D.RK=k .G 0 // for .K=k; G 0 / as above), and let H pred WD H=Ru;k .H / denote its maximal pseudo-reductive quotient. The further quotient H prmt WD H pred =CH pred

(2.3.11)

is pseudo-reductive of minimal type and is maximal among such quotients of H ; its formation commutes with separable extension on k by Lemma 2.3.6. pred

Remark 2.3.12. If Hks has a reduced root system then there is a useful alternative description of H prmt : we claim that it is the image of the natural map fH W H ! RE=k .HE =Ru;E .HE //

26

Preliminary notions

for any finite extension E=k such that Ru .Hk / descends to an E-subgroup of HE . Since Ru;k .H /E  Ru;E .HE /, for H 0 WD HE =Ru;E .HE / we have naturally H 0 D .H pred /0 as quotients of HE and fH D fH pred ı qH for the quotient map qH W H  H pred . Hence, we may replace H with H pred so that now H is pseudo-reductive with a reduced root system over ks . If K=k is the minimal field of definition for the geometric unipotent radical of H then K  E and Ru;E .HE / descends to the K-subgroup Ru;K .HK / of HK . Hence, fH is the composition of iH and the inclusion RK=k .HKred / ,! RK=k .RE=K .HEred // D RE=k .HEred /; so ker fH D ker iH and therefore CH D .ker fH / \ ZH .T / for a maximal ktorus T in H . Via the identification of H 0 and .H=CH /0 we may thereby replace H with H=CH D H prmt (which has the same root system over ks as H ) to reduce to the case where H is also of minimal type. But then ker fH .D ker iH / is trivial since Hks has a reduced root system (see Proposition 2.3.4). Since the quotient map Hk  H red kills any smooth connected unipotent k normal subgroup as well as any central unipotent subgroup scheme, .H prmt /k dominates H red as quotients of Hk . In particular, H ! H prmt induces an isok morphism between maximal geometric reductive quotients. If N is a smooth connected normal k-subgroup of H then its image in H prmt is pseudo-reductive of minimal type by Lemma 2.3.10, so the map N ! H prmt factors through N  N prmt , yielding a canonical k-homomorphism N prmt ! H prmt . It is very important that N prmt ! H prmt has trivial kernel: Proposition 2.3.13. Let H be a smooth connected affine group over a field k, and N a smooth connected normal k-subgroup. The canonical map N prmt ! H prmt has trivial kernel. In particular, if K=k is a purely inseparable finite extension of fields and G 0 is a pseudo-semisimple K-group of minimal type then the natural surjective homomorphism D.RK=k .G 0 //K D D.RK=k .G 0 /K /  D.G 0 / D G 0

(2.3.13)

is the maximal pseudo-reductive quotient over K of minimal type, so naturally D.RK=k .G 0 //ss ' .G 0 /ss and we can canonically reconstruct G 0 from k k D.RK=k .G 0 // and the extension K=k. pred

If Hks has a reduced root system then this result admits an elementary proof via Remark 2.3.12 (using [CGP, Prop. A.4.8]). However, we will later

2.3 Minimal type

27

need the case of non-reduced root systems. See Proposition B.3.5 for a stronger maximality property of (2.3.13) without assuming G 0 is of minimal type, pro0 vided that if char.k/ D 2 then Œk W k 2  6 8 and GK has a reduced root system. s Proof. We may assume k D ks , so H.k/ is schematically dense in H (in the sense of [EGA, IV3 , 11.10.2]). Since the (possibly non-smooth) schematic kernel U D ker.N  N prmt / is normalized by H.k/, it follows that U is a normal k-subgroup scheme of H (normality says that for every k-algebra A and u 2 U.A/, the A-morphism HA ! .H=U /A defined by h 7! huh 1 mod U is trivial, which can be checked on H.k/; see [EGA, IV3 , 11.10.9]). We have seen that U  ker.H  H prmt /, so by passing to the quotient k-groups N=U and H=U we may assume N is pseudo-reductive of minimal type. The commutator subgroup .N; Ru;k .H // is a smooth connected normal ksubgroup of N contained in Ru;k .H /, so .N; Ru;k .H //  Ru;k .N /. But as N is pseudo-reductive, its k-unipotent radical is trivial. Thus, Ru;k .H / commutes with N , so Ru;k .H / \ N is a central unipotent k-subgroup scheme of N . But as N is of minimal type, such a subgroup scheme is trivial (Proposition 2.3.7). Hence, by passing to the quotient H=Ru;k .H / we can assume that H is pseudo-reductive. Now both H and N are pseudo-reductive and N is of minimal type. The intersection CH \ N is a central unipotent k-subgroup scheme of N , so (as N is of minimal type) this subgroup scheme is trivial. Therefore, the homomorphism N ! H=CH D H prmt has trivial kernel.  Remark 2.3.14. The minimal-type condition on the quotients of N and H in Proposition 2.3.13 cannot be dropped: over any imperfect field k of characteristic p > 0 it can happen that the k-homomorphism N pred ! H pred has nontrivial kernel. To build such examples with N even pseudo-reductive (hence necessarily not of minimal type, due to Proposition 2.3.13), let k 0 =k=k be the tower of purely inseparable extensions of degree p as in Example 2.3.5 and define N to be the absolutely pseudo-simple k-group G as defined there. Note that N contains the central k-subgroup scheme Q WD Rk 0 =k .p2 /=Rk=k .p2 /. (By [CGP, Ex. 5.3.7] we have ker iN D Q and dim Q D p 1 > 0, so indeed N is not of minimal type.) The k-group N WD Rk=k .Rk 0 =k .SLp2 /=p2 / has the smooth connected central k-subgroup U WD Rk=k .Rk 0 =k .p /=p / D Rk=k .Rk 0 =k .GL1 /=GL1 / that is unipotent of dimension p.p 1/. Left-exactness of Rk=k provides a central inclusion j W Q ,! N , and j.Q/ contains Z WD Rk 0 =k .p /=Rk=k .p /  U , so the smooth connected central pushout H WD N Q N contains N and N as normal k-subgroups with N \ N D Q and N \ Ru;k .H /  N \ U  Z ¤ 1. Hence, ker.N D N pred ! H pred / is nontrivial.

3 Field-theoretic and linear-algebraic invariants

3.1 A non-standard rank-1 construction Below we give a construction of non-standard pseudo-split absolutely pseudosimple k-groups with root system A1 over any imperfect field k of characteristic 2 for which Œk W k 2  > 2. 3.1.1. Let k be an imperfect field of characteristic 2 and let K=k be a nontrivial purely inseparable finite extension (so K ¤ kK 2 ). Let V be a nonzero ksubspace of K. Define khV i  K (resp. khV 2 i  K) to be the k-subalgebra generated by ratios v=v 0 for v; v 0 2 V f0g (resp. ratios v 2 =v 0 2 for v; v 0 2 V f0g), so khV i D kŒV  if 1 2 V . As K is an algebraic extension of k, both khV i and khV 2 i are subfields of K. Assume V is a khV 2 i-subspace of K, so for any v 2 V f0g the k-subspace 2 v V is independent of v; we denote it as V . If khV i D K and ŒK W kK 2  D 2 then V D K, whereas if ŒK W kK 2  > 2 then there exists a nonzero proper kK 2 subspace V  K satisfying khV i D K. If Œk W k 2  D 2 then ŒK W kK 2  D 2 in all cases, so what follows is of most interest when Œk W k 2  > 2. Definition 3.1.2. Let HV;K=k be the k-subgroup of RK=k .SL2 / generated by the k-subgroup UVC of the upper triangular unipotent k-subgroup of RK=k .SL2 / corresponding to the k-subspace V  K and the k-subgroup UV of the lower triangular unipotent subgroup of RK=k .SL2 / corresponding to the k-subspace V  K. Let PHV;K=k denote the image of HV;K=k inside RK=k .PGL2 / (equivalently, it is the k-subgroup of RK=k .PGL2 / generated by the k-groups UVC and UV viewed inside the upper and lower triangular subgroups of RK=k .PGL2 /). Let L denote either the k-group SL2 or PGL2 and H denote the k-subgroup HV;K=k of RK=k .LK / if L D SL2 and the k-subgroup PHV;K=k of RK=k .LK /

3.1 A non-standard rank-1 construction

29

if L D PGL2 . Clearly 1 2 V if and only if H contains the canonical k-subgroup L  RK=k .LK /. The detailed structure of the k-group H is worked out in [CGP, §9.1] (where a fixed choice of L within fSL2 ; PGL2 g is made, so the notation “HV;K=k ” is used there for each choice of L without confusion; we shall use the distinct notations HV;K=k and PHV;K=k below to distinguish the two cases). The subfield khV i  K is unaffected by K  -scaling of V , and the K-automorphisms of SL2 and PGL2 defined by diag.; 1/ 2 PGL2 .K/ carry HV;K=k to HV;K=k and carry PHV;K=k to PHV;K=k . Thus, by scaling V so that V  khV i (it is equivalent that V \ khV i ¤ 0), the k-groups HV;khV i=k and PHV;khV i=k make sense and respectively equal HV;K=k and PHV;K=k . Hence, for general V we have HV;K=k ' HV;khV i=k and PHV;K=k ' PHV;khV i=k as k-groups, so the main case of interest is when khV i D K. If khV i D K and V 0  K is a nonzero kK 2 -subspace then HV 0 ;K=k ' HV;K=k if and only if V 0 D V for some  2 K  (see [CGP, Prop. 9.1.7], noting that khV 0 i D K if V 0 D V with  2 K  ). This scaling relation likewise characterizes when PHV 0 ;K=k ' PHV;K=k . The split diagonal k-torus D  L is a maximal k-torus in H . To describe the Cartan subgroup ZH .D/, we introduce the following notation.  Definition 3.1.3. We define VK=k to be the Zariski closure of the subgroup of   RK=k .GL1 /.k/ D K generated by the ratios v=v 0 for v; v 0 2 V f0g; VK=k will be considered as a k-subgroup of RK=k .SL2 / via the map t 7! diag.t; 1=t/. When working with subgroups of RK=k .PGL2 / we use the same notation to denote the image of that Zariski closure under the natural map RK=k .SL2 / ! RK=k .PGL2 /; the context will always make the intended meaning clear.

The following result, which is [CGP, Prop. 9.1.4], gives the main properties of HV;K=k and PHV;K=k ; we reproduce it for the convenience of the reader. Proposition 3.1.4. Let H be HV;K=k or PHV;K=k inside RK=k .LK / .with L equal to SL2 or PGL2 respectively/. The k-group H is absolutely pseudo-simple, its root groups with respect to the diagonal torus D . RK=k .DK // are UVC and  UV as in Definition 3:1:2, and ZH .D/ D VK=k . In particular, the k-subgroup H  RK=k .LK / determines the k-subspace V  K. Moreover, the natural map  W HK ! LK is a K-descent of the maximal reductive quotient of Hk , and the minimal field of definition over k for the geometric unipotent radical Ru .Hk / of H is khV i. .In particular, for a nonzero k-subspace V 0  K that is a khV 0 2 i-subspace of K and the corresponding k-subgroup H 0  RK=k .LK /, if H ' H 0 as k-groups then khV i D khV 0 i as purely inseparable extensions of k./

30

Field-theoretic and linear-algebraic invariants

 The structure of the Cartan k-subgroup VK=k (even its dimension) is rather mysterious; see [CGP, 9.1.8–9.1.10].

Proof. Conjugation by diag.; 1/ for  2 K  shows that the problems for V   and V are equivalent. (Note that .V /K=k D VK=k inside RK=k .GL1 /.) Thus, we may and do assume 1 2 V . In particular, the following properties hold: khV i D kŒV , khV 2 i D kŒV 2  (so V is a kŒV 2 -subspace of K), V D V ,  L  H , V 2  V , and VK=k contains a Zariski-dense subset generated by the k-points diag.v; 1=v/ for v 2 V f0g. By [CGP, Cor. A.5.16], L is a Levi k-subgroup of RK=k .LK / via the natural inclusion. As L  H , Proposition 2.1.2(i) implies that H is pseudo-reductive and L is a Levi k-subgroup of H such that  is a K-descent of the maximal reductive quotient of Hk . Hence, the minimal field of definition over k for Ru .Hk / is contained in K and D.H / is absolutely pseudo-simple of type A1 . For the D-root groups U ˙ of L, the D-root groups of H are given by ˙  UH WD H \ RK=k .UK˙ /. We will show that UH˙ D UV˙ and ZH .D/ D VK=k .   We first check that VK=k  ZH .D/, or equivalently that VK=k  H . The subset V f0g  K  is stable under inversion (since 1 2 V and V is a subspace of K over khV 2 i D kŒV 2 ), so it suffices to apply the identity        t 0 1 1 1 0 1 1=t 1 0 D 0 t 1 0 1 t 1 1 0 1 t.1 t/ 1 that expresses a diagonal matrix in SL2 as a product of upper and lower triangular unipotent matrices. (Note that for t 2 V , t.1 t/ D t t 2 2 V because V contains  V 2 .) Since V 2
V  V , it also follows from the description of VK=k as a Zariski ˙ closure that it normalizes the smooth connected k-subgroup UV  RK=k .UK˙ /. Hence, it makes sense to define the following subgroups of H.k/:  Z D VK=k .k/; U D UVC .k/; P D Z n U :   Note that Z is Zariski-dense in VK=k , due to the definition of VK=k as a Zariski closure. The Bruhat decomposition of the group RK=k .LK /.k/ D L.K/ (with L equal to SL2 or PGL2 ) gives that for   0 1 n WD ; 1 0

the map U  P ! H.k/ defined by .u; p/ 7! unp is injective and we have the

3.1 A non-standard rank-1 construction

31

disjoint union containment U nP

[

P  H.k/:

S It is easy to check that H WD U nP P is stable under inversion and multiplication, so it is a subgroup of H.k/. As it contains the subgroups U and nU n 1 that are Zariski-dense in UVC and UV respectively, and H is generated by UVC and UV , we see that H is Zariski-dense in H . By [CGP, Prop. 2.1.8(2),(3)] applied to the standard parameterization  W GL1 ' D, the multiplication map n

1

UHC n  ZH .D/  UHC ! H

is an open immersion, and clearly its left-translate by n meets the Zariski-dense H in U nP. Hence, n 1 U nP is Zariski-dense in H . Therefore, its closure   n 1 UVC n
VK=k
UVC inside n 1 UHC n
ZH .D/
UHC is full, so ZH .D/ D VK=k and UHC D UVC ; hence UH D nUHC n 1 D nUVC n 1 D UV and H is generated by the vector groups UV˙ on which D acts linearly through nontrivial characters, so H is perfect. Now it follows from [CGP, Lemma 3.1.2] that H is absolutely pseudo-simple (with root groups UV˙ D UH˙ relative to D). Our remaining task is to show that the minimal field of definition over k for ker   HK is kŒV . For this, we may (and do) assume, after replacing K by kŒV , that K D kŒV . We will now show that if F  K is a subfield containing k such that ker  descends to an F -subgroup scheme R  HF then F D K. Since the Levi k-subgroup L in H yields a Levi F -subgroup LF in HF , this F -subgroup maps isomorphically onto the quotient HF =R. Hence, the Kmap  W HK ! LK descends to an F -map 0 W HF  LF . (The point is that we have identified the target of the F -descent 0 of  with the canonical F -descent of the target LK of .) Passing to Lie algebras of root groups relative to both the maximal F -torus DF of HF and the diagonal F -torus of the target of 0 , we get an abstract F linear map F ˝k V ! F that descends the canonical K-linear map K ˝k V ! K induced by Lie./. The map  is induced by the canonical quotient map RK=k .LK /K ! LK that on Lie algebras is the natural multiplication map K ˝k l ! l

32

Field-theoretic and linear-algebraic invariants

(with l WD Lie.LK /), so the map K ˝k V ! K induced by Lie./ is c ˝ v 7! cv. Hence, the existence of 0 implies that F
V  F inside of K, and in particular V  F . But F is a subfield of K containing k, so F contains kŒV  D K.  Remark 3.1.5. Since SL2 is a Levi k-subgroup of H WD HV;K=k , the schematic center of H is nontrivial. However, the smooth connected Cartan k-subgroup  VK=k in H may not contain the non-smooth center RK=k .2 / of RK=k .SL2 /, due to dimension reasons. For example, suppose K 2  k and let d D dimk V > 1,  so dim VK=k 6 1 C d.d 1/=2 by [CGP, Prop. 9.1.9]. Since dim RK=k .2 / D ŒK W k 1, it suffices to find V such that 1 C d.d 1/=2 < ŒK W k 1. If V is the k-span of 1 and a 2-basis of K=k then ŒK W k D 2d 1 , so any such V works when ŒK W k > 16. Example 3.1.6. The behavior of HV;K=k with respect to purely inseparable Weil restriction exhibits some subtleties when V ¤ K, essentially because the  formation of the Cartan k-subgroup VK=k can fail to commute with such Weil restriction. To explain this, let k0  k be a subfield over which k is purely inseparable of finite degree > 1, so we have a k0 -subgroup inclusion Rk=k0 .HV;K=k /  Rk=k0 .RK=k .SL2 // D RK=k0 .SL2 / that contains the diagonal k0 -torus D0 and has the same root groups as HV;K=k0 . Since the derived group D.Rk=k0 .HV;K=k // is pseudo-semisimple and hence is generated by its D0 -root groups, which coincide with those of HV;k=k0 , we have the general equality D.Rk=k0 .HV;K=k // D HV;K=k0 :

(3.1.6)

(Note that the minimal field of definition over k0 for the geometric unipotent radical of HV;K=k0 is k0 hV i, and this is equal to khV i inside K since V is a nonzero k-subspace of K.) By the same reasoning, the analogous assertions hold in the PGL2 -case using PH ’s. In the special case V D K (which is the only possibility when Œk W k 2  D 2 and khV i D K), which is to say the standard case when khV i D K, the inclusions   VK=k  RK=k .GL1 /; VK=k  RK=k0 .GL1 / 0 are both equalities and so the intervention of a derived group in (3.1.6) is unnecessary; i.e., Rk=k0 .HV;K=k / is perfect when V D K. But such perfectness can fail in cases with V ¤ K (assuming khV i D K and V is a kK 2 -subspace of K). More specifically, for any k and K  k 1=2 satisfying ŒK W k D 16, in

3.1 A non-standard rank-1 construction

33

 [CGP, 9.1.8–9.1.11] explicit examples are given such that VK=k D RK=k .GL1 / but Rk=k0 .HV;K=k / is never perfect for proper subfields k0  k as above (so  VK=k is a proper k0 -subgroup of RK=k0 .GL1 / for all k0 ¤ k); some such V 0 are recorded in (4.2.2). In these examples, dimk V D 6 and the maximal ksubalgebra F  K over which V is an F -submodule of K coincides with k.

Our interest in the groups HV;K=k and PHV;K=k is due to [CGP, Prop. 9.1.5], which we reproduce here (allowing any characteristic): Proposition 3.1.7. Let k be an infinite field of arbitrary characteristic and let K=k be a purely inseparable finite extension. Let L be either SL2 or PGL2 , and let D be the diagonal k-torus of L. Let G be a pseudo-semisimple k-subgroup of RK=k .LK / that contains the diagonal k-torus D . RK=k .DK //. (i) If char.k/ D 2 then there exists a unique nonzero k-subspace V  K that is a khV 2 i-subspace of K and for which G D HV;K=k or PHV;K=k according as
L D SL2 or PGL2 . In particular, 1 2 V if and only if the element 10 11 2 L.k/ lies in G.k/. (ii) If char.k/ ¤ 2 then there exists a subfield F of K over k and  2 K  such that G is the conjugate of the k-subgroup RF =k .LF / of RK=k .LK / under diag.; 1/. Moreover, G
coincides with the k-subgroup RF =k .LF / if and only if the element 10 11 2 L.k/ lies in G.k/. Proof. The root groups of .G; D/ are the intersections UG˙ of the root groups RK=k .UK˙ / of RK=k .LK / with G. These k-subgroups of G are stable under the natural conjugation action of D and are k-subgroups of the respective upper and lower triangular unipotent subgroups of RK=k .LK /. Thus, by the Zariski-density of .k  /2 in GL1 , the k-groups UG˙ correspond to k-subspaces of K. Hence, V ˙ WD UG˙ .k/ are k-subspaces of RK=k .Ga /.k/ D K. We have V ˙ ¤ 0 since the split unipotent k-groups UG˙ are nontrivial, due to the pseudo-simplicity of G. (In assertion (i), if G equals HV;K=k or PHV;K=k for some V then necessarily V D V C , so uniqueness of V is clear in (i). The problem is to prove existence.) Applying conjugation by diag. 1 ; 1/ for  2 K  has no effect on D and when char.k/ D 2 it has no effect on whether or not G has the form HV;K=k for a nonzero k-subspace V of K that is a khV 2 i-subspace, and it replaces V ˙ with 1 V ˙ . Thus, by
choosing  D v for v 2 V C f0g we may reduce to the case 1 2 V C ; i.e., 10 11 2 G.k/. By [CGP, Prop. 3.4.2], there is a unique element in NG .D/.k/ ZG .D/.k/
of the form  D u0 uu0 , with u D 10 11 and u0 2 UG
.k/. By explicit computation
in SL2 .K/ or PGL2 .K/ we see that u0 D 11 01 and  D 01 01 . Thus, the
standard Weyl element n WD 01 10 lies in G.k/. Using conjugation by n we see

34

Field-theoretic and linear-algebraic invariants

that V C D V . We will denote this common k-subspace of K by V , so 1 2 V . Inside the upper and lower triangular unipotent subgroups of RK=k .LK /, let UV˙ denote the k-subgroups corresponding to V  K. Clearly UG˙ D UV˙ . Next, we prove that the subset V f0g  K  is closed under inversion and that v 2 V  V for all v 2 V . Let Z D ZG .D/. The Bruhat decomposition of G.k/ relative to the minimal pseudo-parabolic k-subgroup Z n UV is provided by [CGP, Thm. 3.4.5], and it says ` G.k/ D UV .k/nZ.k/UV .k/ Z.k/UV .k/:

(3.1.7.1)

Since UVC \ .Z n UV / D f1g, we have UVC .k/ f1g  UV .k/nZ.k/UV .k/. Multiplication defines an open immersion UVC  Z  UV ! G by [CGP, Prop. 2.1.8(2),(3)], so the equality UVC D n 1 UV n and left-translation by n imply that the map UV  Z  UV ! G defined by .u0 ; z; u00 / 7! u0 nzu00 is an open immersion. In particular, for u0 ; u00 2 UV .k/ and z 2 Z.k/ the product u0 nzu00 2 G.k/ uniquely determines u0 , u00 , and z. Thus, for each nonzero x 2 V there exist unique y1 ; y2 2 V and z 2 Z.k/ such that         1 x 1 0 0 1 1 0 D

z
(3.1.7.2) 0 1 y1 1 1 0 y2 1 in SL2 .K/ or PGL2 .K/. As all terms in (3.1.7.2) aside from z come from SL2 .K/, when L D PGL2 we have that z uniquely arises from an element of SL2 .K/ such that (3.1.7.2) holds as an identity in SL2 .K/. Subject to this latter
1 condition in the PGL2 -case, z arises from a unique matrix of the form t0 0t with t 2 K  . We wish to compute y1 , y2 , and t in terms of x. Multiplying out the right side of (3.1.7.2) gives     1 x ty2 t D 0 1 ty1 y2 t 1 ty1 in SL2 .K/, so t D x and y1 D 1=t D y2 . Hence, 1=x D y1 2 V . This shows that the subset V f0g  K  is stable under inversion and moreover that if x 2 V f0g then   x 0 2 G.k/ (3.1.7.3) 0 x 1 inside SL2 .K/ or PGL2 .K/. Thus, for any x 0 2 V the product      1    x 0 1 x0 x 0 1 x2x0

D 0 x 1 0 1 0 x 0 1

3.1 A non-standard rank-1 construction

35

lies in G.k/. In other words, x 2 x 0 2 V for all x; x 0 2 V (as the case x D 0 is trivial). This says exactly that v 2
V  V for all v 2 V . Assume char.k/ ¤ 2, so for x; x 0 2 V both .1 C x/2 x 0 and x 2 x 0 lie in V (recall that 1 2 V ). Thus, xx 0 2 V , so V is a k-subspace of K which contains 1 and is closed under multiplication. Hence, V is a subfield of K containing k; we denote it as F . It is obvious that in this case G D RF =k .LF / (as G, being pseudo-semisimple, is generated by its root groups due to [CGP, Lemma 3.1.5]). This proves assertion (ii) of the proposition. Finally, assume char.k/ D 2. Again since G is pseudo-semisimple, it is generated by the root groups UG˙ D UV˙ , but by definition HV;K=k (resp. PHV;K=k ) is the subgroup of RK=k .LK / generated by these two subgroups because V D v 2 V D V for any v 2 V f0g. Hence, G D HV;K=k (resp. G D PHV;K=k ).  The constructions in Definition 3.1.2 have an intrinsic characterization: Proposition 3.1.8. Let G be an absolutely pseudo-simple group over a field k. Assume the rank of Gks is 1. Let K=k be the minimal field of definition for the geometric unipotent radical of G, and let G 0 D GK =Ru;K .GK /. Let iG W G ! RK=k .G 0 / be the natural homomorphism. (i) The minimal field of definition for the geometric unipotent radical of iG .G/ is K=k, and if G is of minimal type with root system A1 over ks then iG W G ! iG .G/ is an isomorphism. If char.k/ ¤ 2 then iG W G ! RK=k .G 0 / is surjective. (ii) Assume char.k/ D 2 and G is pseudo-split, and fix a split maximal k-torus T in G as well as an isomorphism of G 0 onto L D SL2 or PGL2 carrying TK onto the diagonal K-torus. There exists a nonzero kK 2 -subspace V of K, unique up to K  -scaling, such that iG .G/ D HV;K=k in the SL2 -case and iG .G/ D PHV;K=k in the PGL2 -case, and moreover khV i D K. Proof. By Proposition 2.1.2(ii), the k-subgroup iG .G/ of RK=k .L/ is pseudosimple and the subgroup Ru .iG .G/k /  iG .G/k has minimal field of definition K=k. The isomorphism property for G ! iG .G/ when G is of minimal type with root system A1 over ks is part of Proposition 2.3.4, and to prove that iG is surjective if char.k/ ¤ 2 it suffices to work over ks (so G is pseudo-split). We may now assume G is pseudo-split. Since iG .G/ contains the split diagonal k-torus, by Proposition 3.1.7 it follows that if char.k/ ¤ 2 then there exists a subfield F of K containing k such that iG .G/ ' RF =k .LF / whereas if char.k/ D 2 then iG .G/ has the asserted form for a nonzero k-subspace V  K that is a khV 2 i-subspace. The minimal field of definition for the geometric

36

Field-theoretic and linear-algebraic invariants

unipotent radical of iG .G/ is K=k, but this field of definition for the geometric unipotent radical of RF =k .LF / is F [CGP, Cor. A.5.16] and for both HV;K=k and PHV;K=k it is khV i (Proposition 3.1.4). We conclude that if char.k/ ¤ 2 then F D K, settling (i), and if char.k/ D 2 then khV i D K. Thus, if char.k/ D 2 then V is a kK 2 -subspace of K and the uniqueness of V up to K  -scaling is [CGP, Prop. 9.1.7] (as reviewed in the discussion preceding Definition 3.1.3).  Proposition 3.1.9. Let k be a field with char.k/ ¤ 2 and let G be an absolutely pseudo-simple k-group with root system A1 over ks . Let K=k be the minimal field of definition for Ru .Gk /  Gk and define G 0 D GK =Ru;K .GK /. The map iG W G ! RK=k .G 0 / is an isomorphism, so G is standard and of minimal type. This is [CGP, Thm. 6.1.1] but the proof below is simpler. Proof. We may assume k D ks , so G 0 is K-isomorphic to SL2 or PGL2 . By Proposition 3.1.8, iG is surjective. Thus, by Proposition 2.3.4, Z WD ker iG is a central k-subgroup of G and if G 0 D SL2 then iG is an isomorphism. Suppose instead that G 0 D PGL2 and consider the central extension 1 ! Z ! G ! RK=k .G 0 / ! 1: The central k-subgroup Z contains no nontrivial smooth connected k-subgroup since G is pseudo-reductive and Zk  Ru .Gk /. We have to prove that Z is trivial. e0 D SL2 . For the étale isogeny  W G e0 ! G 0 , the Weil restriction f D Let G RK=k ./ is also an étale isogeny. Hence, the f -pullback central extension e0 / ! 1 1 ! Z ! E ! RK=k .G e0 is simply connected, has middle term E étale over G, so E is smooth. Since G the central extension E is uniquely split by [CGP, Prop. 5.1.3, Ex. 5.1.4]. The ke0 / forces Z to be smooth. But Z contains no nonisomorphism E D Z RK=k .G e0 /. trivial smooth connected k-subgroup, so Z 0 is trivial. Hence, E 0 D RK=k .G 0 e0 / The étale map E ! G must carry E onto G, so G is a quotient of RK=k .G e0 ! G 0 / D modulo a k-subgroup C of ker f D RK=k ./ where  WD ker.G ZG e0 ' 2 is finite étale of order 2 over K (as char.k/ ¤ 2). Thus, RK=k ./ is finite étale of order 2 since K=k is purely inseparable (see [CGP, Prop. A.5.13]). Clearly C ¤ 1, so C D RK=k ./ D ker f and hence iG is an isomorphism. 

3.2 Minimal field of definition for Ru .Gk /

37

3.2 Minimal field of definition for Ru .Gk / 3.2.1. Let G be a pseudo-split pseudo-reductive group over an arbitrary field k, with K=k the minimal field of definition for the geometric unipotent radical red of G, and define G 0 WD GK D GK =Ru;K .GK /. Let iG W G ! RK=k .G 0 / be the natural homomorphism and  W GK ! G 0 the natural projection. Fix a split maximal k-torus T in G, and view T 0 WD TK as a maximal K-torus in G 0 , so ˆ0 WD ˆ.G 0 ; T 0 / is identified with the set of non-multipliable roots in ˆ WD ˆ.G; T / by [CGP, Thm. 2.3.10]. For each c 2 ˆ, let Uc denote the associated root group of .G; T / as in [CGP, Def. 2.3.13], so by [CGP, Prop. 2.3.11] the Lie algebra Lie.Uc / is the c-weight space in Lie.G/ when c is non-multipliable and is the span of the weight spaces for c and 2c when c is multipliable. By [CGP, Prop. 3.4.1], the k-subgroup Gc WD hUc ; U c i is absolutely pseudo-simple with (split) maximal k-torus T \ Gc of dimension 1 and we have a useful alternative description of Gc in terms of the codimension-1 torus Tc D .ker c/0red killed by c: if c is not divisible then Gc D D.ZG .Tc // whereas if c is divisible (so k is imperfect of characteristic 2) then Gc D D.Hc / for the pseudo-reductive identity component Hc of the centralizer in ZG .Tc / of .T \ Gc /Œ2 ' 2 . In all cases ˙c are the non-divisible roots in ˆ.Gc ; T \ Gc / (see [CGP, Prop. 3.4.1(1)]), so ˆ.Gc ; T \ Gc / is equal to f˙cg when c is not multipliable and it is equal to f˙c; ˙2cg when c is multipliable. Define c 0 D c when c is not multipliable and define c 0 D 2c when c is multipliable, so c 7! c 0 is a bijection from the set of non-multipliable roots of ˆ onto the set ˆ0 and ..Uc /K / is the c 0 -root group of .G 0 ; T 0 / (by [CGP, Cor. 2.1.9]). Thus, ..Gc /K / is equal to the subgroup Gc0 0 of G 0 generated by the ˙c 0 -root groups. 3.2.2. For each c 2 ˆ we want to relate the minimal field of definition K=k for the geometric unipotent radical of G to the minimal field of definition Kc =k for the geometric unipotent radical of the k-subgroup Gc of rank 1. In view of the explicit description of Gc in 3.2.1, it is immediate from [CGP, Prop. A.4.8] (for non-divisible c) and [CGP, Prop. A.8.14(2)] (for divisible c) red that Ru ..Gc /k / D Ru .Gk / \ .Gc /k for all c 2 ˆ. Hence, for G 0 WD GK there red 0 is a natural identification of .Gc / with .Gc 0 /k and it also follows that Kc  K k for all c 2 ˆ. For each c 2 ˆ let Gc0 WD .Gc /Kc =Ru;Kc ..Gc /Kc / be the Kc -descent of the maximal geometric reductive quotient of Gc , so naturally .Gc0 /K D Gc0 0 and hence there are natural inclusions:

38

Field-theoretic and linear-algebraic invariants

RKc =k .Gc0 / ,! RKc =k .RK=Kc ..Gc0 /K // D RK=k ..Gc0 /K /

D

RK=k .Gc0 0 /

,! RK=k .G 0 /: Since iGc is the natural homomorphism Gc ! RKc =k .Gc0 /, so iG jGc and iGc are compatible via the universal property of Weil restriction, we obtain a natural identification of iGc .Gc / with iG .Gc /. By Proposition 2.1.2(ii), the minimal field of definition over k for the geometric unipotent radical of iGc .Gc / is Kc , so the minimal field of definition over k for the geometric unipotent radical of iG .Gc / is also Kc . This establishes the following useful lemma via Proposition 3.1.8: Lemma 3.2.3. If char.k/ ¤ 2 or k is perfect then for every c 2 ˆ the k-group iG .Gc / D iGc .Gc / is isomorphic to RKc =k .SL2 / or RKc =k .PGL2 /. If k is imperfect with char.k/ D 2 then for every c 2 ˆ there exists a nonzero kKc2 -subspace Vc  Kc , unique up to Kc -scaling, such that khVc i D Kc and iGc .Gc / is isomorphic to HVc ;Kc =k or PHVc ;Kc =k . Remark 3.2.4. Assume in addition that G is absolutely pseudo-simple with a reduced root system (and continue to allow k to be arbitrary). If char.k/ ¤ 2 or k is perfect then define Vc D Kc . If k is imperfect with characteristic 2 then define Vc  Kc to be a member of the Kc -homothety class of nonzero kKc2 -subspaces of Kc determined by iGc .Gc / as in Lemma 3.2.3. The image iG .G/ is perfect and hence is contained in D.RK=k .G 0 //. The map iG thereby induces a homomorphism G W G ! D.RK=k .G 0 // identifying T with a split maximal k-torus of G .G/. The k-groups RK=k .G 0 / and D.RK=k .G 0 // have the same root groups with respect to T , and by [CGP, Lemma 3.1.5] these root groups generate D.RK=k .G 0 //. By consideration of open cells it follows that G is surjective if and only if iG .Uc / exhausts the c-root group of RK=k .G 0 / for all c 2 ˆ; i.e., if and only if Vc D K for every c. Here is another application of 3.2.2: Proposition 3.2.5. Assume G is pseudo-semisimple with a split maximal k-torus T . The extension K=k is generated by the subextensions Kc for c 2 ˆ.G; T /. This is part of [CGP, Prop. 6.3.1(1)], but we provide a simpler proof.

3.2 Minimal field of definition for Ru .Gk /

39

Proof. We use the notation T 0 D TK  G 0 and identify T with the maximal k-torus iG .T / of the k-group iG .G/ (note that T \ ker iG is trivial since ker iG is unipotent). By Proposition 2.1.2, the k-group iG .G/ is pseudo-semisimple and the minimal field of definition over k for its geometric unipotent radical is K. Since iG .G/ contains T as a split maximal k-torus, by [CGP, Thm. 3.4.6] we may choose a Levi k-subgroup L of iG .G/ containing T , so ˆ.L; T / D ˆ.G 0 ; T 0 / DW ˆ0 . We naturally identify G 0 with LK . For c 2 ˆ let Lc 0 be the k-subgroup of L generated by the ˙c 0 -root groups of .L; T /, so Lc 0 is a Levi k-subgroup of Gc (as .Gc /red D Gc0 0 ˝K k D Gc0 ˝Kc k/. Hence, the k-group iG .Gc / D iGc .Gc / is k contained in RKc =k ..Lc 0 /Kc /  RKc =k .LKc / inside RK=k .LK / D RK=k .G 0 /. Let K 0 be the subfield of K generated over k by the subextensions Kc for c 2 ˆ and define L0 D LK 0 , so iG .G/  RK 0 =k .L0 / inside RK=k .G 0 /. Hence, L  iG .G/  RK 0 =k .L0 /; so by Proposition 2.1.2(i) and [CGP, Cor. A.5.16] the extension K 0 =k is a field of definition for the geometric unipotent radical of iG .G/. But K=k is the minimal such extension, so the containment K 0  K over k is an equality.  Proposition 3.2.6. Let f W H ! H be a central quotient homomorphism between perfect smooth connected affine groups over a field k. The minimal fields of definition over k for the geometric unipotent radicals of H and H coincide. ss

Proof. Since fk carries R.Hk / onto R.H k /, the induced map H ss ! H k has ad

k

central kernel. Hence, fk induces an isomorphism H ad ' H k between maximal k geometric adjoint semisimple quotients (given as the maximal central quotients of the maximal geometric semisimple quotients). More specifically, if we define ad

R D ker.Hk  H ad /; R D ker.H k  H k /; k

then R D f

k

1

.R/ and .ker f /k  R with R=.ker f /k D R inside the k-group

.H= ker f /k D H k . Hence, if K is the minimal field of definition over k for R  H k then the fK -preimage of the K-descent of R is a K-subgroup of HK that is a K-descent of R, so K is also a field of definition over k for R  Hk . It must be minimal as such because if F  K is a subfield over k such that R descends to an F -subgroup R0  HF then .ker f /F  R0 and the F -subgroup R0 =.ker f /F  H F is an F -descent of R, forcing F D K.

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Field-theoretic and linear-algebraic invariants

We have established the equality over k of minimal fields of definition for projection onto maximal geometric adjoint semisimple quotients. It therefore remains to show that if G is a perfect smooth connected affine k-group then the minimal field of definition K=k for Ru .Gk / D ker.Gk  G ss / coincides with k the minimal field of definition F=k for the kernel of the natural homomorphism Gk  G ss =ZG ss . This equality is [CGP, Prop. 5.3.3], but for the convenience of k k the reader we now give a more elementary proof. Clearly F  K, and to prove that this inclusion of purely inseparable extensions of k is an equality it suffices to check after scalar extension to ks . By Galois descent, the formation of F and K commutes with such scalar extension. Thus, we may assume k D ks . Let GFad denote the quotient of GF that descends the quotient G ss =ZG ss of Gk . Since F D Fs , we see via consideration of the k

k

simply connected central cover of GFad that every connected semisimple central cover of GFad ˝F K uniquely descends to a connected semisimple central cover of GFad . Hence, we get a central isogeny GF0 ! GFad that descends the natural ss central isogeny GK ! GFad ˝F K between quotients of GK , and we have to ss show that the K-homomorphism q W GK  GK D GF0 ˝F K descends to an 0 F -homomorphism GF ! GF (i.e., q is “defined over F ”). By perfectness, GF is generated by its maximal F -tori [CGP, Prop. A.2.11]. ss For any maximal F -torus T  GF , the image q.TK /  GK is a maximal Kad torus and so it is the preimage of its image in GF ˝F K (as for maximal tori relative to any central isogeny between connected semisimple groups). Thus, q.TK / arises from a maximal F -torus T 0  GF0 . The resulting quotient map TK  TK0 is defined over F since the F -tori T and T 0 are split (as F D Fs ), so q.T .F //  GF0 .F /. The subgroup †  G.F / generated by the T .F /’s is Zariski-dense in G since such T ’s generate G. Since q.†/  GF0 .F /, it follows that q is defined over F .  Proposition 3.2.7. Let G be an absolutely pseudo-simple k-group whose root system over ks is reduced. Then G is standard if and only if the natural map ss ss G W G ! D.RK=k .GK // is surjective. If GK is simply connected and G is surjective then iG is an isomorphism. This is the main content of [CGP, Thm. 5.3.8]; we now give a simpler proof. Proof. The necessity of surjectivity of G is elementary (see [CGP, Rem. 5.3.6]), so now assume G is surjective. If G is of minimal type then ker G is trivial, ss so standardness is clear in such cases. If instead GK is simply connected then ss RK=k .GK / is perfect by [CGP, Cor. A.7.11], so in such cases G D iG and (by Proposition 2.3.4) this map is an isomorphism when it is surjective.

3.3 Root field and applications

41

It remains to show in the general case that G is standard when G is surjective. The formation of K=k is unaffected by passing to the central quotient G=CG (by Proposition 3.2.6), so the formation of the target of G is unaffected by passing to G=CG . Hence, clearly G=CG is surjective, so the settled minimaltype case ensures that the central quotient G=CG is standard. We may conclude by applying Lemma 3.2.8 below.  Lemma 3.2.8. For a pseudo-reductive k-group G and central extension 1!C !G!G!1

(3.2.8)

with a pseudo-reductive k-group G, if G is standard then so is G. Proof. Standardness of G is equivalent to that of D.G/ [CGP, Prop. 5.2.1], so we may replace G and G with their derived groups to arrange that each is perfect. As G is a standard pseudo-semisimple group, it is isomorphic to Rk 0 =k .G 0 /=Z for a nonzero finite reduced k-algebra k 0 , a semisimple k 0 -group G 0 whose fibers over the factor fields of k 0 are absolutely simple and simply connected, and a central k-subgroup Z  Rk 0 =k .G 0 /. The pullback of (3.2.8) along  W Rk 0 =k .G 0 /  Rk 0 =k .G 0 /=Z D G is a central extension E of Rk 0 =k .G 0 / by C . Note that any smooth connected k-subgroup of C is unipotent, due to the finiteness of the center of the semisimple G red , and k hence is trivial since G is pseudo-reductive. Thus, the central extension E is (uniquely) split by [CGP, Prop. 5.1.3, Ex. 5.1.4]. The splitting provides a homomorphism q W Rk 0 =k .G 0 / ! G with central kernel such that C  Rk 0 =k .G 0 / ! G is surjective, so the perfectness of Rk 0 =k .G 0 / and G and the commutativity of C imply that q is surjective. This central quotient presentation of G can be upgraded to a standard presentation; see [CGP, Rem. 1.4.6]. 

3.3 Root field and applications 3.3.1. Let G be an absolutely pseudo-simple k-group with a reduced root system over ks . Let K=k be the minimal field of definition for the geometric uniposs tent radical of G, and define G 0 D GK . For a maximal k-torus T in G, there is a unique T -stable linear complement g.T / to the subspace gT of g D Lie.G/. Since ker iG is central by Proposition 2.3.4, so Lie.ker iG /  Lie.ZG .T // D gT , Lie.iG / defines a k-linear inclusion of g.T / into Lie.RK=k .G 0 // D Lie.G 0 /. Thus, it makes sense to consider the set FT of elements  2 K such that multiplication by  on Lie.G 0 / preserves the k-subspace g.T /. This is an intermediate field between K and k, and the

42

Field-theoretic and linear-algebraic invariants

analogous such field for the pair .Gks ; Tks / is ks ˝k FT D .FT /s . All maximal ks -tori in Gks are G.ks /-conjugate, so .FT /s is independent of T and hence so is FT . The following definition is therefore intrinsic to G (whose root system over ks has been assumed to be reduced): Definition 3.3.2. The root field of G is the subextension F of K=k that satisfies F D f 2 K j 
g.T /  g.T / inside Lie.G 0 /g for some (equivalently, every) maximal k-torus T  G. In the pseudo-split case we can describe F in more explicit terms as follows. Let T be a split maximal k-torus and ˆ WD ˆ.G; T / D ˆ.G 0 ; TK /. For each c 2 ˆ the c-root group Uc  G satisfies Uc \ ker iG D 1 since ZG .Uc / D 1 (as even UcT is trivial), so iG W Uc ! RK=k .Uc0 / has trivial kernel, where Uc0 ' Ga is the croot group of G 0 . The T -action on Uc is compatible with the RK=k .TK /-action on RK=k .Uc0 /, so Uc corresponds to a nonzero linear subgroup of the vector group RK=k .Uc0 /. Let Gc denote the pseudo-split absolutely pseudo-simple ksubgroup of G generated by Uc and U c , so Gc has split maximal k-torus c _ .GL1 / with weight spaces Lie.U˙c / for its nontrivial weights. The restriction iG jGc factors through iGc , and conjugation by any representative nc 2 NGc .c _ .GL1 //.k/ of the reflection rc 2 W .Gc ; c _ .GL1 // swaps c and c and acts K-linearly on Lie.G 0 /. Thus, the set Fc of elements  2 K whose scaling action on RK=k .Uc0 / preserves Uc is contained in the minimal field of definition Kc =k for the geometric unipotent radical of Gc (as Lie.Uc / lies inside a Kc -line contained in the K-line Lie.Uc0 /) and F c D Fc . In particular, Fc coincides with the root field of Gc . Clearly Fc as a subextension of K=k only depends on c through its orbit under NG .T /.k/=ZG .T /.k/ D W .ˆ/, and \ FD Fc : (3.3.2) c2ˆ

Remark 3.3.3. Let G be an absolutely pseudo-simple k-group with a reduced root system over ks . Define K=k to be the minimal field of definition for its geometric unipotent radical and define F  K to be the root field of G. Two useful properties of F are that k 0 ˝k F is the root field of Gk 0 for any separable extension k 0 =k and that the root field of any pseudo-reductive central quotient G of G coincides with F (as a purely inseparable extension of k). Let us establish these properties. The behavior with respect to separable extension on k is immediate from the definition of F and the compatibility of the formation of both K=k and

3.3 Root field and applications

43

the quotient G 0 of GK with such extension on k. The invariance under passage to a pseudo-reductive central quotient G of G reduces to two other invariance properties: (i) the minimal field of definition over k for the geometric unipotent radical of G coincides with K=k, and (ii) for a maximal k-torus T in G and its isogenous image T in G, the natural map g ! g (which is generally neither injective nor surjective) induces an isomorphism g.T / ' g.T / (and so likewise 0 for the induced central quotient map G 0 ! G over K). Assertion (i) follows from Proposition 3.2.6. To prove (ii) we may assume k D ks , so T and T are split. By centrality and consideration of open cells we see that ˆ WD ˆ.G; T / D ˆ.G; T / inside X.T /Q D X.T /Q and that G  G restricts to an isomorphism between c-root groups for all c 2 ˆ, so (ii) is clear. The following two examples illustrate the root field in A1 -cases. Example 3.3.4. Suppose Gks has root system A1 and either char.k/ ¤ 2 or k is perfect, so G ' RK=k .G 0 / (see Proposition 3.1.9). In particular, G is of minimal type and its root field is K. For a subfield F  K over k, the natural map GF ! RK=F .G 0 / is surjective with a smooth connected unipotent kernel by pred [CGP, Prop. A.4.8]. Thus, the maximal pseudo-reductive quotient GF of GF is pred prmt RK=F .G 0 /. As the latter group is of minimal type, GF D GF (with notation prmt as in (2.3.11)) and the natural map G ! RF =k .GF / is an isomorphism. Example 3.3.5. Suppose Gks has root system A1 and k is imperfect with characteristic 2. The central quotient iG .G/ of G has the same root field as G (by Remark 3.3.3) and is of minimal type (as we saw in the discussion preceding Example 2.3.5), so also assume that G is of minimal type. Denote the root field prmt of G as F , and define G WD GF (notation as in (2.3.11)), so G is absolutely pseudo-simple with root system A1 over Fs . If G is pseudo-split over k then by Proposition 3.1.8 there is a nonzero kK 2 subspace V  K such that khV i D K and G is isomorphic to either HV;K=k or PHV;K=k , with V unique up to K  -scaling. Clearly kK 2  F D f 2 K j V  V g  K; so G is F -isomorphic to either HV;K=F or PHV;K=F by Proposition 2.3.13. In general, without a pseudo-split hypothesis on G, the following properties are straightforward to verify via reduction to the pseudo-split case over ks :  the root field of G D GFprmt is F ,  K is the minimal field of definition over F for Ru .GF /  GF ,  the natural map G ! D.RF =k G / is an isomorphism.

44

Field-theoretic and linear-algebraic invariants

Likewise, for any subfield F 0  F over k, the natural k-homomorphism G ! prmt D.RF 0 =k .GF 0 // is an isomorphism. The isomorphisms built in Examples 3.3.4 and 3.3.5 are special cases of: Proposition 3.3.6. Let G be an absolutely pseudo-simple k-group of minimal type with a reduced root system over ks . Let K=k be the minimal field of definition for the geometric unipotent radical of G, and let F 0 be a subfield of K containing k and contained in the root field F of G. The natural map prmt  W G ! D.RF 0 =k .GF 0 // is an isomorphism. Proof. To prove that  is an isomorphism, we may replace k with ks (as the prmt formation of the root field and GF 0 is compatible with separable extension on k) so that k is separably closed. In particular, G is pseudo-split and so admits a Levi k-subgroup L by [CGP, Thm. 3.4.6]. Let T  L be a maximal k-torus. ss The inclusion L ,! G identifies LK with G 0 WD GK . Since G is of minimal type and its root system is reduced, the homomorphism iG W G ! RK=k .G 0 / D RK=k .LK / has trivial kernel (Proposition 2.3.4) and so carries G isomorphically onto its image. In particular, GF 0 is naturally an F 0 -subgroup of RK=k .LK /F 0 . Let G  RK=F 0 .LK / denote the image of GF 0 under the natural map q W RK=k .LK /F 0 D RF 0 =k .RK=F 0 .LK //F 0 ! RK=F 0 .LK /: Since the composite map GF 0 ! RK=F 0 .LK / is visibly identified with iGF 0 , by prmt Remark 2.3.12 the quotient map GF 0 ! G identifies G with GF 0 . By Proposition 2.1.2, LF 0 is a Levi F 0 -subgroup of G and the inclusion of G into RK=F 0 .LK / is iG ; in particular, the reduced and irreducible root system ˆ D ˆ.G; T / coincides with ˆ.L; T / and ˆ.G ; TF 0 /. The inclusion of G into RK=k .LK / D RF 0 =k .RK=F 0 .LK // clearly factors through RF 0 =k .G /, so we get an inclusion G ,! D.RF 0 =k .G // that recovers . Hence, ker  D 1 and we have to show that G exhausts the target. The root system ˆ.D.RF 0 =k .G //; T / coincides with ˆ.RF 0 =k .G /; T / D ˆ.G ; TF 0 / D ˆ. Thus, since any pseudo-split pseudo-semisimple k-group is generated by its root groups relative to a split maximal k-torus, it suffices to show that for each c 2 ˆ the c-root groups of D.RF 0 =k .G // and G coincide. For c 2 ˆ, let Gc  G be the k-subgroup generated by the ˙c-root groups; define Lc  L and Gc  G similarly. The pseudo-semisimple D.RF 0 =k .Gc // of rank 1 is generated by the ˙c-root groups of D.RF 0 =k .G //, so it suffices to show that the evident inclusion jc W Gc  D.RF 0 =k .Gc // is an equality for all c. The

3.3 Root field and applications

45

case of perfect k is trivial, so we now assume k is imperfect with characteristic p. By (3.3.2), the root field F of G is contained in the root field Fc  Kc of Gc for all c, so F 0  Fc for all c. The compatibility of iG and iGc implies that the inclusion iG of G into RK=k .LK / carries Gc into RKc =k ..Lc /Kc /, so prmt Gc D .Gc /F 0 by Remark 2.3.12 applied to .Gc /F 0 . Thus, our problem is now intrinsic to the separate groups Gc (and the subextension F 0 =k of the root field Fc =k) for each c. That is, we may assume ˆ is A1 . This in turn is handled by Examples 3.3.4 (for p ¤ 2) and 3.3.5 (for p D 2).  Definition 3.3.7. Let G be an absolutely pseudo-simple k-group with a reduced root system over ks and minimal field of definition K=k for the geometric unipotent radical. Let T be a maximal k-torus of G, and define ˆ D ˆ.Gks ; Tks /. The long root field of G is the unique subextension F>  K over k such that ks ˝k F> is the common root field of .Gks /a for all roots a 2 ˆ when all roots are of equal length, and for all long roots a 2 ˆ otherwise. (As NG .T /.ks / acts transitively on the set of roots of a given length, F> exists.) Define the short root field F<  K over k similarly (so F< D F> if ˆ is simply laced). It is clear that the subextensions F> and F< of K=k are independent of the choice of T . These root fields arise in the description of automorphisms of G that act trivially on a chosen Cartan subgroup; see Proposition 8.5.4. By (3.3.2) the intersection of the root fields of the .Gks /a ’s is Fs , where F=k is the root field of G. In general, if G is an absolutely pseudo-simple kgroup then we claim that F>  F< , so F D F> ; this is a consequence of the following theorem that gives restrictions on field-theoretic and linear-algebraic data associated to pseudo-split absolutely pseudo-simple groups with a reduced root system. Theorem 3.3.8. Let G be a pseudo-split absolutely pseudo-simple group with rank n > 2 over an imperfect field k of characteristic p, and let T  G be a split maximal k-torus. Assume the irreducible root system ˆ WD ˆ.G; T / is reduced. Let K=k be the minimal field of definition for the geometric .unipotent/ radical of G, and for each c 2 ˆ let Kc =k be the minimal field of definition for the geometric .unipotent/ radical of Gc . Let F be the root field of G, and let F> and F< respectively denote the long and short root fields. (i) Assume the Dynkin diagram of ˆ does not have an edge of multiplicity p .as is automatic when p ¤ 2; 3/. Then F D Fc D Kc D K for all c 2 ˆ. (ii) Assume the Dynkin diagram of ˆ has an edge of multiplicity p .so either p D 3 with ˆ of type G2 or p D 2 with ˆ of type Bn , Cn , or F4 /. Then

46

Field-theoretic and linear-algebraic invariants

Kc D K for all short roots c and there is a subfield K>  K containing kK p such that Kc D K> for all long roots c. If p D 3 then F< D K and F> D K> .so F D F> /. (iii) Assume p D 2, and let Vc be as in Lemma 3:2:3 for each c 2 ˆ. If there is no edge of multiplicity 2 in the Dynkin diagram then Vc D K for all c. Assuming that there is an edge of multiplicity 2 in the diagram, Vc is a nonzero K> -subspace of K for c short and Vc is a nonzero kK 2 -subspace of K> for c long, so kK 2  F D F>  K>  F< :

(3.3.8)

Moreover, if ˆ is of type F4 or Bn with n > 3 then Vc is a K> -line for all long c .so F> D K> /, and if ˆ is of type F4 or Cn with n > 3 then Vc D K for all short c .so F< D K/. This result includes [CGP, Prop. 6.3.2] as a special case (and has a simpler proof). It is used in the proof of Theorem 3.4.1, which classifies (in the spirit of the Isomorphism Theorem for split connected semisimple groups) pseudo-split absolutely pseudo-simple groups G of minimal type with a reduced root system of rank > 2. (The rank-1 case of that classification is handled in Propositions 3.1.8 and 3.1.9, as Proposition 2.3.4 ensures that ker iG D 1 for such G.) Proof. By [CGP, Thm. 3.4.6], G has a Levi k-subgroup L containing T . As red ˆ is reduced, ˆ.L; T / D ˆ. The natural projection GK ! G 0 D GK maps 0 LK isomorphically onto G and we identify the latter with the former. Via this identification, the homomorphism iG is a homomorphism from G to RK=k .LK /. By Proposition 2.1.2(ii), the k-group iG .G/ is absolutely pseudo-simple with K=k as the minimal field of definition for its geometric unipotent radical, and it is of minimal type (as we noted immediately above Example 2.3.5). Moreover, it has been observed in 3.2.2 that for c 2 ˆ, iGc .Gc / has a natural identification with iG .Gc / and the minimal field of definition for the geometric unipotent radical of iG .Gc / is Kc . K/. The root fields Fc of Gc and F of G are likewise unaffected by passing to the central quotient iG .G/ (see Remark 3.3.3), so we can replace G with iG .G/ to arrange that G is a k-subgroup of RK=k .LK / containing L (and the inclusion G ,! RK=k .LK / is iG ). To reduce our problem to the case where L is simply connected, let  W e ! L be the simply connected central cover and let T eL e be the unique L e e k-torus over T . Via the identification of ˆ.L; T / with ˆ,  restricts to an isomorphism between corresponding root groups. The same therefore holds for

3.3 Root field and applications

47

RK=k .K /, so in this way we may identify each root group of G with a keK / containing the corresponding root group of L. e Let G e subgroup of RK=k .L eK / generated by these root groups of G, so G e be the k-subgroup of RK=k .L e contains L and admits G as a central quotient. e lies between RK=k .L eK / and L, e so it is pseudo-reductive by The k-group G Proposition 2.1.2(ii), and it is generated by the root groups Uc of G. These Uc ’s e acts linearly with only nontrivial weights, so each are vector groups on which T e e lies inside D.G/. Hence, G is perfect. It follows from Proposition 3.2.6 that e the minimal field of definition over k for the geometric unipotent radical of G coincides with the analogous such field K=k for its central quotient G. For each ec  Gc implies that replacing G with G e has no effect c 2 ˆ, the centrality of G on the extension Kc =k or on the root field Fc =k, nor on the kKc2 -subspace e to reduce to the case that L Vc  Kc if p D 2. Thus, we may replace G with G is simply connected. Fix a basis  of ˆ and a pinning of .L; T /. For each c 2  the k-subgroup Lc of L generated by its ˙c-root groups is thereby identified with SL2 carrying T \ Lc to the diagonal k-torus. Choose c 2 . By Proposition 3.1.7, Proposition 3.1.8, and 3.2.2, if p ¤ 2 then Gc is the subgroup RKc =k ..Lc /Kc / of RKc =k .RK=Kc ..Lc /K // D RK=k ..Lc /K /  RK=k .LK /. By applying the same results if p D 2, in such cases there exists a kKc2 -subspace Vc of Kc containing 1 such that kŒVc  D Kc and (via the identification of Lc with SL2 and the corresponding identification of RKc =k ..Lc /Kc / with the subgroup RKc =k .SL2 / of RK=k .SL2 /) we have Gc D HVc ;Kc =k inside RKc =k .SL2 /. To make the subsequent arguments more uniform, we adopt the convention that if ˆ is simply laced then all roots are considered to be both long and short. The action of W .ˆ/ is transitive on the set of roots of a given length, and the natural map NL .T /.k/ ! W .ˆ/ is surjective. Thus, the subfields Kc  K for short roots c coincide with a common subfield K< , and likewise the subfields Kc  K for long roots c coincide with a common subfield K> . The fields K> and K< generate K over k by Proposition 3.2.5, so in the simply laced case K> D K< D K. Likewise, if p D 2 then the k-subspaces Vc  Kc for all long (resp. short) roots c 2  are equal to each other. For p ¤ 2 we define Vc D Kc for all c 2 . For any p we have kŒVc  D Kc for all c and the k-group Gc is generated by the k-subgroups UV˙c of the ˙c-root groups RK=k .Ga / of RK=k .LK / cor responding to the k-subspace Vc  K. For c 2 , let .Vc /K be the Zariski c =k  closure in RKc =k .GL1 / of the subgroup of Kc generated by v 2 Vc f0g. (If  p ¤ 2 then clearly .Vc /K D RKc =k .GL1 /.) By Proposition 3.1.4 when p D 2, c =k and directly for p ¤ 2, the Cartan subgroup ZGc .T \ Gc / of Gc is the image of

48

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 under the map .Vc /K c =k _ RKc =k .cK / W RKc =k .GL1 / ! RKc =k .TKc / ,! RK=k .TK /: c

For any a; b 2 , the action of _  ZGa .T \ Ga /.k/ D RKa =k .aK /..Va /K /.k/  T .K/ a a =k

preserves the root groups U˙b of .G; T /, so it preserves the k-subspace Vb of K. _ In particular, for any nonzero v 2 Va , the aK .v/-action preserves Vb , which is to _ hb;a i  say that scaling by v 2 K on K preserves Vb , so such powers lie in Kb . _ Similarly, for any nonzero v 2 Vb , the bK .v/-action preserves Va , so scaling by _ ha;b i  v 2 K on K preserves Va . Let a and b be simple roots such that hb; a_ i D 1 (so a and b correspond to adjacent vertices in the Dynkin diagram and either they are of same length or a is long and b is short). As kŒVa  D Ka , we see that Vb is a nonzero Ka subspace of Kb inside K, so Ka  Kb . If ˆ contains roots of unequal lengths, by choosing a to be long and b to be short we conclude that K> D Ka  F<  Kb D K< : But as K> and K< generate K over k in all cases (even when ˆ is simply laced), we infer that K< D K in general. Thus, the final assertion in (ii) is settled by the observation that if p ¤ 2 then F< D K< and F> D K> by Proposition 3.1.9. Suppose we can choose a and b of equal length; i.e., ˆ is simply laced, or ˆ is of type F4 or Bn with n > 3 (so there exist non-orthogonal long simple roots), or ˆ is of type F4 or Cn with n > 3 (so there exist non-orthogonal short simple roots). Since Ka D Kb and Vb is a nonzero Ka -subspace of Kb , we see that Vb is equal to Kb , so Fb D Kb . This settles the simply laced case (as part of (i)) due to (3.3.2) since Kc D K< D K for all c in such cases. It remains to consider cases where ˆ has roots of different lengths. Now we may choose a to be long and b to be short, so ha; b _ i D m with m 2 f2; 3g. Let K be the subfield of K generated over k by the mth powers of nonzero elements of Vb ; note that K=K inherits pure inseparability from K=k. Since K D K< D Kb D kŒVb , if m ¤ p then the purely inseparable extension K=K is also separable and hence K D K in such cases. If instead m D p then K D kK p . For any nonzero v 2 Vb the scaling by v m 2 K  on K preserves Va , so Va is a K-subspace of K. Hence, if m ¤ p then Va is a nonzero K-subspace of K and so Va D K. This implies that K> D K and F> D K if m ¤ p (so F D K

3.3 Root field and applications

49

since F<  K> D K). We saw earlier that Vb is a nonzero K> -subspace of K, so likewise Vb D K if m ¤ p. This settles (i) if ˆ is not simply laced. Let us assume now that m D p, so K D kK p and hence Va is a kK p -subspace of Ka . Thus, kK p  F>  K> . We have proved all the assertions of the theorem.  Corollary 3.3.9. Let k be an imperfect field with characteristic p 2 f2; 3g. For any pair .G ; K=k/ as in Definition 2:2:3, the k-group G D RK=k .G / determines .G ; K=k/ uniquely up to unique isomorphism. Proof. By the uniqueness of the desired isomorphism, it suffices to treat the case Q k D ks . Writing K D Ki for fields Ki and letting Gi denote the Ki -fiber of G , the k-subgroups RKi =k .Gi / are precisely the pseudo-simple k-subgroups of G. Thus, it suffices to treat the case where K is a field (so K has no nontrivial kautomorphism and G is a basic exotic K-group). In such cases, since RK=k .G / is perfect we see via Example 2.3.3 and the criterion in Proposition 2.3.13 that the natural map GK ! G is the maximal pseudo-reductive quotient of minimal type. Hence, we just need to reconstruct K=k from G; we claim it is the root field of G. Let T  G be a maximal K-torus, and let T be the maximal k-torus of RK=k .T /, so T is a maximal k-torus of G. By parts (ii) and (iii) of Theorem 3.3.8, the root field of G coincides with that of Gc for long roots c in ˆ.G; T / D ˆ.G ; T /. For such c, the construction of basic exotic K-groups gives Gc ' SL2 _ as K-groups (with cK .GL1 / corresponding to the diagonal K-torus of SL2 ), so Gc D D.RK=k .Gc // D RK=k .SL2 /. Hence, the root field of Gc is K.  Proposition 3.3.10. Let G be a pseudo-split pseudo-semisimple group of minimal type over a field k, and let T be a split maximal k-torus in G. Assume ˆ WD ˆ.G; T / is reduced. The k-isomorphism class of G is determined by G ss k and the k-groups fGc gc2ˆ up to isomorphism. Note that the isomorphism class of Gc only depends on the orbit of c under W .ˆ/ D NG .T /.k/=ZG .T /.k/, and G ss determines ˆ. k

Proof. Let K=k be the minimal field of definition for the geometric unipotent radical of G, so G 0 WD GK =Ru;K .GK / is semisimple and split (with TK a split maximal K-torus). By Proposition 2.3.4, the natural map iG W G ! RK=k .G 0 / has trivial kernel. Let L  G be a Levi k-subgroup containing T (as exists by [CGP, Thm. 3.4.6] since T is split), so LK ' G 0 . By the Isomorphism Theorem for split connected semisimple groups, L is unique up to k-isomorphism since Lk ' G ss . k

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Let Kc =k be the minimal field of definition for the geometric unipotent radical of Gc , so K is generated over k by its subfields Kc =k (Proposition 3.2.5). Consider the k-subgroup Gc  RK=k ..Lc /K / that contains Lc , so Lc is a Levi k-subgroup of Gc . Hence, Gc  RKc =k ..Lc /Kc / inside RK=k ..Lc /K /. If char.k/ ¤ 2 or k is perfect then Gc D RKc =k ..Lc /Kc / for all c (see Proposition 3.1.9), so in such cases G is determined inside RK=k .LK / since it is generated by the k-subgroups Gc . Now we may assume k is imperfect with characteristic 2, so Gc is determined up to k-isomorphism by Lc (D SL2 or PGL2 ) and the Kc -homothety class of a nonzero kKc2 -subspace Vc  Kc satisfying khVc i D Kc : if Lc D SL2 (carrying T \ Lc to the diagonal k-torus D) then Gc ' HVc ;Kc =k and if Lc D PGL2 (carrying T \ Lc to the diagonal k-torus D) then Gc ' PHVc ;Kc =k . Let G be another pseudo-split pseudo-semisimple k-group of minimal type with a reduced root system such that G ss ' G ss , so G has root system ˆ. Upon k k choosing an isomorphism between the root systems, we may compatibly identify L as a Levi k-subgroup of G , so G is a k-subgroup of RK=k .LK / containing L (as G is of minimal type and ˆ is reduced). Assuming Gc ' Gc for all c 2 ˆ, we will build an automorphism of RK=k .LK / that carries G onto G . Let  be a basis of ˆ, so G is generated by fGc gc2 (as fLc gc2 generates L, and NL .T /.k/=T .k/ D W .ˆ/). Upon identifying the pair .Lc ; T \ Lc / with .SL2 ; D/ or .PGL2 ; D/, Gc equals HVc ;Kc =k or PHVc ;Kc =k respectively, where Vc is a nonzero kKc2 -subspace of Kc such that khVc i D Kc . The k-isomorphism Gc ' Gc implies that Vc D c
Vc for some c 2 Kc . Since  is a basis of X.T =ZL /, there exists a unique t 2 .T =ZL /.K/ such that c.t / D c for all c 2 . The action on RK=k .LK / by t 2 RK=k ..T =ZL /K /.k/ carries G to a k-subgroup that contains Gc  Lc for all c 2  and hence contains L. As G is generated by fGc gc2 , it is clear that t carries G onto G . 

3.4 Application to classification results Inspired by Theorem 3.3.8 and Proposition 3.3.10, we now establish (and use) a classification of the isomorphism classes of pseudo-split pseudo-simple groups G over an imperfect field k of characteristic p subject to the hypothesis that G is of minimal type. The associated irreducible root datum, which is sufficient to classify isomorphism classes in the semisimple case, needs to be supplemented with additional field-theoretic and (if p D 2) linear-algebraic data. The possibilities for such G of minimal type with a non-reduced root system are fully described in terms of field-theoretic and linear-algebraic data in [CGP, Thm. 9.8.6, Prop. 9.8.9], so we shall focus on pseudo-split G of minimal type

3.4 Application to classification results

51

with a reduced (and irreducible) root system ˆ. The case of type A1 is settled by Propositions 3.1.8 and 3.1.9, so we assume ˆ has rank n > 2. Let L be the unique split connected absolutely simple k-group with the same root datum as G ss , and let K=k be the minimal field of definition for k the geometric unipotent radical of G (so K=k is a purely inseparable finite ss is simple and extension); note that the connected semisimple group G 0 WD GK split. By [CGP, Thm. 3.4.6], L occurs as a Levi k-subgroup of G. Clearly LK ! G 0 is an isomorphism, so L  G  RK=k .LK / (as ker iG is trivial, by Proposition 2.3.4). Conversely, consider a perfect smooth connected k-subgroup H  RK=k .LK / containing L. By Proposition 2.1.2(i), H is pseudo-semisimple with L as a Levi k-subgroup and the minimal field of definition K=k for its geometric unipotent radical is a subextension of K=k. The given inclusion of H into RK=k .LK / is the composition of iH and the inclusion RK=k .LK / ,! RK=k .LK /. In particular, ker.iH / is trivial, so H is of minimal type. Upon specifying the root datum for G ss and a purely inseparable finite exk tension K=k, and letting L be the unique split connected absolutely simple k-group with the same root datum as G ss , our task amounts to describing the kk isomorphism classes of perfect smooth connected k-subgroups G of RK=k .LK / that contain L and are “large enough” that the minimal field of definition for Ru .Gk /  Gk is K=k (rather than a proper subextension). Theorem 3.4.1. Let k be an imperfect field of characteristic p, and let K=k be a purely inseparable finite extension. Let L be an absolutely simple and split connected semisimple k-group whose root system ˆ is of rank n > 2. The isomorphism classes of pseudo-split absolutely pseudo-simple k-groups G of minimal type with a reduced root system such that K=k is the minimal field of ss definition for the geometric unipotent radical and GK ' LK are given by: (i) If the Dynkin diagram of ˆ does not have an edge of multiplicity p .as is automatic if p ¤ 2; 3/ then D.RK=k .LK // is the unique such G. In particular, such a G is standard. (ii) Assume p D 3 and ˆ is of type G2 . Let K> be a subfield of K containing kK 3 . Up to isomorphism there is exactly one G such that K> =k is the minimal field of definition for the geometric unipotent radical of Gc for long roots c 2 ˆ. .The standard case is K> D K, and the basic exotic case is K> D k with K ¤ k./

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(iii) Assume p D 2 and ˆ is of type Bn , Cn , or F4 . Let K> be a subfield of K containing kK 2 . Let V be a nonzero K> -subspace of K such that khV i D K, and let V> be a nonzero kK 2 -subspace of K> such that khV> i D K> . Assume V D K if ˆ is of type F4 or Cn with n > 3, and assume V> D K> if ˆ is of type F4 or Bn with n > 3. Up to isomorphism there is exactly one G such that: if c 2 ˆ is long then K> =k is the minimal field of definition over k for the geometric  -homothety class of V classifies the unipotent radical of Gc and the K> > isomorphism class of Gc .via Proposition 3:1:8/, and if c 2 ˆ is short then the K  -homothety class of V classifies the isomorphism class of Gc . Proof. Let T be a split maximal k-torus of L. We may and do identify any such G with a k-subgroup of RK=k .LK / containing L. First consider the case where the diagram does not have an edge of multiplicity p, so by Theorem 3.3.8(i) we have Kc D K for all c 2 ˆ and the root field F is also equal to K. Thus, for each c 2 ˆ the c-root group Uc of .G; T / as a k-subgroup of the c-root group RK=k .Uc0 / of RK=k .LK / corresponds to a nonzero k-subspace of K that is a Ksubspace. Hence, the inclusion G  D.RK=k .LK // between pseudo-semisimple groups is an equality on root groups and thus is an equality. Now assume that the diagram has an edge of multiplicity p (so p 2 f2; 3g). ss Since GK ' LK , Proposition 3.3.10 shows that G is uniquely determined up to isomorphism by the isomorphism classes of the k-subgroups Gc for representatives c from the W .ˆ/-orbits on ˆ. Suppose that either p D 3 (so ˆ is of type G2 ) or that p D 2 with ˆ of type F4 . Thus, L is simply connected, so Lc D SL2 for all c. In particular, RKc =k ..Lc /Kc / is perfect for all c. By parts (ii) and (iii) of Theorem 3.3.8, we have Fc D Kc for all c, so the inclusion Gc  RKc =k ..Lc /Kc / between pseudo-semisimple groups (arising from the identification of Lc as a Levi ksubgroup of Gc  RK=k ..Lc /K /) is forced to be an equality on root groups and thus an equality. This shows that G is determined up to k-isomorphism by the subextension K> of K containing kK p . If K> ¤ K then applying RK> =k to a pseudo-split basic exotic K> -group with root system ˆ ensures existence for K> =k, and RK=k .LK / settles existence for K> D K. It remains to address ˆ of type Bn or Cn (n > 2), so p D 2. Since L has been specified and .Gc /ss ' .Lc /k , the k-isomorphism class of Gc is determined k by the Kc -homothety class of a nonzero kKc2 -subspace Vc of Kc D khVc i by Proposition 3.1.7. By Theorem 3.3.8(iii) we have Kc D K for short c, so the Vc ’s for short c lie in a common K  -homothety class of nonzero K> -subspaces V  -homothety of K satisfying khV i D K. The Vc ’s for long c lie in a common K>

3.4 Application to classification results

53

class of nonzero kK 2 -subspaces V> of K> that satisfy khV> i D K> . If n > 3 then Theorem 3.3.8(iii) ensures that V> D K> for type B and V D K for type C. Hence, every possibility for G gives rise to a triple .K> =k; V; V> / that determines G up to k-isomorphism. To show that every triple .K> =k; V; V> / actually arises from some G, we may assume that both V and V> contain 1 by moving each of V and V> within their homothety class. In particular, kŒV>  D K> . It is sufficient to settle exe of L (equipped with its split istence for the simply connected central cover L e over T ). Indeed, if G e  RK=k .L eK / is such a solution unique maximal k-torus T e containing L then its image G in RK=k .LK / is a central quotient that contains e ! L induces an isomorL and has the desired root-space invariants because L e phism between root groups for T and T (and this G has the associated field invariants K=k and K> =k by Proposition 3.2.6). Taking L to be simply connected, fix a basis  of ˆ D ˆ.L; T / and a pinning of .L; T / to identify Lc with SL2 carrying T \ Lc to the diagonal ktorus and the c-root group to the upper triangular unipotent subgroup for every c 2 . Define the k-groups Hc  RK=k ..Lc /K / for c 2  to be HV;K=k for short c and HV> ;K> =k for long c, so Lc  Hc since 1 2 V and 1 2 V> . A basis of X .T / is given by _ since L is simply connected, so Y _ C WD RK=k .TK / D RK=k .cK .GL1 //: c2

By hypothesis, V> D K> for type B with rank > 3 and V D K for type C with rank > 3. Thus, since V> is a kK 2 -subspace of K> and V is a K> subspace of K, it follows that Hc is normalized by C for all c 2 . For c 2 , the intersection Cc WD C \ Hc coincides with the pseudo-reductive Cartan k-subgroup ZHc .c _ .GL1 // of Hc , so the k-group C
Hc D .Hc o C /=Cc is pseudo-reductive by [CGP, Prop. 1.4.3]. Consider the k-subgroup H of RK=k .LK / generated by the k-groups C
Hc for c 2 . We have L  H since L is generated by fLc gc2 , so H is pseudoreductive with L as a Levi k-subgroup and ˆ.H; T / D ˆ. Since ZC
Hc .T / D C for all c 2 , it follows from [CGP, Thm. C.2.29(iii)] that H has c-root group equal to that of C
Hc for every c 2 . The pseudo-semisimple derived group G WD D.H / containing L has the same root system and root groups as H relative to T , so for each c 2  its c-root group inside that of RKc =k ..Lc /Kc / corresponds to V when c is short and V> when c is long.  We now obtain several nice consequences:

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Field-theoretic and linear-algebraic invariants

Theorem 3.4.2. Let G be a pseudo-reductive group over a field k. (i) If char.k/ ¤ 2; 3 then G is standard. (ii) Assume char.k/ D p 2 f2; 3g and that the root system ˆ of Gks is reduced .automatic when p D 3/ with Dynkin diagram having no edge of multiplicity p. If p D 2 then assume moreover that ˆ is not A1 . The k-group G is standard. Proof. It suffices to check standardness over ks (by [CGP, Cor. 5.2.3]), so we may and do assume k D ks . By [CGP, Prop. 5.3.1] it is enough to prove standardness of the normal pseudo-simple k-subgroups of G, so we may assume G is (absolutely) pseudo-simple. Note that ˆ is automatically reduced when char.k/ ¤ 2, by [CGP, Thm. 2.3.10]. We may also assume k is imperfect with characteristic p or else there is nothing to do. Our task is to show that a pseudo-simple group over k D ks with a reduced root system is standard except possibly when the Dynkin diagram either has an edge of multiplicity p or is a single point and p D 2. Proposition 3.1.9 settles the rank-1 cases away from p D 2, so assume ˆ has rank n > 2. If G is of minimal type then we may conclude by Theorem 3.4.1(i). Hence, the central quotient G=CG of minimal type is standard. By Lemma 3.2.8, standardness of G=CG implies standardness of G.  Proposition 3.4.3. An absolutely pseudo-simple k-group G with root system F4 or G2 over ks is either standard or exotic .see Definition 2:2:3/. Proof. Let us first treat the case when k D ks , so G is pseudo-split. Let K=k be the minimal field of definition for the geometric unipotent radical of G. By Theorem 3.4.1, G is standard except possibly if char.k/ D p > 0 with p D 2 for type F4 and p D 3 for type G2 , and in these latter cases G is classified up to isomorphism by the subfield K>  K over kK p that can be arbitrary. The cases with K> ¤ K are obtained as Weil restrictions of (pseudo-split) basic exotic K> groups relative to K=K> . The case K> D K corresponds to the standard case (i.e., RK=k applied to a split K-group of type F4 or G2 , depending on p 2 f2; 3g). This settles the case when k D ks . Now consider general k, so Gks is either standard or exotic. If Gks is standard then G is standard by [CGP, Cor. 5.2.3]. Suppose instead that Gks is exotic, so k is imperfect with characteristic p 2 f2; 3g where p D 2 when the root system is F4 and p D 3 when the root system is G2 . The preceding arguments p construct a proper subfield F 0  Ks containing ks Ks and a basic exotic F 0 0 0 group G such that Gks ' RF 0 =ks .G /. More specifically, as we saw in the proof

3.4 Application to classification results

55

of Corollary 3.3.9, F 0 =ks is the root field of Gks and the natural map GF 0 ! G 0 is the maximal pseudo-reductive quotient of minimal type. For any absolutely pseudo-simple k-group, the formation of its root field commutes with separable extension on k. Likewise, the formation of the maximal pseudo-reductive quotient of minimal type (for any smooth connected affine group) commutes with separable extension on the ground field (due to Lemma 2.3.6). Thus, the purely inseparable root field F=k of G satisfies F ˝k ks D F 0 , and the quotient G WD .GF /prmt of GF satisfies GF 0 D G 0 as quotients of GF 0 . But G 0 is basic exotic over F 0 D Fs , so G is basic exotic over F [CGP, Prop. 7.2.7(3)], and the natural map G ! RF =k .G / is an isomorphism because over ks it recovers the identification Gks D RF 0 =ks .G 0 /.  In [CGP, Thm. 5.1.1], standardness is characterized among pseudo-reductive groups in terms of dimensions of root spaces in the Lie algebra over ks , assuming Œk W k 2  6 2 if char.k/ D 2. (This is only interesting if k is imperfect of characteristic 2 or 3, as otherwise standardness always holds.) This degree restriction in characteristic 2 can now be eliminated: Proposition 3.4.4. Let G be a pseudo-reductive group over a field k, T a maximal k-torus of G, and ˆ D ˆ.Gks ; Tks /. The k-group G is standard if and only if the following conditions all hold: ˆ is reduced, the ks -dimension of the a-root space .gks /a in Lie.G/ks depends only on the irreducible component of a in ˆ, and dim.gks /a D ŒKa W ks  for isolated points a in the Dynkin diagram of ˆ .where Ka =ks is the minimal field of definition for the geometric unipotent radical of .Gks /a /. The additional requirement on isolated points of the Dynkin diagram always holds if char.k/ ¤ 2 (by Proposition 3.1.9) or if char.k/ D 2 with Œk W k 2  6 2 (by [CGP, Prop. 9.2.4]), so it is only relevant when char.k/ D 2 with Œk W k 2  > 2. In such cases this additional condition is necessary: for any such k we may choose K  k 1=2 of degree 4 over k and a nonzero k-subspace V  K of dimension 3 (so khV i D K), and the non-standard absolutely pseudo-simple k-group HV;K=k of rank 1 satisfies all requirements except for the final one. Proof. We may replace G with D.G/ [CGP, Prop. 5.2.1], so G is perfect, and we may assume G ¤ 1. Also, by [CGP, Cor. 5.2.3] we may assume k D ks . Since standardness of G is equivalent to that of its pseudo-simple normal k-subgroups [CGP, Prop. 5.2.6, Prop. 5.3.1], we may also assume that G is pseudo-simple. For a finite extension field k 0 =k and connected semisimple k 0 -group G 0 , any pseudo-reductive central quotient of Rk 0 =k .G 0 / is standard [CGP, Rem. 1.4.6].

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Thus, if G is standard then so is any pseudo-reductive central quotient. Likewise, by Lemma 3.2.8, G is standard if its pseudo-simple central quotient G=CG is standard. Since the root system and root groups of a pseudo-semisimple group are inherited by any pseudo-reductive central quotient (see [CGP, Prop. 2.3.15]), as is the minimal field of definition for the geometric unipotent radical (Proposition 3.2.6), we may replace G with G=CG so that G is of minimal type. Let K=k be the minimal field of definition for Ru .Gk /  Gk . The root system is reduced in the standard case, and the standard pseudo-simple cases satisfy the dimension conditions on root spaces (as standard groups are a central quotient of RK=k .G 0 / for a simple semisimple K-group G 0 ). It remains to prove the sufficiency of the given set of conditions, so we can assume ˆ is reduced. The possibilities for (minimal type and pseudo-simple) G with rank > 2 are classified up to isomorphism in Theorem 3.4.1. We shall check that the dimension conditions on root spaces are not satisfied in the nonstandard cases on that list. Case (i) in Theorem 3.4.1 is standard. In cases (ii) and (iii), V is a nonzero K> -subspace of K and V>  K> , so the dimension condition on root spaces forces V> D K> and V to be a K> -line. But then K D khV i D K> , so all root spaces have dimension ŒK W k. That forces the inclusion G ,! D.RK=k .LK // between pseudo-simple k-groups to be an equality on root groups and thus an equality of k-groups. Finally, we can assume that ˆ is of type A1 and (by Proposition 3.1.9) that char.k/ D 2. Then G is either HV;K=k or PHV;K=k with V  K a nonzero kK 2 -subspace satisfying khV i D K (by Proposition 3.1.8(ii) and Proposition 2.3.4). The dimension condition on root spaces forces V D K, so G is either RK=k .SL2 / or D.RK=k .PGL2 //, each of which is standard. 

4 Central extensions and groups locally of minimal type

4.1 Central quotients The following general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups generalizes a familiar fact in the connected reductive case and will frequently be useful in our subsequent work. Lemma 4.1.1. For any central extension of k-group schemes f

1!Z !G !G !1 with pseudo-reductive G and G , the inclusion ZG  f

1 .Z

G/

is an equality.

Recall that ZG (resp. ZG ) denotes the scheme-theoretic center of G (resp. G ). Proof. We may assume k D ks . Let T be a maximal k-torus in G , so its image T  G is a maximal k-torus. Let C D ZG .T / and C D ZG .T / denote the associated Cartan k-subgroups, which are commutative. The map C ! C is surjective, hence faithfully flat, and Z  C , so the scheme-theoretic preimage f 1 .C / is equal to C . Thus, f 1 .ZG /  C . Our problem is therefore to show that for any k-algebra A, any g 2 C.A/ whose image in G .A/ centralizes G A also centralizes GA . Since G D C
D.G / with C commutative, and D.G / is generated by the T -root groups [CGP, Lemma 3.1.5], ZG represents the functor of points of C centralizing all T -root groups in G . By [CGP, Cor. 2.1.9] and the centrality of Z , each T -root group in G maps isomorphically onto a T -root group in G . Thus, for points of C we can detect the centralizing property against a given

58

Central extensions and groups locally of minimal type

root group for .G ; T / by passing to .G ; T / (as C normalizes every T -root group of G ).  4.1.2. Let G be a smooth connected affine group over a field k. Let T be a maximal k-torus in G, with C WD ZG .T / the associated Cartan subgroup of G. To study the automorphism functor of G, two useful functors on k-algebras are: AutG;C W A

ff 2 AutA .GA / j f jCA D idCA g;

AutG;T W A

ff 2 AutA .GA / j f jTA D idTA g:

The following properties of these functors are known by [CGP, Thm. 2.4.1, Cor. 2.4.4]: the functor AutG;C is represented by an affine finite type k-group scheme AutG;C , if G is generated by tori (for example, if G is perfect) then the functor AutG;T is represented by an affine finite type k-group scheme AutG;T for which the natural map AutG;C ! AutG;T is a closed immersion, and if moreover G is pseudo-reductive (and generated by tori) then the maximal smooth closed k-subgroups of AutG;C and AutG;T coincide. We denote the maximal smooth closed k-subgroup of AutG;C by ZG;C . By [CGP, Thm. 2.4.1], if G is pseudo-reductive then ZG;C is commutative with pseudo-reductive identity component. Assuming that G is pseudo-reductive, we will prove later (see Proposition 6.1.4) that ZG;C is in fact connected. Proposition 4.1.3. The central quotient G=ZG of a pseudo-reductive group G is pseudo-reductive with trivial center. In particular, it is of minimal type. Proof. The natural conjugation action of C on G extends the identity on C , and 0 so is classified by a k-homomorphism C ! ZG;C . Thus, we can form the non0 0 commutative pushout G D .G o ZG;C /=C as a central quotient of G o ZG;C (using the central anti-diagonal inclusion of C ), and by [CGP, Prop. 1.4.3] this pushout is pseudo-reductive. The image of G in G is normal, hence also pseudo0 reductive, and it is the quotient of G modulo ker.C ! ZG;C / D ZG , so G=ZG is pseudo-reductive. Lemma 4.1.1 now implies that G=ZG has trivial center.  Corollary 4.1.4. For a pseudo-reductive k-group G and closed k-subgroup scheme Z  ZG , the central quotient G=Z is pseudo-reductive if and only if ZG =Z does not contain a nontrivial smooth connected unipotent k-subgroup. Proof. The quotient G=ZG of G=Z modulo ZG =Z is pseudo-reductive by Proposition 4.1.3, so the smooth connected k-group Ru;k .G=Z/ is contained in

4.1 Central quotients

59

ZG =Z. Since any smooth connected unipotent k-subgroup of ZG =Z is central in G=Z (and so is contained in Ru;k .G=Z/), we are done.  Note that in contrast with the reductive case, a pseudo-reductive group with trivial center can fail to be perfect (i.e., it may not be pseudo-semisimple). For example, if k 0 =k is a nontrivial purely inseparable finite extension of fields with characteristic p > 0 then G D Rk 0 =k .PGLp / has trivial center by [CGP, Prop. A.5.15(1)] but is not perfect (due to [CGP, Prop. 1.3.4, Ex. 1.3.5]). Proposition 4.1.5. Let f W H ! H be a central quotient homomorphism between pseudo-semisimple groups over a field k, and let K=k be the common minimal field of definition for their geometric unipotent radicals. The homomorphism H W H ! D.RK=k .HKss // is surjective if and only if H is surjective. The extension K=k is the same for H and H due to Proposition 3.2.6. Proof. We may assume k D ks . Let T be a maximal k-torus of H and T its image in H . By consideration of open cells and centrality of the kernel we see that ˆ.H; T / D ˆ.H ; T / via the inclusion X.T / ,! X.T /. Let H 0 D HKss 0 ss 0 and H D H K , so H is a central quotient of H 0 and the K-tori T 0 WD TK and 0 0 T WD T K are maximal in H 0 and H respectively. 0 0 The root system ˆ0 WD ˆ.H 0 ; T 0 / coincides with ˆ.H ; T / via the inclu0 sion X.T / ,! X.T 0 / and they both equal the set of non-multipliable roots in ˆ.H; T /. By [CGP, Ex. 2.3.2], ˆ0 naturally coincides with the root systems of 0 RK=k .H 0 / and RK=k .H / relative to the maximal k-tori T  RK=k .T 0 / and 0 T  RK=k .T /, and the associated root groups are given by applying RK=k to 0 0 root groups of .H 0 ; T 0 / and .H ; T /. Hence, by consideration of open cells, cen0 trality of the kernel of H 0 ! H implies that each T -root group of RK=k .H 0 / 0 maps isomorphically onto the corresponding T -root group of RK=k .H /. Every pseudo-semisimple k-group is generated by the root groups relative to a maximal k-torus (recall that k D ks ), so in the commutative diagram H

H

f





H

/ D.RK=k .H 0 //

H

f0

/ D.RK=k .H 0 //

the map f 0 with central kernel is surjective. (The map analogous to f 0 between Weil restrictions, without the intervention of derived groups, can fail to be sur-

60

Central extensions and groups locally of minimal type

jective.) Thus, if H is surjective then so is H . Conversely, assuming that H is surjective, it follows that the natural k-homomorphism H  ker.f 0 / ! D.RK=k .H 0 // is surjective on k-points. Thus, by passing to derived groups of k-points we see that H is surjective (as H and D.RK=k .H 0 // are perfect).  Proposition 4.1.6. Let G and G be pseudo-semisimple k-groups. Assume that G is of minimal type and G ss is simply connected. Let f W G ! G be a central k quotient homomorphism. Then f is an isomorphism and G is of minimal type. Proof. Let K=k be the minimal field of definition for Ru .Gk /  Gk . By Proposition 3.2.6, K=k is also the minimal field of definition for the geometric uniposs tent radical of G . The central quotient homomorphism fKss W GKss ! GK is an ss isomorphism since GK is simply connected. Let T be a maximal k-torus in G and define C D ZG .T /. Since f is a central quotient homomorphism we have ker f  C , and since G is of minimal type the map iG jC has trivial kernel. In the commutative diagram G

iG

ss RK=k .fK /

f



G

/ RK=k .G ss / K

iG

 / RK=k .G ss / K

the right map is an isomorphism, so the triviality of ker.iG jC / implies that ker.f jC / is trivial. But ker.f jC / D ker.f /, so f is an isomorphism. 

4.2 Beyond the quadratic case Let k be an imperfect field of characteristic 2. There are several phenomena that arise when Œk W k 2  > 2 but not when Œk W k 2  D 2. We now describe four of them, as each underlies a problem that we have to overcome. 4.2.1. Let G be absolutely pseudo-simple over k, with K=k the minimal field of definition for the geometric unipotent radical Ru .Gk /  Gk , and assume Gks has root system A1 . Let G 0 D GK =Ru;K .GK / (a K-form of SL2 or PGL2 ). Suppose Œk W k 2  D 2. By [CGP, Prop. 9.2.4], the k-group G is standard and if 0 moreover GK ' SL2 (rather than PGL2 ) then G ' RK=k .G 0 /. Due to the role of s

4.2 Beyond the quadratic case

61

SL2 in the structure of connected semisimple groups, this standardness for type A1 underlies the proof (when Œk W k 2  D 2) that any pseudo-reductive k-group with a reduced root system over ks admits a “generalized standard” form using exotic constructions for types F4 , Bn , and Cn (n > 2) [CGP, Thm. 10.2.1(2)]. This breaks down when Œk W k 2  > 2: there are many non-standard pseudosplit absolutely pseudo-simple k-groups G with a reduced root system such 0 ' SL2 . Fortunately, such G of minimal type can be classified (see that GK s Proposition 3.1.8). The abundance of these rank-1 groups G leads to the classes of “generalized basic exotic” and rank-2 “basic exceptional” k-groups in Chapter 8 that go beyond the “basic exotic” k-groups in [CGP, Ch. 7–8]. 4.2.2. If Œk W k 2  > 16 then there exist non-standard pseudo-split absolutely ss pseudo-simple k-groups G of minimal type with root system A1 such that G k ' SL2 and G admits a smooth connected central extension G by ˛2 or Z=.2/ (so G is absolutely pseudo-simple) with G not of minimal type. (By Proposition B.3.1, no such examples exist if Œk W k 2  6 8.) In Examples B.4.1 and B.4.3 we give analogous examples over k with root system BCn (n > 1) whenever Œk W k 2  > 2. To build such G with root system A1 , choose K  k 1=2 with k-degree 16 and a 2-basis fe1 ; e2 ; e3 ; e4 g. Let v D e1 e2 C e3 e4 (resp. v D e1 e2 e3 e4 ). Define V D k ˚ ke1 ˚ ke2 ˚ ke3 ˚ ke4 ˚ kv  K:

(4.2.2)

The subfield F D f 2 K j V  V g over k is equal to k. Indeed, since dimk V D 6 the only other possibility is ŒF W k D 2. Since F  V (as 1 2 V ), if ŒF W k D 2 P then F D k ˚ kx where x D ai ei C av for ai ; a 2 k not all 0. But xej 2 V , so the 2-basis property of fe1 ; e2 ; e3 ; e4 g yields a contradiction. Hence, F D k. In terms of Definitions 3.1.2 and 3.3.2, this says that HV;K=k has root field k. Let q W V ,! K ! k be the squaring map and C D Sym.V /=.v
v q.v//, so C is a finite local k-algebra with residue field K. Since char.k/ D 2, the definition of C shows that all squares in C lie inside k; i.e., C 2  k. Consider the k-subgroup G D HV;C =k  RC =k .SL2 / generated by the ksubspace V  C viewed inside both the upper and lower triangular unipotent k-subgroups. By [CGP, 9.1.9–9.1.11], G is a pseudo-split absolutely pseudoss simple central extension of G WD HV;K=k by ˛2 (resp. Z=.2/). Since G k ' SL2 , by Proposition 4.1.6 the absolutely pseudo-simple k-group G is not of minimal type and does not admit an absolutely pseudo-simple central extension of minimal type. In §B.1–§B.2 we construct higher-rank pseudo-split absolutely pseudo-simple k-groups (with root system Bn or Cn for any n > 2) that admit no absolutely pseudo-simple central extension of minimal type.

62

Central extensions and groups locally of minimal type

In view of such examples, when Œk W k 2  > 2 we will give a structure theorem only for pseudo-reductive k-groups that are locally minimal type in the sense of Definition 4.3.1 (also see Remark 4.3.2 and Proposition 5.3.3). 4.2.3. Consider a pseudo-split absolutely pseudo-simple k-group G whose root system is BCn with n > 1. The classification of all such G of minimal type with a specified minimal field of definition K=k for the geometric unipotent radical is the main result of [CGP, Ch. 9] (see [CGP, Thm. 9.8.6, Prop. 9.8.9]) and involves nonzero K 2 -subspaces of K satisfying nontrivial conditions. If Œk W k 2  D 2 then (without assuming G to be of minimal type at the outset) G admits no nontrivial pseudo-semisimple central extension (see [CGP, Prop. 9.9.1]). For such k there are very few possibilities for the relevant K 2 subspaces of K (see [CGP, Cor. 9.8.11]). This implies that if Œk W k 2  D 2 and G is absolutely pseudo-simple over k with Gks having root system BCn then by [CGP, Thm. 9.9.3(1)] two properties hold: G is necessarily pseudo-split and of minimal type over k, and G is determined up to k-isomorphism by n and the minimal field of definition K=k of its geometric (unipotent) radical (which can be any nontrivial purely inseparable finite extension of k). These conclusions about G generally fail when Œk W k 2  > 2. 4.2.4. Let G be a pseudo-reductive k-group. If Œk W k 2  D 2 then by [CGP, Prop. 10.1.6, Thm. 10.2.1(1)] there is a unique decomposition G D G1  G2 where .G1 /ks has a reduced root system and G2 is pseudo-semisimple with root system over ks having only non-reduced irreducible components. This direct product structure without a pseudo-semisimplicity hypothesis on G rests on the fact, specific to the case Œk W k 2  D 2, that any pseudo-simple k-group G with root system BCn over ks satisfies Autk .G / D G .k/ (via conjugation). Without restriction on Œk W k 2  but assuming G is of minimal type and pseudo-split, Autk .G / is computed by Proposition 6.2.2 and [CGP, Prop. 9.8.15] and the natural map G .k/ ! Autk .G / is generally not a surjection when Œk W k 2  > 2 (see [CGP, Ex. 9.8.16]) though it is injective since ZG D 1 by [CGP, Prop. 9.4.9].

4.3 Groups locally of minimal type In our study of the structure and classification of pseudo-reductive groups, we want to assume less than that G is of minimal type, since over any imperfect field there are standard absolutely pseudo-simple groups that are not of minimal type (Example 2.3.5). It is also too strong to assume every .Gks /c is of minimal

4.3 Groups locally of minimal type

63

type, as this also fails, even over k of characteristic 2 for which Œk W k 2  D 2 (see Example 2.3.5 with p D 2). The hypothesis we shall use is: Definition 4.3.1. A pseudo-reductive group G over a field k is locally of minimal type if, for a maximal torus of Gks , the pseudo-simple ks -subgroup of rank 1 generated by any pair of opposite root groups is a central quotient of an absolutely pseudo-simple ks -group of minimal type (necessarily of rank 1). This definition is of most interest when k is imperfect of characteristic 2 because if either k is perfect or char.k/ ¤ 2 then G is necessarily locally of minimal type. Indeed, the case of perfect k is obvious (as then G is reductive), so suppose char.k/ ¤ 2. We may assume k D ks and that G is pseudo-simple of rank 1. The root system is reduced since char.k/ ¤ 2, so it is A1 . Hence, G is of minimal type by Proposition 3.1.9. In general, the description of iGc in 3.2.2 implies that a pseudo-reductive group of minimal type is locally of minimal type. It is a tautology that every commutative pseudo-reductive k-group is locally of minimal type (as the root system is empty in such cases), or this could be regarded as a convention. Remark 4.3.2. Assume G is not locally of minimal type. As we saw above, char.k/ D 2 and k is imperfect, so Œk W k 2  > 2. We claim that Œk W k 2  > 2, and that if moreover Gks has a reduced root system then Œk W k 2  > 16. To prove these claims we may assume k D ks ¤ k and G is pseudo-simple of rank 1. If the root system is non-reduced (so BC1 ) and Œk W k 2  D 2 then G is of minimal type by [CGP, Thm. 9.9.3(1)]; counterexamples whenever Œk W k 2  > 2 are given in §B.4. If the root system is reduced and Œk W k 2  6 8 then G is locally of minimal type by Proposition B.3.1. When Œk W k 2  > 16, there exist pseudo-split absolutely pseudo-simple kgroups G not locally of minimal type that have root system Bn or Cn for any n > 2 (we saw the same for n D 1 in 4.2.2). Indeed, for any subfield K  k 1=2 with degree 16 over k, in Appendix B we give a pseudo-split absolutely pseudosimple k-group G with root system Bn or Cn and minimal field of definition K=k for Ru .Gk /  Gk such that for some root a the subgroup Ga is of the form H WD HV;C =k as in 4.2.2 that is not locally of minimal type, where V is a certain 6-dimensional subspace of K and C is the Clifford algebra of V relative to its squaring map into k. Thus, G is not locally of minimal type. Proposition 4.3.3. Let G be pseudo-semisimple and T  G a maximal k-torus. (i) If T is split and  is a basis of ˆ.G; T / then C WD ZG .T / is generated by fCa ga2 for Ca WD Ga \ ZG .T / D ZGa .T \ Ga /.

64

Central extensions and groups locally of minimal type

(ii) If G is locally of minimal type and G ss is simply connected then G is of k Q minimal type, and if moreover T is split then a2 Ca  C D ZG .T / is an isomorphism for any basis  of ˆ.G; T /. The equality Ga \ ZG .T / D ZGa .T \ Ga / for a 2 ˆ.G; T / follows from the description of Ga provided by [CGP, Prop. 3.4.1(2),(3)] and the behavior of Cartan subgroups under intersection against derived groups in [CGP, Lemma 1.2.5(ii)]. The product structure in (ii) on ZG .T / generalizes the familiar fact that coroots associated to a basis of the root system form a Z-basis of the cocharacter lattice of a split maximal torus in a simply connected semisimple group. Proof. We may assume k D ks . Let  be a basis of ˆ D ˆ.G; T /, and let C  C be the k-subgroup generated by fCa ga2 , so C normalizes Ga for all a 2 . For each a 2 , the k-group Ha D C
Ga contains T and ZHa .T / D C . By [CGP, Thm. C.2.29(i)], for the k-subgroup H  G generated by fHa ga2 we have ZH .T / D C . But NH .T /.k/ contains representatives of reflections that generate W .ˆ/, so H contains Ga for every a 2 ˆ and thus H D G. Hence, C D ZG .T / D C , proving (i). Now we assume that G is locally of minimal type and G ss is simply conk nected. Letting K=k be the minimal field of definition for Ru .Gk /  Gk , the ss K-group G 0 WD GK is simply connected and semisimple with maximal torus 0 T WD TK . To prove that G is of minimal type, we need to show that .ker iG / \ C is trivial for the natural map iG W G ! RK=k .G 0 /. By [CGP, Prop. 3.2.10, Thm. 2.3.10], ˆ0 WD ˆ.G 0 ; T 0 / coincides with the set ˆnm of non-multipliable roots in ˆ. For each a 2 ˆ let a0 denote aK or 2aK depending on whether a is non-multipliable or multipliable respectively, so the quotient map GK  G 0 carries .Ua /K onto Ua0 0 . The set 0 D fa0 ga2 is a basis of ˆ0 , so since G 0 is simply connected we see that the natural map Q 0_ 0 GL a .ta / is a K-isomorphism. In particular, 1 ! T defined by .ta / 7! 0 the coroots associated to  constitute a basis of X .T 0 /. Applying RK=k then implies that the Cartan k-subgroup RK=k .T 0 / of RK=k .G 0 / is the direct product of its intersections RK=k .a0 _ .GL1 // with the pseudo-simple k-subgroups RK=k .Ga0 0 / ' RK=k .SL2 / generated by the ˙a0 -root groups for a0 2 0 . By 3.2.2, the map iG carries Ca into RK=k .a0 _ .GL1 // D RK=k .GL1 / via the composition of iGa with RKa =k .GL1 / ,! RK=k .GL1 /. The composite map Y a2

iG

Ca ! C ! RK=k .T 0 / D

Y a2

_

RK=k .a0 .GL1 //

4.3 Groups locally of minimal type

65

is the direct product of the maps iGa jCa (post-composed with inclusions). By 3.2.2, for every a 2 ˆ we have .Ga /ss D .Ga0 0 /K , and this subgroup is SL2 since k G 0 is simply connected. Thus, by Proposition 4.1.6, Ga is of minimal type for every a (due to the hypothesis that G is locally of minimal type). Hence, the restriction iGa jCa has trivial kernel for all a 2 , so iG jCa has trivial kernel for all a 2 . This implies that the multiplication map Y mW Ca ! C a2

has trivial kernel. We know from (i) that m is onto, so it is an isomorphism. Therefore, .ker iG / \ C is trivial and so G is of minimal type.  Remark 4.3.4. In [CGP, §9.8] the pseudo-split absolutely pseudo-simple kgroups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. Proposition 4.3.3(ii) shows that in this classification there is no effect if we relax the “minimal type” hypothesis to “locally of minimal type”.

5 Universal smooth k-tame central extension

5.1 Construction of central extensions In this chapter we construct canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. Definition 5.1.1. Let k be a field. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. For an affine k-group scheme G of finite type, a central extension 1!Z!E !G!1 with affine E of finite type is k-tame if Z is k-tame. Since Z might not be smooth or connected, the absence of nontrivial unipotent k-subgroups is much weaker than the condition that Z is of multiplicative e0 ! G 0 the type. As an example, let K=k be a finite extension of fields and q W G simply connected central cover of a connected semisimple K-group G 0 . For the finite k-group  WD ker q of multiplicative type, we obtain a central extension e0 / ! D.RK=k .G 0 // ! 1 1 ! RK=k ./ ! RK=k .G

(5.1.1)

by [CGP, Prop. 1.3.4]. There is no nontrivial unipotent k-subgroup j W U ,! RK=k ./ since j corresponds to a K-homomorphism UK !  that must be trivial as there are no nontrivial homomorphisms between unipotent and multiplicative type group schemes over a field [SGA3, XVII, Prop. 2.4]. Thus, (5.1.1) is a k-tame central extension. If K=k is not separable and  is not K-étale then RK=k ./ has positive dimension and is not of multiplicative type.

5.1 Construction of central extensions

67

Proposition 5.1.2. Let Z be a commutative affine k-group scheme of finite type and k 0 =k a separable extension field. Then Z is k-tame if and only if Zk 0 is k 0 -tame. Proof. For any commutative affine k-group scheme Z of finite type we claim that there exists a unipotent k-subgroup scheme Z unip of Z containing all others and that for any separable extension k 0 =k the inclusion .Z unip /k 0  .Zk 0 /unip is an equality. Since Z is commutative and an extension of a unipotent k-group scheme by a unipotent k-group scheme is unipotent [SGA3, XVII, Prop. 2.2(iv)], by choosing a unipotent k-subgroup scheme U  Z with maximal dimension we may pass to Z=U 0 so that all unipotent k-subgroup schemes of Z are 0-dimensional. If char.k/ D 0 (so Z is smooth) then Z 0 D T  U for a k-torus T and (smooth) connected unipotent k-group U , so necessarily U D 1 and Z 0 is a torus. Since all unipotent groups in characteristic 0 are connected, it follows that in such cases Z unip D 1 and likewise after any extension on k. Now we may and do assume char.k/ D p > 0. To prove the existence of Z unip as a finite k-subgroup of Z it suffices to show that for any commutative affine k-group scheme Z of finite type and sequence of unipotent k-subgroup schemes U1  U2  U3  : : : , the closed subscheme U  Z whose ideal is the inS tersection of those of the Ui ’s (denoted as i Ui and called the schematic union of the Ui ’s) is a unipotent k-subgroup scheme of Z. (Indeed, then dim U > 0 if fUi g is strictly increasing, yielding a contradiction if all unipotent k-subgroups of Z are finite.) It is elementary to check (or see [EGA, IV3 , 11.9.7.1]) that U  Ui0 D S S S .U i  Ui0 / for every i0 and U  U D i .U  Ui /, so U  U D i .Ui  Ui /. i Hence, U is a k-subgroup scheme of Z. For the purpose of proving U is S unipotent, we may replace Z with U so that Z D i Ui . For any closed ksubgroup scheme Z 0  Z, the faithful flatness of Z ! Z=Z 0 implies that S Z=Z 0 D i .Ui =.Ui \ Z 0 //, so Z=Z 0 cannot be of multiplicative type unless it is trivial. Setting Z 0 to be the kernel ker.FZ=k;n / of the n-fold relative Frobenius morphism of Z, by [SGA3, VIIA , Prop. 8.3] if we take n sufficiently large then Z=Z 0 is smooth. This commutative smooth affine k-group quotient of Z has no nontrivial quotient of multiplicative type, so it must be unipotent. For the k-groups Ga , ˛p , and Z=pZ, the relative Verschiebung morphism (in the sense of [SGA3, VIIA , 4.2–4.3]) vanishes. Thus, there are integers m; m0 > 0 such that V.Z=Z 0 /=k;m vanishes and every unipotent k-subgroup of the k-finite Z 0 has vanishing m0 -fold relative Verschiebung morphism, so VU=k;mCm0 D 0 for every unipotent k-subgroup U  Z. For example, VUi =k;mCm0 D 0 for every

68

Universal smooth k-tame central extension mCm0

0

.p mCm /

r

/ ! Z kills U i . Thus, VZ=k;mCm0 W Z .p for all i . But Z .p / D i S .pr / for any r > 0, so VZ=k;mCm0 D 0. Hence, Zk cannot contain ` for i Ui any prime ` whatsoever (treat ` D p separately), so Z is unipotent by [SGA3, XVII, Thm. 4.6.1(vi)]. This completes the construction of Z unip in general. It remains to show that the inclusion .Z unip /k 0  .Zk 0 /unip is always an equality (so Z unip is trivial if and only if .Zk 0 /unip is trivial). By Galois descent, unip .Z unip /ks D .Zks /unip and .Zk 0 /ks0 D .Zks0 /unip , so it is equivalent to check unip that the inclusion .Zks /ks0  .Zks0 /unip is an equality. Hence, we may replace k 0 =k with ks0 =ks to reduce to the case that k D ks . By passing to Z=Z unip we may arrange that Z unip is trivial and need to rule out the existence of a nontrivial unipotent k 0 -subgroup U 0  Zk 0 when k D ks . Assume there is such a U 0 . We may descend to the case where k 0 is finitely generated over k, so by separability k 0 is the fraction field of a smooth k-algebra A of finite type. By standard “spreading out” arguments, at the cost of shrinking S WD Spec.A/ around its generic point  D Spec.k 0 / we may find a closed flat S -subgroup scheme U 0  Z  S with generic fiber U 0 . If U 0 has nontrivial unipotent fibers over a dense open subset   S then since .k/ is non-empty (as k D ks and S is k-smooth) we could specialize at such a k-point to obtain a nontrivial unipotent k-subgroup of Z, a contradiction. To complete the proof of the second assertion of the lemma we shall prove that a flat finite type S-group scheme U 0 with nontrivial unipotent -fiber has nontrivial unipotent fibers over all points in a dense open subset of S . By [SGA3, XVII, Thm. 3.5], U0 has a finite composition series f1 D U00 


 Um0 D U0 g over  with m > 1 and Ui0 a normal -subgroup scheme of Ui0C1 such that Ui0C1 =Ui0 is isomorphic to Ga , ˛p , or a nontrivial p-torsion commutative finite étale k 0 -group. This provides a faithfully flat homomorphism from Ui0C1 onto Ga , ˛p , or a nontrivial p-torsion commutative finite étale group such that the kernel of the homomorphism is Ui0 . By shrinking S around , we may arrange that this extends to such a composition series fUj0 g over S . Every fiber Us0 therefore admits such a composition series and so is unipotent and nontrivial. 

Let G be a perfect smooth connected affine k-group, and consider two central extensions 1 ! Z ! E ! G ! 1; 1 ! Z ! E ! G ! 1; with E and E perfect smooth connected affine k-groups. There is at most one k-homomorphism f W E ! E over G since any two are related through multipli-

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cation against a k-homomorphism h W E ! Z and the only such h is the trivial homomorphism since E =.ker h/ is a perfect smooth connected affine k-group that is commutative and hence trivial. Moreover, such an f is surjective since f .E /  Z ! E is visibly surjective on k-points yet E.k/ is perfect. When f exists we say that E dominates E (as central extensions of G). Theorem 5.1.3. Let G be a perfect smooth connected affine group over a field k, ss with K=k the minimal field of definition for Ru .Gk /  Gk . Let G 0 WD GK and 0 0 e let G ! G be its simply connected central cover over K. Consider a k-tame central extension 1 ! Z ! E ! G ! 1 with E a perfect smooth connected affine k-group. (i) The minimal field of definition over k for Ru .Ek /  Ek is K, and the ss natural map EK DW E 0 ! G 0 is a central isogeny between connected semisimple K-groups. (ii) If 1 ! Z ! E ! G ! 1 is a k-tame central extension with E a perfect smooth connected affine k-group then E dominates E over G if and only if E 0 WD EKss dominates E 0 over G 0 .in which case E ! E is unique/. (iii) The functor E E 0 is an equivalence from the category of perfect smooth connected k-tame central extensions of G onto the category of connected semisimple central extensions of G 0 over K. In particular, up to unique isomorphism there is a unique k-tame central extension e!G !1 1!Z!G e is a perfect smooth connected affine k-group and G ess is simply such that G K connected. The formation of this central extension is functorial with respect to isomorphisms in G and separable extension of the ground field. Proof. The determination of the minimal field of definition over k in (i) is a special case of Proposition 3.2.6 since E is perfect. The quotient map E 0 ! G 0 between connected semisimple K-groups has kernel that is a quotient of the central K-subgroup scheme ker.EK ! GK /  EK (since R.Ek / ! R.Gk / is onto), so E 0 ! G 0 is a central isogeny. Part (i) is thereby proved. In the setting of (ii), if there is a k-homomorphism f W E ! E over G then f is surjective due to the perfectness of E since E.k/ D f .E .k//Z.k/ with central Z.k/  E.k/. Such an f is unique since the perfect smooth E has no nontrivial k-homomorphism to a commutative k-group scheme (such as the central Z  E). By surjectivity, f induces a K-homomorphism f 0 W E 0 ! E 0 as connected semisimple central covers of G 0 .

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The main task is to show that if f 0 is given over G 0 then there exists an f over G giving rise to f 0 . This is where we will use the k-tameness of the central extensions. The idea is to give a functorial reconstruction of E from E 0 (as central extensions). More specifically, there is an evident k-homomorphism E W E ! G D.RK=k .G 0 // D.RK=k .E 0 // DW G .E 0 / over G. In general, G .E 0 / might not be smooth, even if G is absolutely pseudosimple. (The difficulty arises from Cartan subgroups; see Example 5.1.6.) Recall that for any finite type k-group scheme H , there is a smooth closed k-subgroup H sm  H that contains all others and is final among smooth k-schemes equipped with a k-morphism to H [CGP, Lemma C.4.1]; note that this “maximal smooth closed k-subgroup” is generally smaller than the underlying reduced scheme of H (see [CGP, Rem. C.4.2]). Using this, the k-group scheme G .E 0 / reconstructs E from E 0 as follows (completing the proof of (ii)): Lemma 5.1.4. The map E carries E isomorphically onto D.G .E 0 /sm /. A refinement of the proof below shows that E represents the sheafified commutator of the (not necessarily smooth) fppf group sheaf G .E 0 /, but we do not need this and so do not discuss it. Proof. As a first step, we check that the scheme-theoretic kernel of E is trivial (so E is a closed immersion). This kernel is contained inside ker.E ! G/ D Z, so it has no nontrivial unipotent k-subgroup due to the k-tameness hypothesis on E as a central extension of G. But ker E is also contained in the kernel of the natural map h W E ! D.RK=k .E 0 //; so if ker h is unipotent then ker E is unipotent and hence trivial. The subgroup Ru .RK=k .E 0 /k /  RK=k .E 0 /k is defined over K by [CGP, Prop. A.5.11(1),(2)], so by [CGP, Prop. A.4.8] the geometric unipotent radical of D.RK=k .E 0 // is defined over K and Ru;K .D.RK=k .E 0 //K / D D.RK=k .E 0 //K \ Ru;K .RK=k .E 0 /K /: This yields a composite map hK

red red EK ! D.RK=k .E 0 //K  D.RK=k .E 0 //K D D.RK=k .E 0 /K / D D.E 0 /

D E0

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that is easily checked to be the natural projection whose kernel is Ru;K .EK /. Thus, ker hK is unipotent, so ker h is unipotent. This completes the proof that ker E D 1. We may now view E as a k-subgroup scheme of G .E 0 /, and hence as a k-subgroup scheme of D.G .E 0 /sm /. To show that D.G .E 0 /sm /  E, it clearly suffices to show that the commutator subgroup of G .E 0 /.k/ lies inside E.k/. But E.k/ ! G.k/ is surjective and the k-group scheme Z WD ker.pr1 W G .E 0 / ! G/ D ker.D.RK=k .E 0 // ! D.RK=k .G 0 ///  RK=k .ker.E 0 ! G 0 // is visibly central in G .E 0 /, so G .E 0 /.k/ D E.k/Z .k/ with Z .k/ central in G .E 0 /.k/. The commutator subgroup of G .E 0 /.k/ therefore coincides with E.k/.  5.1.5. Completion of the proof of Theorem 5.1.3. Since part (ii) of Theorem 5.1.3 has been proved, to establish part (iii) it suffices to show that for any connected semisimple central extension E 0 ! G 0 over K there is a k-tame central extension E of G with E a perfect smooth connected affine k-group red such that EK ' E 0 as central extensions of G 0 . We will show that the smooth connected k-group E WD D..G .E 0 /sm /0 / works. As a first step, we show that E is perfect and a k-tame central extension e0 /=RK=k .E 0 / of G via pr1 . By [CGP, Prop. 1.3.4], D.RK=k .E 0 // D RK=k .G e0  E 0 /, and similarly for G 0 in place of E 0 (with G 0  for E 0 WD ker.G e0 / is perfect since G e0 is simply connected [CGP, E 0 ). (Recall that RK=k .G Cor. A.7.11].) Thus, e0 /=RK=k .E 0 //; G .E 0 / D G RK=k .G e0 /=RK=k . 0 / .RK=k .G G so G .E 0 / ! G is the quotient modulo the central closed k-subgroup 1  Z where Z WD RK=k .G 0 /=RK=k .E 0 /  RK=k .G 0 =E 0 / does not contain any nontrivial unipotent k-subgroup scheme (since G 0 =E 0 is of multiplicative type). Hence, if we show that G WD G .E 0 /sm is mapped onto G by pr1 , it would follow that G 0 is a smooth connected k-tame central extension of G and that G 0 D D.G 0 /
ZG 0 . An immediate consequence of this would be that the k-group E D D.G 0 / is perfect and is a k-tame central extension of G. To show that G is mapped onto G by pr1 , we may replace k by ks and seek

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a smooth connected k-subgroup H of G .E 0 / which maps onto G. To construct such an H , we shall use the “open cell” structure provided by the dynamic method for rather general smooth affine k-groups in [CGP, §2.1]. To this end, e0 be the unique maximal Klet T  G be a (split) maximal k-torus and let T 0 0 e lifting the maximal K-torus T WD TK in the central quotient G 0 of torus in G e0 . The k-group iG .T / is a maximal k-torus of RK=k .G e0 /=RK=k .G 0 / since it G e in RK=k .G e0 /. (Explicitly, T e is the is the image of a (unique) maximal k-torus T 0 e maximal k-torus in RK=k .T /.) For any cocharacter  2 X .T /, if we multiply it by a sufficiently divisible positive integer then we can arrange that the map e/Q D X .T e0 /Q X .T /Q  X .iG .T //Q D X .T e0 /. Choose such a  2 X .T / that does not annihilate any carries  into X .T nonzero T -weights that occur in Lie.G/, so ZG ./ D ZG .T /. Thus, we have an open subscheme  D UG . /  ZG .T /  UG ./  G via multiplication (see [CGP, Prop. 2.1.8]), with UG .˙/ a unipotent smooth connected k-group. The k-groups UG ./ and UG . / generate G. Indeed, the smooth closed ksubgroup N of G generated by UG ./ and UG . / is normalized by the open cell  and hence is normal in G. The composite homomorphism ZG .T / ! G=N is dominant and thus surjective, so G=N is a perfect smooth connected affine k-group with a central maximal k-torus. Such a group is trivial, so N D G as claimed. To simplify the notation, define the central quotient e0 /=RK=k .E 0 / QE 0 D RK=k .G and likewise for QG 0 . There is a natural central quotient map q W QE 0 ! QG 0 , and by [CGP, Cor. 2.1.9] the induced maps UQE 0 .˙/ ! UQG 0 .˙/ are isomorphisms. The restrictions of their inverses to the k-subgroups iG .UG .˙//  UQG 0 .˙/ have associated graph morphisms that define homomorphisms j˙ W UG .˙/ ,! G QG 0 QE 0 D G .E 0 / whose compositions with pr1 are the natural inclusions UG .˙/ ,! G that generate G. Thus, the smooth closed k-subgroup H of G .E 0 / generated by

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73

j˙ .UG .˙// maps onto G. red It remains to build an isomorphism EK ' E 0 as connected semisimple 0 central extensions of G . Via the inclusion E of E into the fiber product G .E 0 / in Lemma 5.1.4, we obtain a K-homomorphism EK ! D.RK=k .E 0 //K  E 0

(5.1.5)

over GK  G 0 . Thus, surjectivity of E ! G and the isogeny property for E 0 ! G 0 imply that EK ! E 0 is surjective (since E 0 is connected), so we obtain a red K-homomorphism f W EK ! E 0 over G 0 . Since f is a map between connected semisimple central extensions of G 0 , ker f must be finite of multiplicative type. To prove f is an isomorphism, it suffices to show that ker f is unipotent. It is equivalent to prove that EK ! E 0 has unipotent kernel. Since the second map in (5.1.5) has unipotent kernel, it suffices to show that the first map in (5.1.5) has unipotent kernel. In other words, for pr2 W G .E 0 / ! E 0 we want pr2 jE to have unipotent kernel. But ker.pr2 / D ker iG is unipotent. This completes the proof of Theorem 5.1.3.  Example 5.1.6. Here are examples for which the G .
/-construction in the proof of Theorem 5.1.3 is not smooth. Let k be imperfect of characteristic p, and 2 choose t 2 k k p . Define k 0 D k.t 1=p / and K D k.t 1=p /. Let G D RK=k .SLp /=Rk 0 =k .p /: As explained in [CGP, Ex. 5.3.7] more generally, this is absolutely pseudosimple and its geometric unipotent radical has minimal field of definition over k ss e0 D SLp . Since RK=k .SLp / equal to K. Clearly G 0 WD GK is equal to PGLp , so G is perfect, we have e0 / D G R .PGL / RK=k .SLp /: G .G p K=k This is isomorphic to Q  RK=k .SLp / for Q WD RK=k .p /=Rk 0 =k .p /. The kgroup Q has positive dimension and no nontrivial ks -points, so it is not smooth. (See [CGP, Ex. 5.3.7] for a detailed discussion of a generalization of Q, including a proof that Q.ks / is trivial.)

5.2 A universal construction For a perfect smooth connected affine group G over a field k, we call the canone of G in Theorem 5.1.3 the universal smooth k-tame ical central extension G

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Universal smooth k-tame central extension

central extension. To see that this terminology is reasonable, consider a k-tame central extension 1!Z!E !G!1 with E a smooth affine k-group. The k-group D j .E 0 / is another such extension of G for any j > 0. For sufficiently large j , D j .E 0 / stabilizes and thus is e as k-tame central extensions of perfect, so it receives a unique map from G e G. This establishes universality for G among all smooth affine k-tame central extensions of G. e is its simply conRemark 5.2.1. If G is a connected semisimple group then G e is pseudo-semisimple. nected central cover. If G is pseudo-semisimple then G e of G e must Indeed, the unipotent smooth connected normal k-subgroup Ru;k .G/ e  Z WD have trivial image in the pseudo-reductive quotient G, so Ru;k .G/ e e ker.G ! G/. By k-tameness, this forces Ru;k .G/ to be trivial. Proposition 5.2.2. Let G be a perfect smooth connected affine group over a field k, and let E be a perfect smooth connected k-tame central extension of G. If G admits a Levi k-subgroup L then E admits a unique Levi k-subgroup L.E/ over L. Proof. Let K=k be the minimal field of definition for the geometric unipotent radical of G, so it is also the minimal such field for E (Theorem 5.1.3(i)). Let red E 0 D EK , so E 0 is a connected semisimple group that is a central extension of red G 0 D GK . By Lemma 5.1.4, the natural map E ! D.G .E 0 /sm / is an isomorphism, where G .E 0 / WD G D.RK=k .G 0 // D.RK=k .E 0 //: The natural map LK ! G 0 is an isomorphism (as L is a Levi k-subgroup of G), so L is a k-descent of G 0 . e ! G be the universal smooth k-tame central extension, so there Let G e ! E over G and the induced map G e0 WD is a unique homomorphism f W G ered ! E 0 is the simply connected central cover (Theorem 5.1.3(iii)). The simG K e ! L is uniquely identified (over L) ply connected semisimple central cover L e0 . Note that C WD ker.G e0 ! G 0 / is of multiplicative type as a k-descent of G e with k-descent ker.L ! L/, and all K-subgroups of C descend uniquely to ke since it suffices to check the analogue for the étale Cartier duals subgroups of L (and the purely inseparable extension K=k induces an isomorphism between absolute Galois groups). Hence, the intermediate central cover E 0 ! G 0 uniquely descends to a central cover L.E/ ! L. In particular, L.E/ is a Levi k-subgroup of D.RK=k .L.E/K // D D.RK=k .E 0 //.

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Likewise, L is identified as a Levi k-subgroup of D.RK=k .G 0 // in an analogous way, yielding a k-subgroup inclusion L.E/ D L L L.E/ ,! G D.RK=k .G 0 // D.RK=k .E 0 // D G .E 0 /; so L.E/  D.G .E 0 /sm / D E. By construction, the resulting composite map red L.E/K ! EK  EK D E 0 is the isomorphism which defines L.E/ as a k0 descent of E , so L.E/ is a Levi k-subgroup of E. Moreover, by design E ! G carries L.E/ into L via a map L.E/ ! L that descends E 0  G 0 , so L.E/ maps onto L. This completes the existence proof. It remains to show that L.E/ is the only Levi k-subgroup of E over L. If N  E is a Levi k-subgroup over L then the resulting map N ! L must be a k-descent of the quotient map E 0 ! G 0 due to the Levi subgroup property, so e over L. the simply connected central cover of N is uniquely identified with L 0 0 e e e Since ker.L ! N / is a k-descent of ker.G ! E /, it equals ker.L ! L.E//. e so Hence, uniquely N ' L.E/ over L and this isomorphism is as quotients of L, 0 via the identification of E as NK and as L.E/K we see that N D L.E/ inside D.RK=k .E 0 //. The perfect smooth N and L.E/ are each carried onto L under the central quotient map E ! G, so N D L.E/ inside D.G .E 0 /sm / D E. 

e 5.3 Properties and applications of G We begin by recalling the following useful result (which is [CGP, Prop. 2.2.12]). Proposition 5.3.1. Let f W G ! G be a surjective homomorphism between smooth affine groups over a field k, and assume ker f is central in G. (i) For every maximal k-torus T  G, the schematic preimage f 1 .T / is commutative and contains a unique maximal k-torus T . This k-torus is maximal in G, and f .T / D T . The map T 7! T defines a bijection between the sets of maximal k-tori of G and G, with inverse T 7! f .T /. (ii) Assume G is connected and perfect. The schematic center ZG coincides with ZG =.ker f /. In particular, if G ! G is a central quotient map then the composite surjection G ! G has central kernel and G=ZG has trivial center. (iii) If G and G are pseudo-reductive then P 7! f .P / is a bijection between the sets of pseudo-parabolic k-subgroups of G and pseudo-parabolic ksubgroups of G, with inverse P 7! f 1 .P /.

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To establish good properties of the universal smooth tame central extension by means of Proposition 5.3.1, we begin with a lemma in the rank-1 case: Lemma 5.3.2. Let G be an absolutely pseudo-simple k-group of rank 1 over ks such that G is a central quotient of an absolutely pseudo-simple k-group H of e of G is of minimal type. Then the universal smooth k-tame central extension G e minimal type and G is pseudo-split if G is pseudo-split. e coincide, so if G is Proof. By Proposition 5.3.1(i) the k-ranks of G and G e is pseudo-split. For the rest of the proof pseudo-split (i.e., its k-rank is 1) then G we may and do assume k D ks . If G ss ' SL2 (as occurs whenever the root k system of G is non-reduced) then by Proposition 4.1.6 we have H D G, so G e D G as G ss is simply connected. Thus, it remains to is of minimal type, and G k ss consider the case G ' PGL2 , so the root system of G is A1 . k

By Proposition 3.2.6, G and H have the same minimal field of definition K over k for their geometric unipotent radicals. The root systems of G and H coincide, by [CGP, Prop. 2.3.15]. By Proposition 3.1.7, if char.k/ ¤ 2 then either (i) H ' RK=k .SL2 / or (ii) H ' RK=k .PGL2 / whereas if char.k/ D 2, there is a nonzero kK 2 -subspace V  K such that khV i D K and either (i0 ) H ' HV;K=k or (ii0 ) H ' PHV;K=k . Consider cases (i) and (i0 ), so the central quotient map H ! G has kernel contained in ZH  RK=k .2 /. Thus, this kernel is k-tame e as central extensions of G. But H ss ' SL2 , and H is uniquely dominated by G k e D H and therefore G e is of minimal type. In cases (ii) and (ii0 ) we have so G e coincides with RK=k .SL2 / or HV;K=k , ZH D 1, so G D H and the k-group G e giving once again that G is of minimal type.  Proposition 5.3.3. Let G be a pseudo-semisimple group over a field k. Then G e is of minimal type. is locally of minimal type if and only if G Proof. The property “locally of minimal type” is inherited by pseudo-reductive e is of minimal type (hence central quotients of pseudo-reductive groups, so if G locally of minimal type) then its central quotient G is locally of minimal type. e Conversely, assuming G is locally of minimal type, it suffices to show that G is locally of minimal type (due to Proposition 4.3.3). We may and do assume eG e be a maximal k-torus, and T  G the maximal k-torus k D ks . Let T e e/ identifies image of T in G. By [CGP, Prop. 2.3.15], the inclusion X.T / ,! X.T eT e/. For each a 2 ˆ.G; T /, let Ga  G and G ea  G e be the ˆ.G; T / with ˆ.G; k-subgroups generated by the ˙a-root groups. Root group considerations show ea maps onto Ga , thereby making G ea a k-tame central extension of Ga . that G

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77

ea /ss can be idene we see that .G Applying 3.2.2 to the pseudo-semisimple G k ess generated by a pair of opposite T e -root groups tified with the subgroup of G k k (relative to a in the non-multipliable case, and 2a in the multipliable case). Any ess is simply connected, such pair of opposite root groups generates SL2 since G k ea is the universal smooth k-tame central extension of Ga . By the very defso G inition of G being locally of minimal type, it then follows from Lemma 5.3.2 ea is of minimal type, so G e is locally of minimal type. that G  The preservation of centrality under composition in Proposition 5.3.1(ii) underlies the centrality of the composite map f in the following result. Proposition 5.3.4. Let G be a pseudo-semisimple group locally of minimal type e its universal smooth k-tame central extension, and fGi g the over a field k, G finite set of pseudo-simple normal k-subgroups of G. The composite central quotient map Y Y ei ! f W G Gi ! G Q ei ' G. e uniquely lifts to an isomorphism G Q

See [CGP, Prop. 3.1.8] for the proof that the Gi pairwise commute and Gi ! G is a central quotient map.

e implies that its formation Proof. The construction (or universal property) of G is compatible with separable extension of the ground field, so it is elementary to ei is absolutely pseudo-simple since k D ks , reduce to the case k D ks . Each G and each is of minimal type by Proposition 5.3.3. Hence, any Cartan k-subgroup ei lies inside a k-group of the form RK =k .Ti / for the minimal field of of G i ei and a maximal Ki definition Ki =k for the geometric unipotent radical of G ei . In particular, Z e is contained inside such a Weil restriction, torus Ti  G Gi Q ei is a perfect so ZG contains no nontrivial unipotent k-subgroup. Thus, G ei smooth connected k-tame central extension of G, and its geometric maximal Q ei /ss that is simply connected. By semisimple quotient is the direct product .G k e near the end of Theorem 5.1.3, it follows that the unique characterization of G Q ei is uniquely isomorphic to G e over G. G  Although inseparable Weil restriction does not generally preserve perfectness, the formation of the universal smooth k-tame central extension interacts well with derived groups of Weil restrictions: Proposition 5.3.5. Let k be a field, G an absolutely pseudo-simple k-group, e ! G the universal smooth k-tame central extension of G. Let K=k and  W G

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be the common minimal field of definition for the geometric unipotent radicals e define G 0 D G ss , and let G e0 ! G 0 be the simply connected central of G and G, K 0 cover of G over K. e is standard. In such cases (i) The k-group G is standard if and only if G 0 e ' RK=k .G e / over the canonical central quotient map RK=k .G e0 /  G. G (ii) Let k0  k be a subfield over which k is purely inseparable of finite degree. Consider the absolutely pseudo-simple k0 -groups G0 WD D.Rk=k0 .G// e0 WD D.Rk=k .G//. e The map G e0 ! G0 is the universal smooth and G 0 k0 -tame central extension of G0 , and if G is of minimal type then so are e0 . G0 and G e is standard then its central quotient G is stanProof. If the pseudo-semisimple G dard by [CGP, Rem. 1.4.6]. Conversely, any pseudo-reductive central extension of a standard pseudo-reductive group is standard (Lemma 3.2.8). Since K=k is the minimal field of definition for the maximal geometric reductive quotient of e and G ess D G e0 , the desired description of G e in the standard case in (i) follows G, K e from Proposition 3.2.7 (applied to G). For (ii), if G is of minimal type then so is Rk=k0 .G/ (Example 2.3.9) and e ! G be hence so is its normal derived group G0 (by Lemma 2.3.10). Let  W G the natural central quotient map, and define 0 D D.Rk=k0 .//. Clearly 0 has central kernel, and it is also surjective (since to prove surjectivity we may assume k D ks , over which surjectivity is verified via consideration of root groups). To prove that 0 is the universal smooth k0 -tame central extension of G0 , it is harmless to extend the ground field k0 to a separable closure, so k D ks . Let  denote the k-tame ker , so 0 WD ker 0 is contained in the k0 -group Rk=k0 ./ that is k0 -tame since  is k-tame. It follows that 0 is k0 -tame, so e0 is a perfect smooth connected k0 -tame central extension of G0 . By [CGP, G e0 /k ! G e induces an isomorProp. A.5.11(2), Prop. A.4.8], the natural map .G e0 has simply phism between the maximal geometric semisimple quotients, so G e connected maximal geometric semisimple quotient. Thus, G0 is the universal smooth k0 -tame central extension of G0 , so Proposition 5.3.3 implies that if G0 e0 . This proves (ii). is of minimal type then so is G 

6 Automorphisms, isomorphisms, and Tits classification

6.1 Isomorphism Theorem The proofs of our main classification results (Theorems 6.3.11, 8.4.5, and 9.2.1) require understanding isomorphisms among pseudo-reductive groups, so we now establish a version of the Isomorphism Theorem for pseudo-split pseudoreductive groups. A pseudo-reductive variant of the Isogeny Theorem for split connected semisimple groups is proved in Appendix A. The key to both proofs is a technique in [CGP, Thm. C.2.29] to construct pseudo-reductive subgroups of an ambient smooth affine group. Theorem 6.1.1 (Isomorphism Theorem). Let k be a field and let the pairs .H; S/ and .H 0 ; S 0 / be pseudo-split pseudo-reductive k-groups equipped with split maximal k-tori. Suppose there is given an isomorphism f W ZH .S/ ' ZH 0 .S 0 / restricting to an isomorphism fS W S ' S 0 carrying a basis 0 of ˆ0 WD ˆ.H 0 ; S 0 / to a basis  of ˆ WD ˆ.H; S/. Assume in addition the following: (i) For every a 2  and the corresponding a0 2 0 via fS , an isomorphism fa W Ha ' Ha0 0 is given that is equivariant via fS for the actions of the k-tori S and S 0 , where Ha D hUa ; U a i and Ha0 0 D hUa0 0 ; U 0 a0 i. (ii) For every a 2 , fS ı a_ D a0 _ . Then there is a unique k-isomorphism H ' H 0 extending f and ffa ga2 . Since any k-automorphism of SL2 or PGL2 that restricts to the identity on the diagonal torus is easily checked to arise from the action of a unique diagonal element of PGL2 .k/, the interested reader may verify without difficulty that the Isomorphism Theorem for pinned split connected reductive groups is a consequence of Theorem 6.1.1.

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Proof. Our proof will use [CGP, Thm. C.2.29], which was suggested by a result of Steinberg that he used to give a new proof of the Isomorphism Theorem in the reductive case. The maximality of S and S 0 as k-tori forces ZH .S/ and ZH 0 .S 0 / to be commutative. By [CGP, Thm. C.2.29], it suffices to show that fa and f glue (necessarily uniquely) to an isomorphism Ha
ZH .S/ ' Ha0 0
ZH 0 .S 0 / for all matching pairs .a; a0 /. Since Za WD Ha \ ZH .S/ is the centralizer of the S -action on Ha , it is smooth and connected and inherits pseudo-reductivity from ZH .S/. Similarly, the centralizer Za0 0 of the S 0 -action on Ha0 0 is pseudo-reductive. The hypothesis in (i) gives S-equivariance of fa , so fa .Za / D Za0 0 . This isomorphism fa W Za ' Za0 0 between commutative pseudo-reductive k-groups carries the unique maximal k-torus S \ Ha of Za onto the unique maximal k-torus S 0 \ Ha0 0 of Za0 0 . By hypothesis (ii), fS also carries a_ .GL1 / D S \ Ha onto a0 _ .GL1 / D S 0 \ Ha0 0 . Now the S -equivariance of fa implies that for all s 2 S \ Ha , fS .s/ 1 fa .s/ belongs to the center of Ha0 0 . But since Ha0 0 is pseudo-simple, we conclude using [CGP, Lemma 1.2.1] that fS .s/ 1 fa .s/ D 1 for all s 2 S \ Ha . Hence, f jS \Ha D fS jS \Ha D fa jS \Ha . Since ZH 0 .S 0 / is a commutative pseudoreductive group, the two homomorphisms f jZa and fa jZa that carry Za into ZH 0 .S 0 / and coincide on the maximal torus S \ Ha of Za are equal by [CGP, Prop. 1.2.2]. Hence, f carries Za into Za0 0 via fa W Za ' Za0 0 . The anti-diagonal embedding of Za into Ha o ZH .S/ has central image, so we have a central quotient presentation .Ha o ZH .S//=Za ' Ha
ZH .S/ and similarly for .H 0 ; S 0 ; a0 /. Since f and fa restrict to the same isomorphism Za ' Za0 0 , to glue them to an isomorphism Ha
ZH .S/ ' Ha0 0
ZH 0 .S 0 / it remains to check that fa is equivariant with respect to f (i.e., fa .zhz 1 / D f .z/fa .h/f .z/ 1 for h 2 Ha and z 2 ZH .S/). We may assume k D ks . It suffices to show for z 2 ZH .S/.k/ that the map Ha ! Ha0 0 defined by h 7! f .z/

1

fa .zhz

1

/f .z/

coincides with fa . By pseudo-reductivity of Ha0 0 and [CGP, Prop. 1.2.2], such an equality of k-homomorphisms is equivalent to equality for the induced isomorphisms .Ha /red ' .Ha0 0 /red between maximal geometric reductive quotients. k

k

The map .Ha /red ! .Ha0 0 /red induced by fa is equivariant with respect to .fS /k , k

k

so it suffices to show that the action of ZH .S/k on .Ha /red factors through its k maximal torus quotient that is also the image of Sk , and likewise for .H 0 ; S 0 ; a0 /.

6.1 Isomorphism Theorem

81

By [CGP, Prop. A.4.8], the quotient .Ha /red of .Ha /k is the image of .Ha /k k

in H red , so it remains to observe that the image of ZH .S/k D ZHk .Sk / under k

the quotient map Hk  H red is the centralizer of Sk , and that this centralizer is k

Sk since H red is connected reductive. k



Let G be a pseudo-reductive group over a field k, and let C be a Cartan k-subgroup of G. We shall now use the Isomorphism Theorem for pseudo-split pseudo-reductive groups to prove in general the connectedness of the (commutative) maximal smooth closed k-subgroup ZG;C of AutG;C (see 4.1.2). This requires the following useful lemma: Lemma 6.1.2. Let G be a pseudo-reductive group over a field k and Z  G a closed central k-subgroup such that G WD G=Z is pseudo-reductive. Let C  G be a Cartan k-subgroup, and C D C =Z its Cartan k-subgroup image in G. (i) If C 0 WD C \ D.G/ is the associated Cartan k-subgroup of D.G/ then the natural map ZG;C ! ZD.G/;C 0 is an isomorphism. (ii) For the natural map ˛ W AutG;C ! AutG;C , the induced map ZG;C ! ZG;C is an isomorphism. By [CGP, Lemma 1.2.5(iii), Prop. 1.2.6] the map C 7! C \ D.G/ is a bijection between the sets of Cartan k-subgroups of G and D.G/. An interesting special case of (ii) is the central quotient G D G=CG as in §2.3. Proof. Since G D .D.G/ o C /=C 0 (using the central anti-diagonal inclusion of C 0 ), it is clear that the restriction map ZG;C ! ZD.G/;C 0 between smooth affine k-groups induces a bijection between points valued in every separable extension of k. Hence, this map is an isomorphism (as explained in the proof of [CGP, Prop. 8.2.6]), so (i) is proved. By (i), the natural map of k-group schemes  W AutD.G/;C 0 ! AutG;C has restriction  sm W ZD.G/;C 0 ! ZG;C between maximal smooth closed ksubgroups that is an isomorphism. (In fact,  is an isomorphism of k-group schemes because D.G/ represents the fppf sheafification of the commutator subfunctor of G on the category of k-schemes, but we do not need this stronger property of .) The same holds compatibly for the pair .G; C /, so for the proof of (ii) we may replace G with D.G/ to reduce to the case that G is pseudosemisimple. Hence, Z.ks / is finite.

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Since k is arbitrary and the formation of ˛ commutes with separable extension on k, to prove (ii) it suffices to prove ˛ is bijective on k-points. By Galois descent we may assume k D ks , so the maximal k-torus T of C is split. It remains to show that any k-automorphism f of G restricting to the identity on C uniquely lifts to a k-automorphism f of G restricting to the identity on C . Let T be the image of T under the central quotient map q W G ! G, so q induces an isomorphism between the root systems ˆ.G; T / and ˆ.G; T / [CGP, Prop. 2.3.15]. If  2 X .T /  X .T / does not annihilate any roots then q carries C D ZG ./ onto C D ZG ./ and carries UG .˙/ isomorphically onto UG .˙/. Since any k-automorphism of G restricting to the identity on C must restrict to an automorphism of each root group, and likewise for .G; C /, the only possibility for f is a k-automorphism of G extending the automorphism f 0 of  WD UG . /  C  UG ./ given by uzu0 7! f .u/zf .u0 / via the isomorphism q W UG .˙/ ' UG .˙/. To construct f extending f 0 , we just have to check that f 0 is a rational homomorphism; i.e., the map h W .  / \ mG1 ./ ! G given by .!1 ; !2 / 7! f 0 .!1 !2 /f 0 .!2 / 1 f 0 .!1 / 1 is the constant map equal to 1. Obviously q ı h D 1, so h is valued in ker.q/ D Z. But Z.k/ is finite whereas .  / \ mG1 ./ is connected with a Zariski-dense set of k-points, so h is the constant map equal to 1.  As a preliminary step towards a description of ZG;C in general, first we need to understand the case where Gks has a rank-1 root system. The BC1 -case is part of [CGP, Prop. 9.8.15], so now we address the A1 -case: Lemma 6.1.3. Let G be absolutely pseudo-simple with minimal field of definition K=k for its geometric unipotent radical, and assume Gks has root system A1 . Let T be a maximal k-torus in G and define C D ZG .T /, T ad D ss T =.T \ ZG /, and G 0 D GK . Let F  K be the root field of G. The natural action on RK=k .G 0 / by the k-group RF =k .TFad /  RK=k .TKad / uniquely lifts through iG to an action on G that is trivial on C , and the khomorphism f W RF =k .TFad / ! ZG;C classifying this action is an isomorphism. Proof. The formation of ZG is compatible with passing to central quotients by Proposition 5.3.1(ii), and likewise for the formation of root fields by Remark 3.3.3. Thus, by Lemma 6.1.2(ii) we can replace G with G=CG (and T with its image in this quotient) to arrange that G is of minimal type. We can then e and replace T replace G with its universal smooth k-tame central extension G e e with the corresponding maximal k-torus T  G via Proposition 5.3.1(i) so that ss GK ' SL2 .

6.1 Isomorphism Theorem

83

In view of the uniqueness assertion, by Galois descent we may now assume k D ks , so T is split. By [CGP, Thm. 3.4.6] there exists a Levi k-subgroup of G containing T , and we may identify it with SL2 over k carrying T onto the diagonal k-torus. Thus, via Proposition 3.1.8, if char.k/ ¤ 2 then iG identifies G with RK=k .SL2 / (carrying T to the diagonal k-torus) whereas if char.k/ D 2 then iG identifies G with HV;K=k  RK=k .SL2 / for a nonzero kK 2 -subspace V  K containing 1 such that kŒV  D K and T is identified with the diagonal k-torus in HV;K=k . Define V D K if char.k/ ¤ 2, so in all cases the T -root groups of G D iG .G/  RK=k .SL2 / are the subgroups UV˙ corresponding to V  K inside the (upper and lower triangular) root groups of RK=k .SL2 /. Under the natural action of RK=k .PGL2 / on RK=k .SL2 /, the diagonal ksubgroup RF =k .GL1 /  RK=k .PGL2 / preserves the root groups UV˙ of iG .G/ that generate iG .G/. This defines an action of RF =k .GL1 / on iG .G/ D G that is trivial on C and compatible with the natural action on RK=k .SL2 /. This is exactly an action of RF =k .TFad / on G extending the identity on C and equivariant through iG with the natural action on RK=k .G 0 /, and it is unique as such because uniqueness holds on k-points (since any k-automorphism of G is uniquely determined by the induced K-automorphism of G 0 [CGP, Prop. 1.2.2]). We have proved existence and uniqueness of the desired f W RF =k .TFad / ! ZG;C and need to prove that f is an isomorphism. Recall the general fact (reviewed in the proof of [CGP, Prop. 8.2.6]) that a k-homomorphism between smooth k-groups of finite type is an isomorphism if it is bijective on points valued in every separable extension k 0 =k, and that we may restrict attention to separably closed k 0 . The formation of f and F=k commutes with separable extension on k, so we may rename such k 0 as k to reduce to checking that f is bijective on k-points. In other words, it suffices to show that for any k-automorphism  of G that is trivial on C , the induced automorphism of RK=k .G 0 / arises from a unique element of T ad .F /. Since jC D idC , the induced K-automorphism  0 of G 0 restricts to the identity on the image TK of CK and so is induced by a unique t 0 2 T ad .K/; i.e., t 0 is diagonal in PGL2 .K/ D AutK .G 0 /. Writing t 0 D diag.; 1/ for  2 K  , the effect of t 0 on the TK -root lines in sl2 .K/ is multiplication by ˙1 , and this has to preserve the subspaces arising from the T -root groups of G. Thus, multiplication by ˙1 on K preserves V ; i.e.,  2 F . Hence, t 0 2 T ad .F /.  By [CGP, Thm. 2.4.1], if G is a pseudo-reductive k-group and C is a Cartan 0 k-subgroup then ZG;C is pseudo-reductive. Here is an improvement: Proposition 6.1.4. The k-group ZG;C is connected. In particular, ZG;C is pseudo-reductive. Moreover, ZG;C is of minimal type .equivalently, ZG;C does

84

Automorphisms, isomorphisms, and Tits classification

not contain a nontrivial unipotent k-subgroup scheme/, and if G is pseudo-split with basis  of ˆ.G; T / for a split maximal k-torus T  G then naturally Y ZG;C ' ZGa ;Ca (6.1.4) a2

for Ga WD hUa ; U a i, Ta WD T \ Ga a split maximal k-torus in Ga , and Ca WD ZGa .Ta / a Cartan k-subgroup of Ga . Proof. Everything is obvious if G is commutative (equivalently, G D C ), so we may and do assume G is non-commutative. By Lemma 6.1.2(i) the formation of ZG;C is unaffected by replacing G with D.G/, so we may assume G is pseudosemisimple. By Lemma 6.1.2(ii) with Z D CG , we may assume G is of minimal type. Likewise, we can pass to the universal smooth k-tame central extension of G so that G ss is simply connected. It suffices to treat the case when the maximal k k-torus T in C is split (via scalar extension to the splitting field of T ). The idea is to reduce to the case where G is pseudo-semisimple of rank 1, and then to carry out a direct analysis in the rank-1 cases. Let  be a basis of ˆ WD ˆ.G; T /, so for a 2  a k-automorphism of Ga restricts to the identity on Ca if and only if it restricts to the identity on Ta (as Ca is commutative and pseudo-reductive). For any k-algebra A and f 2 AutG;C .A/, the functorial characterization of root groups over k-algebras implies that f carries each .Ua /A into itself, hence isomorphically onto itself (by using f 1 ). Thus, likewise f carries .Ga /A isomorphically onto itself, so we obtain a natural Q k-homomorphism AutG;C ! a2 AutGa ;Ca and hence a k-homomorphism as in (6.1.4) between maximal smooth closed k-subgroups. To prove that this latter homomorphism is an isomorphism it suffices to do so on rational points over all separable extensions of k (as explained in the proof of [CGP, Prop. 8.2.6]), and by Galois theory we may restrict attention to such extensions that are separably closed. Renaming such an extension as k reduces the isomorphism property for (6.1.4) to its bijectivity on k-points. This in turn is an immediate special case of the Isomorphism Theorem (Theorem 6.1.1) using the identity map on C since Q C D a2 Ca by Proposition 4.3.3. To show that ZG;C is conncted and of minimal type, it now suffices to treat the pairs .Ga ; Ca / separately, so we may assume that G is absolutely pseudosimple of minimal type and ˆ has rank 1. The k-group ZG;C is described explicitly in Lemma 6.1.3 when ˆ D A1 and in [CGP, Prop. 9.8.15] when ˆ D BC1 . In both cases, we conclude by inspection. 

6.1 Isomorphism Theorem

85

Example 6.1.5. As an application of the connectedness (and hence the pseudoreductivity) of ZG;C in Proposition 6.1.4, we may use the canonical map f W C ! ZG;C , composed with inversion, to obtain a central quotient G WD .G o ZG;C /=C

(6.1.5)

of G o ZG;C that is pseudo-reductive (by [CGP, Prop. 1.4.3]) with Cartan ksubgroup ZG;C . Since ker.f / D ZG , the image of G in G is G=ZG . (In Proposition 6.2.4 we will see that if G D D.G/ then G is the identity component of the maximal smooth k-subgroup of the automorphism scheme of G.) This construction leads to some instructive examples (not used in later proofs) over imperfect fields k of characteristic 2. To explain the context, recall that if Œk W k 2  D 2 then any pseudo-reductive k-group H has the unique decomposition H1  H2 where H1 has a reduced root system (over ks ) and H2 is pseudo-semisimple with root system (over ks ) whose irreducible components are non-reduced [CGP, Prop. 10.1.6]. If Œk W k 2  > 2 then such a unique decomposition exists when H is perfect of minimal type, as we may check over Q ks by using both the central quotient map Hi ! Hks for the pseudo-simple normal ks -subgroups Hi of Hks [CGP, Prop. 3.1.8, Prop. 3.2.10] and the triviality of the center of minimal type pseudo-split pseudo-simple groups whose irreducible root system is non-reduced [CGP, Prop. 9.4.9]. However, generally there is no pseudo-reductive direct factor H2 of H whose root system (over ks ) is the union of the non-reduced irreducible components of the root system of Hks when H is either (i) perfect but not of minimal type, (ii) non-perfect of minimal type. Examples as in (i) can be made over any k satisying Œk W k 2  > 16 as follows. Let G1 be a pseudo-split absolutely pseudo-simple k-group such that CG1 D ˛2 and the root system is reduced (see §B.1, §B.2, or 4.2.2). Let G2 be a pseudosplit absolutely pseudo-simple k-group such that CG2  ˛2 and the root system is non-reduced (see Example B.4.1, Remark B.4.2, and Example B.4.3). Pasting G1 and G2 along the central ˛2 gives a k-group G that is an extension of G1 =CG1 by G2 . The k-group G is clearly pseudo-semisimple, and it is not of minimal type since its normal k-subgroup G2 is not of minimal type. It is more interesting to give examples as in (ii), assuming Œk W k 2  > 2. We will do this via the pushout construction (6.1.5). For m  2 mod 4, define G1 D D.RK=k .PGLm // with a split maximal k-torus denoted as T1 , so ZG1 D 1 and the equality G1 D RK=k .SLm /=RK=k .m / identifies T1 with the maximal k-

86

Automorphisms, isomorphisms, and Tits classification

torus in C1 WD RK=k .T /=RK=k .m / for a unique (split) maximal K-torus T of SLm . The map C1 ,! ZG1 ;C1 and natural inclusion j W RK=k .T /=RK=k .m / ,! RK=k .T =m / are thereby identified. Since Œm.RK=k .GL1 // D RkK 2 =k .GL1 / (as m  2 mod 4), the description of m D ZSLm in terms of T -valued coroots identifies j with RkK 2 =k .GL1 /  RK=k .GL1 /m 2 ! RK=k .GL1 /m 1 : The pseudo-split pseudo-simple k-groups of minimal type with root system BCn (n > 1) are classified in [CGP, Thm. 9.8.6, Prop. 9.8.9] (these groups all have trivial center). Let .G2 ; T2 / be such a group as constructed in [CGP, Ex. 9.8.16]; the minimal field of definition K=k for its geometric unipotent radical may be any nontrivial purely inseparable finite extension of k, and kK 2 is then necessarily a proper subfield of K. Since we assume Œk W k 2  > 4, we may choose K so that ŒK W kK 2  > 4 (e.g., let K D k.u1=2 ; v 1=2 / for fu; vg part of a 2-basis of k). The inclusion C2 WD ZG2 .T2 / ,! ZG2 ;C2 is identified with the natural map RK=k .GL1 /n ,! RF\ =k .GL1 /  RK=k .GL1 /n 1 for a finite extension F\ =K contained in K 1=2 due to [CGP, 9.6.8, (9.7.6), Thm. 9.8.1(2), (9.8.6)]. Explicitly, G2 rests on an arbitrary choice of nonzero kK 2 -subspace V 0  K and a nonzero finite-dimensional K 2 -subspace V .2/  K such that V 0 \ V .2/ D 0 (with the additional requirement that khV .2/ C V 0 i D K when n D 1), and F\2 is the K 2 -finite subfield of elements of K whose multiplication action preserves V 0 and V .2/ ˚ V 0 . In [CGP, Ex. 9.8.16] it is explained how to choose V .2/ and V 0 so that F\2 strictly contains K 2 , and that property underlies the construction of G2 . But we shall require more: F\2 should also not be contained in kK 2 . (This would be impossible to arrange if ŒK W kK 2  D 2, as then V 0 would have to be a kK 2 line, forcing F\2  kK 2 .) Since ŒK W kK 2  > 4, we can choose a proper subfield E of K that strictly contains kK 2 . Let V 0 D E, pick ˛ 2 E kK 2 , and let V .2/ ˚ V 0 correspond to the K 2 .˛/-span of an E-basis of the quotient K=V 0 (so khV .2/ C V 0 i D K). Thus, F\2 contains the field K 2 .˛/ that strictly contains K 2 and is not contained in kK 2 . Define G WD G1  G2 (so ZG D 1), C WD C1  C2 , and consider a smooth connected k-subgroup C  ZG;C D ZG1 ;C1  ZG2 ;C2 (equality by Proposition 6.1.4) with C  C . The smooth connected k-subgroup H WD .G o C /=C  .G o ZG;C /=C

6.1 Isomorphism Theorem

87

is normal and hence pseudo-reductive, with D.H / D G (since ZG D 1). If H D H1  H2 (so each Hj is pseudo-reductive and pseudo-split) where H2 has root system that is the non-reduced component of that of H then D.Hj / D Gj and the Cartan k-subgroup C of H must have the form C1  C2 inside ZG;C for Cj lying between ZGj ;Cj and Cj . Hence, to rule out such a direct product decomposition for H it suffices to choose C that does not have the form C1  C2 . Equivalently, we seek a smooth connected k-subgroup of ZG;C =C D .RK=k .GL1 /=RkK 2 =k .GL1 //  .RF\ =k .GL1 /=RK=k .GL1 // that is not a direct product of k-subgroups of the two factors. The squaring map from the second factor into the first is nontrivial since F\2 6 kK 2 , so its graph € is such a k-subgroup. We will now prove that for any C  ZG;C containing C , the corresponding pseudo-reductive group H is of minimal type. It suffices to show that H does not contain a nontrivial central unipotent k-subgroup scheme (since CH is a central unipotent k-subgroup scheme). For this, first note that the natural projection G o C ! .G o C /=C D H carries C isomorphically onto a Cartan k-subgroup of H . The center of H is contained in any Cartan k-subgroup, and C is a k-subgroup of ZG;C . But ZG;C does not contain a nontrivial unipotent k-subgroup scheme since ZG;C D ZG1 ;C1  ZG2 ;C2 . This proves that H is of minimal type. 6.1.6. As an application of the Isomorphism Theorem and Lemma 6.1.2, we next address the behavior of ZG;C with respect to Weil restriction in the pseudoreductive case. To formulate this, we first review some constructions. Let k 0 be a nonzero finite reduced algebra over a field k and G 0 a smooth affine k 0 -group with pseudo-reductive fibers over the factor fields of k 0 . Let G D Rk 0 =k .G 0 / and define ZG 0 ;C 0 to be the commutative fiberwise maximal smooth k 0 -subgroup of the automorphism scheme AutG 0 ;C 0 over k 0 (built in the evident manner, working separately over the factor fields of k 0 ). By [CGP, Prop. A.5.15(3)], C 0 7! Rk 0 =k .C 0 / is a bijection from the set of Cartan k 0 subgroups of G 0 to the set of Cartan k-subgroups of G WD Rk 0 =k .G 0 /. Moreover, C 7! C \ D.G/ is a bijection between the sets of Cartan k-subgroups of G and D.G/ due to [CGP, Lemma 1.2.5(iii), Prop. 1.2.6]. Hence, each Cartan k-subgroup C of D.G/ has the form Rk 0 =k .C 0 / \ D.G/ for a unique Cartan k 0 subgroup C 0  G 0 and all such intersections are Cartan k-subgroups of D.G/. Fix a C in G, and consider the associated C 0 in G 0 and C WD C \ D.G/ in D.G/. By Lemma 6.1.2(i) ZD.G/;C ' ZG;C . To relate Rk 0 =k .ZG 0 ;C 0 / and

88

Automorphisms, isomorphisms, and Tits classification

ZG;C , observe that for any k-algebra A and Ak 0 -automorphism ' of GA0 0 that k restricts to the identity on CA0 0 , the map RAk0 =A .'/ is an A-automorphism of k RAk0 =A .GA0 0 / D GA that restricts to the identity on CA . This defines a homok morphism Rk 0 =k .AutG 0 ;C 0 / ! AutG;C of k-group schemes that must carry the commutative k-smooth closed subgroup Rk 0 =k .ZG 0 ;C 0 / into the commutative maximal k-smooth closed subgroup ZG;C , yielding k-homomorphisms Rk 0 =k .ZG 0 ;C 0 / ! ZG;C ! ZD.G/;C :

(6.1.6)

Proposition 6.1.7. The maps in .6:1:6/ are isomorphisms. Moreover, using C conjugation on D.G/, the resulting k-homomorphism C ! ZD.G/;C between commutative pseudo-reductive k-groups restricts to an isogeny between their unique maximal k-tori. Proof. By Lemma 6.1.2(i), the second map in (6.1.6) is an isomorphism. By the same reasoning as in the proof of Lemma 6.1.3 (making an arbitrary separable extension on the ground field, and renaming the new ground field as k), the first map in (6.1.6) is an isomorphism if it induces a bijection on k-points in general, and we may moreover assume k D ks (so all factor fields of k 0 are separably closed). By left-exactness of Weil restriction, G and its k-subgroup Rk 0 =k .D.G 0 // have the same derived groups, so we may assume G 0 D D.G 0 /. If fki0 g is the set of factor fields of k 0 and .Gi0 ; Ci0 / is the ki0 -fiber of .G 0 ; C 0 / Q Q then G D Gi and C D Ci for Gi WD Rki0 =k .Gi0 / and Ci D Rki0 =k .Ci0 /. Let T 0  C 0 be the unique maximal k 0 -torus, so if Ti0 is its ki0 -fiber then the maximal Q k-torus T of C is Ti where Ti is the maximal k-torus of Rki0 =k .Ti0 /. For the isomorphism assertion, our task is to show that every k-automorphism f of G restricting to the identity on C has the form Rk 0 =k .f 0 / for a unique k 0 automorphism f 0 of G 0 restricting to the identity on C 0 . Since we have arranged that each Gi0 is perfect and hence is generated by its Ti0 -root groups, any k 0 automorphism of G 0 restricting to the identity on C 0 (equivalently, on T 0 ) must arise from a collection of automorphisms fi0 2 ZGi0 ;Ci0 .ki0 /. Likewise, for the Cartan k-subgroups C WD C \ D.G/ and Ci WD Ci \ D.Gi / we have Y Y ZG;C .k/ D ZD.G/;C .k/ D ZD.Gi /;Ci .k/ D ZGi ;Ci .k/; so for both the isomorphism and torus isogeny assertions it suffices to treat each triple .ki0 =k; Gi0 ; Ci0 / separately; i.e., we can assume k 0 is a field. By Lemma 6.1.2(ii) it is harmless to replace G 0 with the pseudo-reductive central quotient G 0 =Z 0 for Z 0 D CG 0 since for the central k-subgroup scheme

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89

Z D G \Rk 0 =k .Z 0 / in G the inclusion G=Z ,! Rk 0 =k .G 0 =Z 0 / induces an equality on derived groups (as we see by considerations of root groups). Thus, we can assume that G 0 is of minimal type. Likewise, we can replace G 0 with its universal smooth k 0 -tame central extension so that its root datum is simply connected. The root systems of the pairs .G; T / and .G 0 ; T 0 / are naturally identified and the corresponding root groups are related through Weil restriction (see [CGP, Ex. 2.3.2]), so by the Isomorphism Theorem and Proposition 6.1.4 we are reduced to the rank-1 case. We now separately consider when the common root system ˆ of .G; T / and .G 0 ; T 0 / is A1 or BC1 . Suppose ˆ is A1 , and let K 0 =k 0 be the minimal field of definition for the 0 ss geometric unipotent radical of G 0 . Identify .GK with SL2 carrying TK0 0 to 0/ 0 the diagonal K -torus and carrying a chosen coroot to t 0 7! diag.t 0 ; 1=t 0 /. By Proposition 3.1.7, if char.k/ ¤ 2 then G 0 D RK 0 =k 0 .SL2 / whereas if char.k/ D 2 then G 0 D HV 0 ;K 0 =k 0 for a nonzero k 0 K 0 2 -subspace V 0  K 0 satisfying k 0 hV 0 i D K 0 . Letting F 0  K 0 be the root field of G 0 (so k 0  F 0 , and F 0 D K 0 if char.k/ ¤ 2), Lemma 6.1.3 gives that ZG 0 ;C 0 D RF 0 =k 0 .GL1 / with its action on G 0 induced by the diagonal of RK 0 =k 0 .PGL2 / via t 0 7! diag.t 0 ; 1/. If char.k/ ¤ 2 then G D RK 0 =k .SL2 / is perfect. If char.k/ D 2 then (3.1.6) gives D.G/ D HV 0 ;K 0 =k with khV 0 i D k 0 hV 0 i D K 0 . In both cases, by inspection the root field over k is also equal to F 0 and ZG;C is identified with RF 0 =k .GL1 /, under which the map of interest Rk 0 =k .ZG 0 ;C 0 / ! ZG;C is easily seen to be the natural isomorphism Rk 0 =k .RF 0 =k 0 .GL1 // ' RF 0 =k .GL1 / via calculation on k-points. Next consider the BC1 -case, so k is imperfect with characteristic 2. Once again let K 0 =k 0 be the minimal field of definition for the geometric unipotent radical of G 0 . An explicit description of .G 0 ; T 0 / as the derived group of a construction in terms of certain linear algebra data relative to K 0 =k 0 is given in [CGP, Thm. 9.8.6(i)], and its compatibility with passage to D.G/ D D.Rk 0 =k .G 0 // is given in [CGP, Prop. 9.8.13] using “the same” linear algebra data viewed relative to K 0 =k. Upon choosing a basis of the root system, an identification of ZG 0 ;C 0 with RF\0 =k 0 .GL1 / for a subfield F\0  K 0 1=2 over K 0 is given in [CGP, (9.8.7)]. The definition of F\0 in terms of the linear algebra data shows that it is also the field (now viewed over k) which arises in the description of ZG;C , and the construction of these identifications is easily seen to identify the map of interest Rk 0 =k .ZG 0 ;C 0 / ! ZG;C with the natural isomorphism Rk 0 =k .RF\0 =k 0 .GL1 // ' RF\0 =k .GL1 /. 

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6.2 Automorphism schemes Away from the reductive case, the consideration of automorphism schemes for pseudo-reductive groups has to be restricted to the pseudo-semisimple case because commutative pseudo-reductive groups that are not tori generally have a non-representable automorphism functor: Example 6.2.1. Let k be an imperfect field of characteristic p > 0, and let G D Rk 0 =k .GL1 / for a nontrivial finite extension k 0 =k satisfying k 0 p  k. Since Gk 0 D GL1  Gna for n D Œk 0 W k 1, if AutG=k is represented by a k-scheme AutG=k then AutGna =k 0 is represented by a closed subscheme of .AutG=k /k 0 . Thus, non-representability of AutG=k is reduced to the non-representability of AutGna =k 0 (on the category of k 0 -algebras). If this latter functor were representable then the representing scheme would be locally of finite type over k 0 , due to the functorial criterion in [EGA, IV3 , 8.14.2], so the nilradical would locally be killed by a single exponent. Hence, by considering additive polynomials with nilpotent coefficients, we see that representability cannot hold. Proposition 6.2.2. Let G be a pseudo-semisimple k-group. The functor on kalgebras AutG=k W A AutA .GA / is represented by an affine k-group scheme AutG=k of finite type. Proof. Let T be a maximal k-torus of G. Let StabAutG=k .T / be the subfunctor of AutG=k assigning to any k-algebra A the set of A-group automorphisms of GA that carry TA into itself, hence isomorphically onto itself by [EGA, IV4 , 17.9.6]; this is visibly a subgroup functor. Since G=ZG is identified with a normal subgroup functor of AutG=k that meets StabAutG=k .T / in NG .T /=ZG , we can form a non-commutative pushout subgroup functor as in [CGP, Rem. 1.4.5]: .G o StabAutG=k .T //=NG .T /  AutG=k :

(6.2.2)

But this inclusion of fppf group sheaves on the category of k-algebras is an equality due to the fppf-local conjugacy of maximal tori in smooth affine group schemes [SGA3, XI, Cor. 5.4], so it suffices to prove the representability of StabAutG=k .T / by an affine k-group scheme of finite type. There is a natural homomorphism of group functors StabAutG=k .T / ! AutT =k whose kernel is AutG;T . Since AutT =k is represented by an étale k-scheme AutT =k and AutG;T is represented by an affine k-group scheme AutG;T of finite

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91

type, we have a left-exact sequence of group functors f

1 ! AutG;T ! StabAutG=k .T / ! AutT =k on the category of k-algebras. Viewing the target as the automorphism scheme of the Galois lattice X D X.Tks /, the map f lands inside the automorphism scheme of the root system .ˆ; XQ / of .G; T /ks . (Note that ˆ spans XQ because G is pseudo-semisimple.) But this latter automorphism scheme is finite, so by affineness of AutG;T and calculations over a finite Galois splitting field of T we can represent StabAutG=k .T / by the Galois descent of a disjoint union of finitely many AutG;T -torsors.  The k-group AutG=k is generally not smooth when the pseudo-semisimple G is not semisimple, due to the existence of k=k-forms of G that are not pseudoreductive: Example 6.2.3. Let k be an imperfect field of characteristic p > 0 and G D Rk 0 =k .Gk 0 / for a purely inseparable extension k 0 =k of degree p and a nontrivial simply connected semisimple k-group G . For A D kŒx=.x p /, the smooth connected k-group H D RA=k .GA / is an extension of G by a nontrivial smooth connected unipotent k-group. Thus, H is not pseudo-reductive but Hk ' Gk because k 0 ˝k k ' A ˝k k as k-algebras. By descent theory with the affine Aut-scheme of G, there is an Isom-scheme Isom.G; H / that is a right fppf AutG=k -torsor. This non-empty Isom-scheme has no ks -points since H is not pseudo-reductive, so it is not smooth. But it is an fppf AutG=k -torsor, so AutG=k is not smooth. We are mainly interested in the maximal smooth closed k-subgroup scheme Autsm G=k whose group of points valued in any separable extension K=k coincides with AutK .GK /. (See [CGP, Lemma C.4.1] for a general discussion of the maximal smooth closed k-subgroup of a finite type k-group scheme.) Proposition 6.2.4. Let G be a pseudo-semisimple k-group, T a maximal k-torus 0 of G, and C D ZG .T /. Then .G o ZG;C /=C ! .Autsm G=k / is an isomorphism sm 0 0 and .Autsm G=k / is pseudo-reductive with derived group D..AutG=k / / D G=ZG . 0 Moreover, if T is k-split then .G.k/ o ZG;C .k//=C.k/ D .Autsm G=k / .k/. Proof. By Proposition 6.1.4, ZG;C is pseudo-reductive. Since ZG;C is the maximal smooth closed k-subgroup of AutG;T and moreover ZG .C / D C , the construction of AutG=k identifies the smooth connected .G o ZG;C /=C with a (necessarily closed) k-subgroup scheme of AutG=k , so this k-subgroup scheme

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0 lies inside .Autsm G=k / . The pseudo-reductivity of this k-subgroup is a special case of [CGP, Prop. 1.4.3] (using that ZG;C is pseudo-reductive), and to prove that it exhausts the identity component of Autsm G=k we just need to check that its sm set of ks -points has finite index in AutG=k .ks / D Autks .Gks /. Let f be a ks -automorphism of Gks . Since f .Tks / is G.ks /-conjugate to Tks , modifying f by G.ks /-conjugation brings us to the case where f preserves Tks . In other words, f 2 StabAutG=k .T /.ks /;

so it suffices to show that the subgroup ZG;C .ks / D AutG;T .ks / of this stabilizer group has finite index. But the image of f in Aut.Tks / lands inside the finite automorphism group of the based root datum of .G; T /ks , so the finite-index 0 property is established. This completes the determination of .Autsm G=k / , and the computation of its derived group is immediate from the commutativity of ZG;C . 0 It remains to describe .Autsm G=k / .k/ when T is k-split. The transitivity of the G.k/-action on the set of split maximal k-tori [CGP, Thm. C.2.3] and the surjectivity of NG .T /.k/=ZG .T /.k/ ! W .ˆ.G; T // implies that any f 2 0 .Autsm G=k / .k/ has a G.k/-translate that preserves T and a positive system of roots in ˆ.G; T /. We may replace f by that translate. But as a point of 0 .Autsm G=k / .ks / D .G.ks / o ZG;C .ks //=C.ks /

clearly such an f must come from ZG;C .ks /, so f 2 ZG;C .k/.



0 Remark 6.2.5. In general .Autsm G=k / is not perfect (unlike for semisimple G), since the inclusion C =ZG  ZG;C can fail to be an equality. This is seen via the explicit description of ZG;C in various cases: see [CGP, Thm. 1.3.9] for the standard case, Lemma 6.1.3 and Proposition 8.5.4 for some non-standard cases with root system of type B, C, F4 , or G2 over ks , and [CGP, Prop. 9.8.15] for minimal type cases whose root system over ks is BCn (n > 1).

Corollary 6.2.6. Let G be a pseudo-semisimple group over a field k, and Z  G a central closed k-subgroup scheme such that G D G=Z is pseudo-reductive. 0 The action of .Autsm G=k / on G is trivial on Z and the induced homomorphism sm 0 sm ˛ W .AutG=k / ! .AutG=k /0 is an isomorphism. Proof. Let C be a Cartan k-subgroup of G, so its image C D C =Z in G D G=Z is also a Cartan k-subgroup. The action of ZG;C on G is trivial on C and hence trivial on the schematic center ZG of G, and clearly the conjugation action of

6.3 Tits-style classification

93

0 G on itself is trivial on ZG . Thus, the explicit description of .Autsm G=k / shows that this identity component acts trivially on ZG . The natural map ˛ is identified with the natural map

.G o ZG;C /=C ! .G o ZG;C /=C : Since the natural map of coset spaces G=C ! G=C is visibly an isomorphism and C ! ZG;C has kernel ZG , to show that ˛ is an isomorphism it remains to observe that the natural map ZG;C ! ZG;C is an isomorphism due to Lemma 6.1.2(ii). 

6.3 Tits-style classification If G is a split connected reductive k-group then Aut0G=k D G=ZG and the étale component group of AutG=k is a constant k-group naturally identified with the group of automorphisms preserving a choice of pinning (or equivalently, the group of automorphisms of the based root datum). This description, along with the existence of split forms and the Borel–Tits structure theory via relative root systems, underlies the proof of the classification of connected semisimple groups [Tits, 2.7.1] in terms of the Tits index and anisotropic kernel. For every imperfect field k, much of the familiar structure in the proof of the classification for the semisimple case breaks down with pseudo-semisimple k-groups G: (i) there is generally no useful description of the finite étale component group EG of Autsm G=k (see Remark 6.3.6), (ii) the identity component sm 0 .AutG=k / is generally much larger than G=ZG (Remark 6.2.5), (iii) if the Galois theory of k is sufficiently rich then G may not admit a pseudo-split ks =kform (see §C.1), and (iv) in characteristic 2 there is generally no good notion of pinning in the pseudo-split case (due to constructions as in Definition 3.1.2). Despite these deficiencies, remarkably there is a version of the Tits classification theorem in the general pseudo-semisimple case! We shall prove this as an application of relative root systems for pseudo-reductive groups (developed in 0 [CGP, C.2.13–C.2.15]) and the description of .Autsm G=k / in Proposition 6.2.4. We are going to use Dynkin diagrams to express the classification theorem, but such diagrams involve a choice of maximal k-torus T in G (or even in ZG .S / for a maximal split k-torus S in G), and there is no k-rational conjugacy for such choices in general. Thus, we seek a way to work with diagrams that avoids reference to the choice of T and is functorial with respect to all k-isomorphisms among such k-groups G. Moreover, to ensure that conjugation in Gal.ks =k/ will not affect the meaning of our classification result, we have to

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keep track of canonicity with respect to change in ks =k. The notion of Dynkin scheme (over k) in [SGA3, XXIV, 3.2–3.3] is very well-suited to this purpose, but to keep the discussion in a more concrete form we will instead use the equivalent notion of canonical diagram (with Galois action) due to Kottwitz. 6.3.1. We first describe the canonical diagram for pseudo-split pseudo-reductive k-groups G. Let T be a split maximal k-torus in G, and let ˆC be a positive system of roots in ˆ.G; T /. If .T 0 ; ˆ0 C / is another such pair for G then it is carried to .T; ˆC / via conjugation by some g 2 G.k/, thereby defining an isomorphism Dyn.T 0 ; ˆ0 C / ' Dyn.T; ˆC / between Dynkin diagrams. This isomorphism is independent of g because any g 2 G.k/ preserving .T; ˆC / lies in ZG .T /.k/. Indeed, such g lies in NG .T /.k/ with image in W .G; T /.k/ D W .ˆ.G; T // that preserves ˆC and hence is trivial, so g 2 ZG .T /.k/ as desired. Likewise, by Proposition 6.2.4, if G is pseudo-semisimple then the stabilizer 0 of .T; ˆC / in .Autsm G=k / .k/ is ZG;C .k/ for C D ZG .T /, so for such G every 0 0 0C C f 2 .Autsm G=k / .k/ carrying .T ; ˆ / to .T; ˆ / induces the same isomorphism

Dyn.T 0 ; ˆ0 C / ' Dyn.T; ˆC /. Hence, if G is pseudo-reductive we have canonical isomorphisms Dyn.T 0 ; ˆ0 C / ' Dyn.T; ˆC / using any g 2 G.k/ carrying .T 0 ; ˆ0 C / to .T; ˆC /, and if G D D.G/ then the isomorphisms can even be 0 determined using .Autsm G=k / .k/. These canonical isomorphisms are associative (in an evident sense) relative to any third such pair .T 00 ; ˆ00 C / with split T 00 . We may now define the canonical diagram Dyn.G/ attached to any pseudosplit pseudo-reductive G to be the inverse limit of the graphs Dyn.T; ˆC / for split maximal k-tori T and positive systems of roots ˆC  ˆ.G; T /, using as transition maps the canonical isomorphisms induced from the action of G.k/ 0 (or even .Autsm G=k / .k/ when G is pseudo-semisimple). Equivalently, it is the set of compatible systems of vertices and directed edges with multiplicity in such diagrams, with compatibility defined via the canonical diagram isomorphisms 0 Dyn.T 0 ; ˆ0 C / ' Dyn.T; ˆC / arising from G.k/ (or even from .Autsm G=k / .k/ when G is pseudo-semisimple). The formation of Dyn.G/ is clearly functorial with respect to isomorphisms in such G and is compatible with separable extension on k. Thus, if we drop the pseudo-split hypothesis and k 0 =k is a Galois extension such that Gk 0 is pseudo-split then functoriality with respect to both scalar extension along a kautomorphism W k 0 ' k 0 and the canonical k 0 -isomorphism c W .Gk 0 / ' Gk 0 defines a diagram automorphism Œ  W Dyn.Gk 0 / ' Dyn. .Gk 0 //

Dyn.c /

'

Dyn.Gk 0 /

6.3 Tits-style classification

95

that is easily seen to define a continuous left action of Gal.k 0 =k/ on Dyn.Gk 0 /. This Galois action is functorial in such k 0 =k, so it defines a canonical continuous left action of Gal.ks =k/ on Dyn.Gks / that is functorial in the choice of ks =k. Definition 6.3.2. The canonical diagram Dyn.G/ is Dyn.Gks / equipped with the action of Gal.ks =k/ defined above. It is clear from the construction that naturally Dyn.D.G// D Dyn.G/ and the formation of Dyn.G/ equipped with its Galois action is functorial with respect to both k-isomorphisms in G and separable extension on k (equipped with a compatible extension between separable closures defining the Galois groups under consideration). In particular, for pseudo-semisimple G the natural action of Autsm G=k .ks / D AutG=k .ks / on Dyn.Gks / is a Gal.ks =k/-equivariant action 0 on the canonical diagram Dyn.G/ under which .Autsm G=k / .ks / acts trivially. For the finite étale component group EG WD 0 .Autsm G=k /; the finite group EG .ks / thereby acts Gal.ks =k/-equivariantly on Dyn.G/. Here are three ways to interpret the Gal.ks =k/-action on the set of vertices of the canonical diagram. Firstly, it agrees with the action arising from the natural identification of the set of vertices of Dyn.Gks / with the set of G.ks /conjugacy classes of maximal proper pseudo-parabolic ks -subgroups of Gks . Secondly, it agrees with the action on the set of ks -points of the Dynkin scheme defined similarly to the reductive case as in [SGA3, XXIV, 3.2–3.3]. To give a third interpretation (which will be used for calculations in some proofs below), observe that for a maximal k-torus T of G the natural action of Gal.ks =k/ on X.Tks / permutes the set of positive systems of roots ˆC of ˆ.Gks ; Tks /. Hence, since W .Gks ; Tks / D W .ˆ/ acts simply transitively on the set of such ˆC ’s, upon choosing ˆC we see that for every 2 Gal.ks =k/ there is a unique w 2 W .Gks ; Tks / such that w . .ˆC // D ˆC , or equivalently w . .// D  where  is the basis of ˆC . This leads to the following notion: Definition 6.3.3. For a basis  of ˆ, the -action of Gal.ks =k/ on the Dynkin diagram associated to  is defined by  a D w . .a// for a 2 . Under the natural identification of Dyn.G/ with the diagram  associated to the basis of ˆC , the Gal.ks =k/-action on Dyn.G/ coincides with the -action of Gal.ks =k/ on . In particular, . ; a/ 7!  a is an action of Gal.ks =k/ through diagram automorphisms (i.e., it respects the pairing between roots and coroots), as is also easy to check directly. The canonicity of the Galois action on

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Dyn.G/ expresses the fact (which is easily verified directly as well) that if 0 is another basis of ˆ.Gks ; Tks / and 0 denotes the associated Gal.ks =k/-action on 0 then for the unique w 2 W .Gks ; Tks / such that 0 D w./ the action of w 0 WD ww .w/ 1 carries .0 / to 0 , so 0 .w.a// D w.  a/; i.e., the isomorphism defined by w between the diagrams associated to  and 0 is Gal.ks =k/-equivariant when using the -action on  and the 0 -action on 0 . Proposition 6.3.4. Let G be a pseudo-semisimple k-group. (i) The action map EG .ks / ! Aut.Dyn.G// has trivial kernel, 0 (ii) H1 .k; .Autsm G=k / / is the set of k-isomorphism classes of pairs .G ; '/ where G is pseudo-semisimple over k and ' W Dyn.G / ' Dyn.G/ is a Galois-equivariant isomorphism between the canonical diagrams such that ' D Dyn.f / for some ks -isomorphism f W Gks ' Gks , sm 0 1 (iii) the natural map H1 .k; .Autsm G=k / / ! H .k; AutG=k / is the forgetful map sending each pair .G ; '/ to the class of G as a ks =k-form of G. Proof. The meaning of (i) is that the kernel of the action of Autsm G=k .ks / on sm 0 Dyn.Gks / is .AutG=k / .ks /, and this clearly implies the rest. To prove (i) we may assume k D ks . Consider a k-automorphism f of G with trivial action on Dyn.G/; f can be modified by an inner automorphism arising from an element of G.k/ in order to preserve a maximal k-torus T and a positive system of roots ˆC in ˆ.G; T /. The action of f on Dyn.G/ is determined by the natural effect of f on the Dynkin diagram associated to .G; T; ˆC /, so we conclude that f acts trivially on the basis of ˆC . As X.T /Q is generated by that basis (since G is pseudo-semisimple), f acts trivially on T . It follows that f acts trivially on the commutative pseudo-reductive C WD ZG .T / [CGP, Prop. 1.2.2], so f belongs 0 to ZG;C .k/  .Autsm  G=k / .k/ (see Proposition 6.1.4). 0 Remark 6.3.5. Explicitly, (ii) says that H1 .k; .Autsm G=k / / classifies isomorphism classes of pairs .G ; '/ consisting of a pseudo-semisimple k-group G and a -compatible isomorphism ' from the canonical diagram of Gks to that of Gks such that ' is induced by a ks -isomorphism Gks ' Gks .

Remark 6.3.6. If T is a maximal k-torus of G and ˆC is a positive system of roots in ˆ.Gks ; Tks / then by Proposition 6.3.4(i) and [CGP, Thm. 2.3.10] the group EG .ks / is naturally a subgroup of the automorphism group of the C based root datum .R; / attached to .G ss ; Tk ; ˆC nm /, where ˆnm is the set of nonk multipliable roots in ˆC . This inclusion is well-known to be an equality in the reductive case, and in Proposition 6.3.10 we will show that it is an equality in the absolutely pseudo-simple case except for G arising from a special construction

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97

of type D2n (n > 2) over imperfect fields of characteristic 2. Allowing G to be pseudo-semisimple but possibly not absolutely pseudo-simple, over every imperfect field there exist such G for which EG .ks / ,! Aut.R; / fails to be an equality, due to phenomena illustrated in Examples 6.3.8 and C.1.6. For pseudo-semisimple G, a characterization of EG .ks / inside the automorphism group of the based root datum can be obtained by using Theorem 6.1.1. However, such a result is cumbersome to state (and use) due to the presence of “moduli” (finite extensions of ks , and in characteristic 2 some homothety classes of subspaces of such extensions), so we do not record it here. Proposition 6.3.7. If G is a pseudo-split pseudo-semisimple k-group then EG is constant and Autk .G/ ! EG .k/ is surjective .equivalently, Autk .G/ meets every connected component of .Autsm G=k /ks /. Proof. It suffices to show that Autk .G/ ! EG .ks / is surjective. Let S be a split maximal k-torus in G, ˆC a positive system of roots in ˆ.G; S/, and  the basis of ˆC . Since all maximal split k-tori in G are G.k/-conjugate and NG .S /.k/ ! W .ˆ.G; S// is surjective, for every k-automorphism f of G there exists g 2 G.k/ such that the composition of f with conjugation by g preserves S and the induced action on ˆ.G; S/ preserves ˆC . The same likewise holds over ks , so by Proposition 6.3.4(i) it suffices to show that for any ks -automorphism  of .Gks ; Sks ; ˆC / there exists a k-automorphism ' of .G; S; ˆC / such that 'ks and  have the same effect on . To build such a ' we shall use Theorem 6.1.1. The restriction 0 of  to ZGks .Sks / D ZG .S/ks is defined over k. Indeed, its restriction to Sks is defined over k (as S is a split k-torus), so the Galoisequivariance of 0 follows from the fact (part of [CGP, Prop. 1.2.2]) that any isomorphism between commutative pseudo-reductive groups is uniquely determined by its restriction between the unique maximal tori. Let f be the k-automorphism of ZG .S/ descending 0 , and let fS be its restriction to S . Since  preserves ˆC , clearly fS carries  onto itself. For any a 2  and a0 WD X.fS /.a/ 2 , the effect of  on  is a 7! a0 since jSks D .fS /ks . Hence, fS ı a_ D a0 _ for all a 2 . By the Isomorphism Theorem (i.e., Theorem 6.1.1), to build a k-automorphism of .G; S; ˆC / restricting to a 7! a0 on  and restricting to fS on S , it suffices to construct k-isomorphisms fa W Ga ' Ga0 for all a 2  such that each fa is equivariant via fS for the respective actions of S on Ga and Ga0 . (As is shown in the proof of Theorem 6.1.1, any such fa must agree with fS on Ga \ S D a_ .GL1 /.) Since  restricts to a ks -isomorphism a W .Ga /ks ' .Ga0 /ks with the desired Sks -equivariance properties, the obstruction to building fa is given by a degree-

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1 Galois cohomology class for ks =k valued in the group of automorphisms of Ga that restrict to the identity map on a_ .GL1 /, or equivalently on the Cartan subgroup Ca WD ZGa .a_ .GL1 //. Hence, by Hilbert’s Theorem 90 it suffices to show that ZGa ;Ca ' RFa =k .GL1 / for some finite extension field Fa =k. If Ga has root system A1 then we may use Lemma 6.1.3, and if Ga has root system BC1 (so k is imperfect of characteristic 2) then we may use [CGP, (9.8.7)].  We now provide some causes for the phenomenon in Remark 6.3.6. For pseudo-semisimple G, consider kernels of projections from Gk onto the simple factors of its maximal adjoint semisimple quotient G ad . If the minimal fields of k definition over ks for two such kernels are not ks -isomorphic but the corresponding irreducible components of the root system of Gks are isomorphic then there is no automorphism of Gks realizing a swap of those two irreducible components. The combinatorial based root datum does not encode this field-theoretic information. For example: Example 6.3.8. Let H be a split connected semisimple group that is absolutely simple and simply connected over an imperfect field k of characteristic p. Define G D Rk 0 =k .Hk 0 /  H where k 0 =k is a nontrivial purely inseparable finite extension. There are exactly two pseudo-simple normal k-subgroups of G, namely the factors Rk 0 =k .Hk 0 / and H in the definition, and these are preserved under any ks -automorphism since they cannot be swapped. The based root datum of G is the disjoint union of two copies of the one for H , so an automorphism swapping these components cannot arise from Autsm G=k .ks / D Autks .Gks /. The interested reader can make analogous pseudo-simple (but not absolutely pseudo-simple) examples over many k satisfying Œk W k p  > p 2 via the idea in Example C.1.2. To analyze the finite étale component group EG of Autsm G=k , now it is natural to restrict attention to absolutely pseudo-simple G. Over many imperfect k with char.k/ D 2 there are such G for which EG is smaller than expected: for G as in Proposition C.1.4 the based root datum is that of the adjoint central quotient of Spin4n with n > 2 (so its automorphism group is nontrivial) but it is easy to see that EG is trivial. This construction can be generalized: Example 6.3.9. Let k be imperfect of characteristic 2, G the split k-group Spin4n with n > 2, and Z WD ZG ' 2  2 . Let K=k be a nontrivial purely inseparable finite extension, and let Z be a closed k-subgroup of RK=k .ZK / containing Z such that RK=k .ZK /=Z has no nontrivial smooth connected ksubgroup and Z is not stable under the natural action on RK=k .ZK / by the group € WD 0 .AutG =k / of diagram automorphisms of G (i.e., € D Z=.2/ if

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99

n > 2, € D S3 if n D 2); see (the proof of) Proposition C.1.4 and Remark C.1.5 for the existence of such Z over suitable k. The k-group G D RK=k .GK /=Z is pseudo-reductive (and therefore absolutely pseudo-simple) due to the same reasoning (based on [CGP, Lemma 9.4.1]) as in the proof of Proposition C.1.4, and G ss is the adjoint central quotient of Gk k since Z  Z. Thus, the automorphism group of the based root datum of Gks is €. However, EG .ks / is a proper subgroup of € since Z is not stable under the action of € on RK=k .ZK /. It is clear that G is not of minimal type. Example 6.3.9 accounts for all absolutely pseudo-simple groups G for which the ks -points of EG do not exhaust the group of diagram automorphisms: Proposition 6.3.10. Let G be an absolutely pseudo-semisimple group over a field k, T a maximal k-torus of G, R D .X; ˆ; X _ ; ˆ_ / the .semisimple/ root datum for .Gks ; Tks /, and  a basis of the irreducible ˆ. Let EG D 0 .Autsm G=k /. The natural inclusion EG .ks /  Aut.R; / is an equality except precisely when k is imperfect with char.k/ D 2 and Gks arises in Example 6.3.9 over ks . Proof. We may and do assume k is separably closed. If there are no nontrivial diagram automorphisms then there is nothing to do, so the only cases we need to consider are those with diagram An (n > 2), Dn (n > 4), or E6 . In particular, by Theorem 3.4.2, G is standard! Now G D RK=k .GK /=Z for a purely inseparable finite extension field K=k, a (necessarily split) connected absolutely simple k-group G that is (absolutely) simple and simply connected, and a closed k-subgroup Z  RK=k .ZK / for Z WD ZG such that RK=k .ZK /=Z does not contain a nontrivial smooth connected unipotent k-subgroup. (The final condition encodes that the central quotient G is pseudo-reductive, by Corollary 4.1.4.) We may assume K ¤ k (so k is imperfect), as otherwise G is in the well-known absolutely simple case. Since 1 ! Z ! RK=k .GK / ! G ! 1 is a smooth k-tame central extension, it is the universal smooth k-tame central extension of G since G is simply connected. Hence, every k-automorphism of G lifts (uniquely) to an automorphism of RK=k .GK / that preserves Z, and conversely every automorphism of RK=k .GK / preserving Z descends to an automorphism of G. By [CGP, Thm. 1.6.2(2)], every k-automorphism of RK=k .GK / has the form RK=k .'/ for a unique K-automorphism ' of GK . Away from the D2n -cases (with n > 2), the action of diagram automorphisms on Z is through the identity or inversion, so Z is preserved by RK=k .'/ for any ' and hence Autsm G=k D RK=k .AutGK =K /. This clearly has the expected component group.

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Now suppose the diagram is D2n with n > 2. If char.k/ ¤ 2 then Z is étale, so Z D RK=k .ZK0 / for a k-subgroup Z 0  Z [CGP, Prop. A.5.13] and G D RK=k ..G =Z 0 /K / with G red D .G =Z 0 /k . Thus, in such cases Autsm G=k D k RK=k .Aut.G =Z 0 /K =K /, so its finite component group is the same as that of the k-group Aut.G =Z 0 /=k ; hence, this component group is the automorphism group of the common based root datum of G and G =Z 0 . Finally, assume char.k/ D 2 (and the diagram is D2n with n > 2). If Z is stable under the action by 0 .AutG =k / D 0 .RK=k .AutGK =K // on RK=k .ZK / then the same reasoning as above may be applied, so we may assume that Z is not stable under this action. If Z  Z then we are exactly in the cases from Example 6.3.9, so we may assume that Z \ Z is a proper k-subgroup of Z D 2 2 . Recall that over a field, there are no nontrivial homomorphisms between unipotent group schemes and multiplicative type group schemes [SGA3, XVII, Prop. 2.4]. We may assume Z is nontrivial, so Z is not unipotent (as RK=k .ZK / does not contain a nontrivial unipotent k-subgroup). Thus, Z \ Z ¤ 1 since the k-subgroup scheme RK=k .ZK /=Z  .RK=k .GL1 /=GL1 /2 is unipotent (as the k-group RK=k .GL1 /=GL1 is unipotent). Hence, the proper k-subgroup  WD Z \ Z  Z D 2  2 is one of the three copies of 2 . Since Z= inherits unipotence from RK=k .ZK /=Z , the natural map ZK ! ZK corresponding to the inclusion Z ,! RK=k .ZK / induces a map .Z=/K ! .Z =/K that must be trivial (as Z= is unipotent and Z = is of multiplicative type), so   Z  RK=k .K / and G red D .G =/k . k We may assume that the common based root datum .R; / of G and G = has a nontrivial automorphism (or else there is nothing to do), so the quotient G = of G D Spin4n is isomorphic to SO4n as a k-group (even if n D 2) and the automorphism group of the based root datum has order 2 with trivial action on  (and so also on Z). Thus, the evident map Autsm G=k ! RK=k .Aut.G =/K =K / is an isomorphism, so 0 .Autsm / D  .Aut  0 .G =/=k / D Aut.R; / as desired. G=k Now we prepare to define the anisotropic kernel of a pseudo-reductive group over a field k. Recall (see [CGP, C.2.3–C.2.5]) that if G is a smooth connected affine k-group then the minimal pseudo-parabolic k-subgroups P and the maximal k-split tori S each constitute a single G.k/-conjugacy class (any pseudoparabolic k-subgroup contains such an S ), and if G is pseudo-reductive and S  P then P D ZG .S / n U for the k-split U D Ru;k .P /. Fix a minimal pseudo-parabolic k-subgroup P of G and a maximal k-split torus S  P . Let M D ZG .S/. Every maximal k-torus T of M contains S and is clearly maximal in G. Assume G is pseudo-reductive, so M is pseudoreductive. By [CGP, Lemma 1.2.5], T is an almost direct product of the maximal

6.3 Tits-style classification

101

central k-torus of M and the maximal k-torus T 0 WD T \ D.M / of D.M /, so the pseudo-semisimple derived group D.M / is k-anisotropic (i.e., it does not contain a nontrivial k-split torus); we call D.M / the anisotropic kernel of G. For our purposes this will only matter through its maximal central quotient (which coincides with D.M=ZM / since the triviality of the center of M=ZM via Proposition 4.1.3 ensures that D.M=ZM / has trivial center, due to [CGP, Lemma 1.2.5(ii), Prop. 1.2.6]). There is an evident Galois-equivariant diagram inclusion Dyn.D.M // D Dyn.M /  Dyn.G/: In more explicit terms, if T is a maximal k-torus of M and T 0 WD T \ D.M / then for a positive system of roots ˆC  ˆ.Gks ; Tks / contained in ˆ.Pks ; Tks / and the positive system of roots ‰ C D ‰ \ ˆC in ‰ WD ˆ.Mks ; Tks /, the diagram of .D.M /ks ; Tk0 s ; ‰ C / is naturally a subdiagram of that of .Gks ; Tks ; ˆC / compatibly with the -action of Gal.ks =k/ on each (seen classically by the fact that NM .T /.ks / acts transitively on the set of choices of ˆC that lie inside ˆ.Pks ; Tks /, as P D M n U ). The vertices of Dyn.D.M // D Dyn.M / are thereby identified with the non-distinguished vertices of Dyn.G/ (i.e., those corresponding to the simple roots whose restriction to Sks is trivial). 0 The use of Dynkin diagrams to interpret H1 .k; .Autsm G=k / / in Proposition 6.3.4(ii) and Remark 6.3.5 leads to a pseudo-semisimple generalization of the Tits classification of connected semisimple groups: Theorem 6.3.11 (Relative Isomorphism Theorem). Let .G; S/ be as above with G pseudo-semisimple, and let M D ZG .S/. Naturally identify Dyn.D.M // D Dyn.D.M=ZM // with a subdiagram of Dyn.G/. If .G ; S / is another such pair and M WD ZG .S / then G ' G if and only if there exists a Galois-equivariant diagram isomorphism  W Dyn.G / ' Dyn.G/ restricting to an isomorphism 0 W Dyn.D.M // ' Dyn.D.M // such that  arises from a ks -isomorphism f W Gks ' Gks and 0 arises from a kisomorphism f0 W D.M =ZM / ' D.M=ZM /. In particular, G is determined up to k-isomorphism by the isomorphism class of the triple .Gks ; D.M=ZM /; /, where  is the Gal.ks =k/-equivariant inclusion of Dyn.D.M // D Dyn.D.M=ZM // as a subdiagram of Dyn.G/. In the final assertion of the theorem, an isomorphism between two such triples is a pair .f; f0 / as above (so Dyn.f / is Gal.ks =k/-equivariant and the isomorphisms Dyn.f / and Dyn.f0 / intertwine the given diagram inclusions). If G is semisimple then  uniquely extends to an isomorphism between “indexed

102

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root data” (see [Spr, 16.2.1]), so the semisimple case of Theorem 6.3.11 is exactly the well-known theorem of Tits. The necessity in Theorem 6.3.11 is obvious, and to prove sufficiency we assume that ; f; f0 are given. Let P  G and P  G be minimal pseudoparabolic k-subgroups containing S and S respectively. Let T be a maximal ktorus of M , so T contains S and is maximal in G. The relationship between ˆ WD ˆ.Gks ; Tks / and ‰ WD ˆ.Mks ; Tks / D ˆ.D.M /ks ; Tk0 s / for T 0 WD T \ D.M / works out as follows. Choose a positive system of roots ˆC of ˆ contained in the parabolic subset ˆ.Pks ; Tks /  ˆ and define ‰ C D ‰ \ ˆC . Let  be the basis of ˆC , so 0 WD ‰ \  is the basis of ‰ C . The root system ‰ consists of the elements of ˆ that have trivial restriction to Sks . The restriction map  0 ! X.Sks / D X.S / (whose fibers are orbits under the -action of Gal.ks =k/) has image k  that is a basis of the relative root system k ˆ WD ˆ.G; S/ (as shown in the proof of [CGP, Thm. C.2.15]). For c 2 ‰, the c-root groups of .Mks ; Tks / and .Gks ; Tks / coincide. In particular, for each c 2 ‰ the subgroup .Gks /c generated by the ˙c-root groups of Gks coincides with .D.M /ks /c . The set of pseudo-parabolic ks -subgroups of Gks containing Tks is in natural bijective correspondence with the set of parabolic subsets of ˆ D ˆ.Gks ; Tks / via Q 7! ˆ.Q; Tks / [CGP, Prop. 3.5.1(2)], so the set of pseudo-parabolic ks subgroups containing Tks and corresponding to parabolic subsets containing  is in natural inclusion-preserving bijection with the set of subsets of . The restriction map X.Tks / ! X.Sks / D X.S/ carries  0 onto k , so minimality of P implies that ˆ.P; S / is the positive system of roots in k ˆ with basis k . Thus, Pks corresponds to 0  , and similarly for .G ; P/. Hence, we may compose f with conjugation by an element of G.ks / so that f .Pks / D Pks ; note that composing with such conjugation has no effect on Dyn.f /. 0 To simplify notation, let A D .Autsm G=k / . The k-group G is isomorphic to the twist of G by the continuous 1-cocycle c W Gal.ks =k/ ! A.ks / defined by

c. / D f ı . f /

1

(where f denotes the -twist of f ); this is valued in A.ks / due to Proposition 6.3.4(i) since Dyn.f / D  is Gal.ks =k/-equivariant. By Proposition 6.2.4, A.ks / D .G.ks / o ZG;C .ks //=C.ks / where C WD ZG .T /. Since f .Pks / D Pks , c is valued in the stabilizer of Pks in A.ks /. The triviality of the ZG;C -action on T implies that ZG;C preserves P , so the stabilizer of P in A is .P o ZG;C /=C since NG .P / D P [CGP, Prop. 3.5.7].

6.3 Tits-style classification

103

The ZG;C -action on G preserves S, so it preserves M D ZG .S/. Hence, it makes sense to consider the inclusion of k-groups j W .M o ZG;C /=C ,! .P o ZG;C /=C: The retraction P  P =U ' M onto M (for U WD Ru;k .P /) is clearly ZG;C equivariant and so defines a retraction r W .P o ZG;C /=C  .M o ZG;C /=C whose kernel is identified with the k-split smooth connected unipotent k-group U . Every ks =k-form of U is k-split [CGP, Thm. B.3.4], so the twisting method in Galois cohomology implies that H1 .r/ is a bijection. Hence, H1 .j / is bijective, so by replacing c with a cohomologous 1-cocycle we may and do assume that c is valued in .M.ks / o ZG;C .ks //=C.ks /. The ZG;C -action on M preserves D.M /, so we get a natural map 0 q W .M o ZG;C /=C ! .Autsm D.M /=k / : 0 Corollary 6.2.6 identifies the target of q with .Autsm D.M=ZM /=k / , and in this way via Proposition 6.3.4(ii) the map H1 .q/ carries the cohomology class Œc to the class of the pair .D.M =ZM /; 0 /. The existence of the k-isomorphism f0 therefore implies that H1 .q/.Œc/ D 1. To exploit this triviality, we shall use:

Proposition 6.3.12. The map q is surjective with kernel of the form RF =k .GL1 / for a nonzero finite reduced k-algebra F . In particular, q is a smooth surjection. Granting the proposition, let us conclude the proof of Theorem 6.3.11. Since q is a smooth surjection and H1 .q/.Œc/ D 1, Œc comes from H1 .k; ker q/. But the description of ker q as a Weil restriction implies that H1 .k; ker q/ D 1 by Shapiro’s Lemma and Hilbert’s Theorem 90, so Œc D 1. Thus, by the construc0 tion of c, the class of .G ;  / in H1 .k; .Autsm G=k / / is trivial, so G ' G. To prove Proposition 6.3.12, we first describe q in more concrete terms. Let T 0 be the maximal k-torus T \ D.M / of D.M / and let Z be the maximal central k-torus of M , so T D Z
T 0 by [CGP, Lemma 1.2.5(ii)]. Thus, the intersection C 0 WD C \ D.M / D ZD.M / .T 0 / is a Cartan k-subgroup of D.M /. Since M D C
D.M /, we have .M o ZG;C /=C D .D.M / o ZG;C /=C 0 . Also, by Proposition 6.2.4 (applied to D.M /) we have 0 0 .Autsm D.M /=k / D .D.M / o ZD.M /;C 0 /=C ;

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so q is identified with the canonical map .D.M / o ZG;C /=C 0 ! .D.M / o ZD.M /;C 0 /=C 0 induced by the natural restriction map  W ZG;C ! ZD.M /;C 0 D ZM;C (arising from the property that ZG;C preserves M and D.M / inside G; the equality is Lemma 6.1.2(i)). The surjectivity of q is equivalent to that of , and ker q ' ker . Hence, to prove Proposition 6.3.12 it is equivalent to prove the surjectivity of  and to determine its kernel. Let E=k be a finite Galois extension that splits T . The surjectivity of  is verified by computing over E via applying Proposition 6.1.4 to the pairs .G; C / and .M; C /: since ˆ.ME ; TE / has 0 as a basis and .ME /a D .GE /a for each a 2 ˆ.ME ; TE /, clearly E is the natural projection Y Y Z.GE /a ;.CE /a ! Z.GE /a ;.CE /a ; a2

a20

so  is surjective and ker  is commutative and pseudo-reductive of minimal type (Proposition 6.1.4). It remains to show ker  ' RF =k .GL1 / for some F . To analyze ker  as a k-group we will first reduce to the case where G is absolutely pseudo-simple. Consider G D G=ZG ; this is pseudo-semisimple with trivial center by Proposition 4.1.3, and the image S of S is a maximal k-split torus (due to Proposition 5.3.1(i)). Thus, the central quotient M D M=ZG of M coincides with ZG .S /. By Lemma 6.1.2(ii), the formation of  is unaffected by passage to G, C , S. Hence, we may (and will) assume that ZG is trivial. Lemma 6.3.13. There exists a nonzero finite étale k-algebra k 0 and a smooth affine k 0 -group G 0 with trivial center and absolutely pseudo-simple fibers over the factor fields of k 0 such that G ' Rk 0 =k .G 0 /. If .k 00 =k; G 00 / is a second such pair then any k-isomorphism Rk 0 =k .G 0 / ' Rk 00 =k .G 00 / arises from a unique pair .'; ˛/ consisting of a k-algebra isomorphism ˛ W k 0 ' k 00 and a group isomorphism ' W G 0 ' G 00 over ˛. Proof. By Galois descent and the uniqueness assertions we may assume k D ks . If fGi0 gi 2I is the set of pseudo-simple normal k-subgroups of G then they pairQ wise commute and multiplication m W Gi0 ! G is a central quotient map [CGP, Prop. 3.1.8], so the triviality of ZG forces all ZGi0 to be trivial and hence m is ` an isomorphism. This provides existence (using k 0 D k I and G 0 D Gi0 ), and

6.3 Tits-style classification

105

Q uniqueness follows from the fact that the product Gi0 uniquely characterizes its k-subgroups Gi0 as its pseudo-simple normal k-subgroups.  Let k 0 and G 0 be as in the preceding lemma. We identify G with Rk 0 =k .G 0 /. By [CGP, Prop. A.5.15], the Cartan k-subgroup C has the form Rk 0 =k .C 0 / for a unique Cartan k 0 -subgroup C 0  G 0 and there is a unique maximal k 0 -split torus S 0  G 0 for which S is the maximal k-split torus in Rk 0 =k .S 0 /, with M 0 WD ZG 0 .S 0 / satisfying Rk 0 =k .M 0 / D ZG .Rk 0 =k .S 0 //  ZG .S/ D M . To prove the reverse inclusion Rk 0 =k .M 0 /  M it suffices to treat the case where k 0 is a field, and in that case it suffices to show that the natural map Mk 0 ! G 0 lands inside M 0 . But the natural map Sk 0 ! S 0 induced by  W Gk 0 ! G 0 is easily seen to be an isomorphism, so  carries Mk 0 D ZGk0 .Sk 0 / into ZG 0 .S 0 / D M 0 as desired. Proposition 6.1.7 applied to G 0 and M 0 identifies  with Rk 0 =k .0 /, where 0  is the analogous map over k 0 associated to .G 0 ; S 0 ; C 0 / (working over each factor field of k 0 separately). This permits us to replace .G; S; C / with the fibers of .G 0 ; S 0 ; C 0 / over the factor fields of k 0 , so now we have arranged that G is absolutely pseudo-simple. Consider the identification Y .ker /E D Z.GE /a ;.CE /a : (6.3.13) a2 0

Note that all elements of  0 in a common orbit under the -action have the same length in the irreducible (possibly non-reduced) root system ˆ.GE ; TE /. To use (6.3.13), we need to describe the factors in this direct product: Lemma 6.3.14. Let H be a pseudo-split absolutely pseudo-simple E-group of rank 1, and define C WD ZH .T / for a split maximal E-torus T  H . For a basis fag of ˆ.H ; T / there is a unique purely inseparable finite extension F=E and unique E-isomorphism ZH ;C ' RF =E .GL1 /

(6.3.14)

carrying the natural map T ! ZH ;C over to the composition of a W T ! GL1 with the natural inclusion GL1 ,! RF =E .GL1 /. This isomorphism is functorial with respect to isomorphisms in the triple .H ; T ; fag/ and compatible with separable extension on E. Note that F=E is independent of the choice of T (by working over Es ). Proof. By Lemma 6.1.2(ii), we may replace H with its maximal quotient of minimal type, and then pass to the universal smooth E-tame central extension

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so that H ss is simply connected. The uniqueness assertions are clear, assuming k existence. If the root system is A1 then we may conclude by Lemma 6.1.3, and for the root system BC1 we may use the rank-1 case of [CGP, Prop. 9.8.15].  6.3.15. Completion of the proof of Proposition 6.3.12. Let  W  0  k  be the natural map induced by X.TE /  X.SE / D X.S/, so the fibers of  are the -orbits. Applying Lemma 6.3.14 to the E-groups .GE /a for a 2 ˆ D ˆ.GE ; TE / defines purely inseparable finite extensions Fa0 =E for all a 2 ˆ. Any two elements of the irreducible (possibly non-reduced) root system ˆ with the same length are conjugate under NG .T /.E/, so Fa0 only depends on a through its length. But clearly if 2 Gal.E=k/ then canonically

.F 0 / ' F 0 D Fa0 as E-algebras (induced by ..GE /a / ' .GE / .a/ carrying a

.a/

..C / / onto .C / 0 E a E .a/ ), thereby defining an E=k-descent datum on Fa . Let Fa =k be the associated Galois descent, For each ˛ 2 k  and a in the -orbit  1 .˛/ inside  0 , clearly the purely inseparable finite extension Fa =k only depends on ˛. Denote this extension as F˛ , so (6.3.13) gives a natural isomorphism .ker /E '

Y



RF˛ =k .GL1

1 .˛/

/E :

(6.3.15)

˛2k 

It suffices to show that the natural Gal.ks =k/-action on .ker /.ks / (or alternatively the E=k-descent datum on .ker /E ) corresponds to the permutation action on the right side through the -action on each index set  1 .˛/. Indeed, once this is established, if L˛ =k is the finite separable subextension of E=k corresponding to the Gal.E=k/-stabilizer of a chosen element of  1 .˛/ then Y ker  ' RF˛ =k .RL˛ F˛ =F˛ .GL1 // D RF =k .GL1 / ˛2k 

Q with F WD ˛2k  L˛ F˛ . (It is for absolutely pseudo-simple G that each factor field of F is a compositum of a separable and a purely inseparable extension.) ` Clearly S  ker  (as M D ZG .S/), and for a 2  0 D ˛2k   1 .˛/ the 1-dimensional maximal E-torus inside the a-factor of (6.3.15) is by definition identified with GL1 via the nontrivial character a W SE  GL1 . Since an isomorphism between commutative pseudo-reductive groups is uniquely determined by its restriction between maximal tori, for any a 2  0 and

2 Gal.E=k/ it suffices to show: (i) .a/jSE D .  a/jSE ,

6.3 Tits-style classification

107

(ii) Z.GE / .a/ ;.CE / .a/ D Z.GE / a ;.CE / a inside ZGE ;CE , where the left side is embedded using the basis ./ of ˆ and the right side is embedded using the basis   D  of ˆ. By definition,  a D w . .a// D .a/ ı Int n 1 for n 2 NG .T /.E/ representing the unique w 2 W .ˆ/ carrying .ˆC / onto ˆC . But ˆC was chosen inside the Gal.E=k/-stable parabolic subset ˆ.PE ; TE / D ˆ.Pks ; Tks /  ˆ, so

.ˆC /  ˆ.PE ; TE /. Thus, ˆC and .ˆC / correspond to two minimal pseudoparabolic E-subgroups of GE that are contained in PE D ME n UE and contain the split maximal E-torus TE . By [CGP, Lemma 3.5.5], these correspond to minimal pseudo-parabolic E-subgroups of PE =UE D ME containing TE , so W .ME ; TE / carries one to the other. Hence, we can choose n 2 NM .T /.E/, so n centralizes SE and thus (i) holds. Also, Z.GE /a ;.CE /a viewed inside ZGE ;CE via  acts trivially on ME by definition of  since a 2  0 , so likewise its twist Z.GE / .a/ ;.CE / .a/ inside ZGE ;CE D .ZG;C /E (embedded via ./!) acts trivially on ME too. Since n 2 M.E/, we obtain (ii) as claimed, completing the proof of Proposition 6.3.12 (and so also of Theorem 6.3.11). Remark 6.3.16. The k-isomorphism class of a pseudo-semisimple k-group G is determined by a 4-tuple .G; ; M0 ; / where G is a pseudo-semisimple ks -group (a root datum in the semisimple case),  is a continuous action of Gal.ks =k/ on Dyn.G/, M0 is a k-anisotropic pseudo-semisimple k-group with trivial center, and  is a Galois-equivariant diagram inclusion Dyn.M0 / ,! Dyn.G/. The discussion preceding Lemma C.2.2 shows that G is quasi-split (i.e., its minimal pseudo-parabolic k-subgroups are solvable) precisely when M0 D 1, so G is pseudo-split precisely when M0 D 1 and  is the trivial action. The special case that a pseudo-split pseudo-semisimple k-group G is determined by Gks is immediate from Theorem 6.3.11 and is generalized to pseudo-reductive G in Proposition C.1.1. (See Proposition C.2.8 for variant in the quasi-split case.) Theorem 6.1.1 determines G in terms of: C D ZG .T/ for a maximal ks -torus T, the root system ˆ D ˆ.G; T/, and the T-equivariant rank-1 pseudo-simple subgroups associated to roots in a basis of ˆ. That does not address the existence problem over ks when such data are given. For irreducible ˆ, the pseudo-split possibilites of minimal type over k are classified alongside an existence result: see Theorem 3.4.1 for reduced ˆ and [CGP, Thm. 9.8.6] for non-reduced ˆ. What about the general existence problem over k? Beyond the absolutely pseudo-simple case this is nontrivial since for many k (occurring in any positive characteristic) Examples C.1.2 and C.1.6 provide standard pseudo-semisimple (but not absolutely pseudo-simple) k-groups with no pseudo-split ks =k-form. The absolutely pseudo-simple case is addressed in Remark C.2.13.

7 Constructions with regular degenerate quadratic forms

For an imperfect field k with char.k/ D p, the non-standard pseudo-split absolutely pseudo-simple k-groups G of minimal type with a reduced root system ˆ of rank n > 2 are classified in Theorem 3.4.1(ii),(iii) in terms of field-theoretic and linear-algebraic data. These only exist when ˆ has an edge of multiplicity p (so p 2 f2; 3g). For type G2 in characteristic 3 and type F4 in characteristic 2 this involves just field-theoretic data: a nontrivial purely inseparable finite extension K=k and a proper subfield K>  K containing kK p . The non-standard G2 -cases and non-standard F4 -cases as handled in [CGP, Ch. 7–8] amount to the basic exotic construction reviewed in §2.2, and for the convenience of the reader we reproved what we need about G2 and F4 in §3.4. Non-standard G of types Bn and Cn are more subtle: they are classified by a subfield K>  K containing kK 2 and a pair of nonzero kK 2 -subspaces V; V>  K satisfying certain conditions (with V D K for type Cn with n > 3, and V> D K> for type Bn with n > 3). For any such G, by Proposition 4.1.3 the quotient G=ZG is absolutely pseudo-simple with trivial center, its root system over ks is clearly ˆ, and it is non-standard because any pseudo-reductive central extension of a standard pseudo-reductive group is standard (Lemma 3.2.8). The aim of this chapter is to use degenerate quadratic forms and quadrics in Severi–Brauer varieties to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks such that ZG D 1 and the Cartan k-subgroups of G are tori. (Even in the pseudo-split case this will provide more information than Theorem 3.4.1(iii).) Our inspiration is the fact that the adjoint connected semisimple groups of type Bn over a field F are precisely SO.q/ for non-degenerate quadratic spaces .V; q/

7.1 Regular degenerate quadratic forms

109

over F of dimension 2n C 1, with .V; q/ determined up to conformal isometry by the isomorphism class of SO.q/. In Chapter 8 we will use universal smooth k-tame central extensions of these adjoint type-B constructions, as well as fiber products in the spirit of the basic exotic construction in §2.2, to build non-standard G of minimal type such that G ss is simply connected of type B or C. (The link to type C rests on the unipotent k isogeny SO2nC1 ! Sp2n from type B to type C in characteristic 2.)

7.1 Regular degenerate quadratic forms Let k be an imperfect field with char.k/ D 2, and let .V; q/ be a quadratic space over k such that d WD dim.V / is finite and q ¤ 0. Non-degeneracy of q is defined to mean k-smoothness of the quadric hypersurface Hq WD .q D 0/  P.V  /. Let Bq .v; w/ D q.v C w/ q.v/ q.w/ be the associated symmetric bilinear form (also alternating since char.k/ D 2), and let V ? be the defect space, so (as we saw in Example 1.3.1) dim.V =V ? / D 2n is even. Assume .V; q/ is regular in the sense that the 2-linear map q W V ? ! k is injective. Regularity of q has geometric meaning: it amounts to regularity (or equivalently, k-smoothness) of the projective quadric Hq at its k-points. Indeed, such regularity is equivalent to that of the affine quadric fq D 0g at its nonzero k-points v0 2 V and the tangent space at such v0 has dimension d 1 precisely when the degree-2 polynomial q.v C v0 / q.v0 / D Bq .v; v0 / C q.v/ on V has nonzero linear part. This linear part is Bq .v; v0 /, so it vanishes if and only if v0 2 V ? . Hence, regularity at all k-points is precisely the condition that ker.qjV ? / D f0g, as asserted. Lemma 7.1.1. If k 0 =k is a separable extension field then .Vk 0 ; qk 0 / is regular. Proof. The formation of V ? commutes with any extension on k. Fix a k-basis fe1 ; : : : ; ed 2n g of V ? . In the coordinates determined by this basis, the 2-linear qjV ? is given by c1 x12 C


C cd 2n xd2 2n with ci 2 k. The regularity hypothesis says that fci g is k 2 -linearly independent inside k, and we just have to check that fci g is k 0 2 -linearly independent inside k 0 . Since k 0 2 is separable over k 2 , and k is purely inseparable algebraic over k 2 , the tensor product k 0 2 ˝k 2 k is a field. Thus, the natural map of k-algebras k 0 2 ˝k 2 k ! k 0 is injective, so we are done.  Assume 0 < dim V ? < dim V , which is to say Bq is degenerate but nonzero, so d > 2n C 1 > 3. By regularity qjV ? ¤ 0, so q is non-degenerate if and only if dim V ? D 1 (equivalently, d D 2n C 1). In the non-degenerate case the group

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Constructions with regular degenerate quadratic forms

scheme O.q/ is identified with SO.q/  2 where SO.q/ is an adjoint connected semisimple group that is absolutely simple of type Bn , so in such cases SO.q/ is the maximal smooth closed subgroup of O.q/. Motivated by the non-degenerate case, in general we define SO.q/ to be the maximal smooth closed k-subgroup of the k-group scheme O.q/; this is the ks =k-descent of the Zariski closure of O.q/.ks / in O.q/ks . Proposition 7.1.2. The SO.q/-action on V is trivial on V ? . The k-group SO.q/ is absolutely pseudo-simple of type Bn with trivial center and its Cartan ksubgroups are tori. For a linear complement V 0 to V ? in V , the quadratic space .V 0 ; qjV 0 / is non-degenerate of dimension 2n and if qjV 0 is split .i.e., hyperbolic/ then SO.q/ is pseudo-split. Moreover, SO.q/ admits a pseudo-split ks =k-form and every maximal k-torus T of SO.q/ is contained in a Levi k-subgroup. As in the semisimple adjoint type-B case, Proposition 7.2.2 will show that if .V 0 ; q 0 / is a second such regular quadratic space then all k-isomorphisms SO.q/ ' SO.q 0 / arise from conformal isometries .V; q/ ' .V 0 ; q 0 /. (See §7.2 for the notion of conformal isometry.) Also, Proposition 7.1.6 will relate the k-rank of SO.q/ and the dimension of maximal hyperbolic subspaces of V . Proof. It is clear that V ? is stable under the action of O.q/ on V . For g 2 SO.q/.ks / D O.q/.ks / and v 2 Vk?s , q.g.v/

v/ D q.g.v//

q.v/ D q.v/

q.v/ D 0:

But g.v/ v 2 Vk?s and q W Vk?s ! ks is injective, so g.v/ v D 0. The Zariskidensity of SO.q/.ks / in SO.q/ks then implies that SO.q/ acts trivially on V ? . For any linear complement V 0 to V ? in V , the natural map V 0 ! V =V ? is an isomorphism identifying Bq on V 0 with the non-degenerate symplectic form B q on V =V ? induced by Bq , so dim V 0 D 2n and qjV 0 is non-degenerate. For a line `  V ? and W WD V 0 ˚ `, qjW is non-degenerate and SO.qjW / can be realized as a type-Bn adjoint connected semisimple k-subgroup of SO.q/ acting trivially on V ? . Thus, any maximal torus of SO.qjW / has dimension n. Letting T be a maximal k-torus of SO.q/, by taking V 0 to be a T -equivariant complement to V ? in V we see that the k-subgroup SO.qjW /  SO.q/ contains T as a maximal torus (since T is a maximal torus of SO.q/). Hence, no nonzero vector of V 0 is fixed under T , and V 0 is the unique T -equivariant linear complement to V ? in V . It also follows that dim T D n. Note that if qjV 0 is split then SO.qjW / is k-split; i.e., SO.qjW / contains a k-split maximal k-torus. Any such torus is then a maximal k-torus of SO.q/, so in such cases SO.q/ contains a

111

7.1 Regular degenerate quadratic forms

k-split maximal k-torus. To show in general (allowing non-split T ) that SO.q/ is absolutely pseudo-simple with trivial center and admits SO.qjW / as a Levi k-subgroup, it suffices to work over ks . We will show at the end of this proof that SO.q/ admits a pseudo-split ks =kform, so for now we may and do assume k D ks . Hence, SO.q/ is the Zariski closure in O.q/ of the group O.q/.k/ of k-points. The action of O.q/ on V preserves V ? so it defines a k-homomorphism  W O.q/ ! Sp.B q / ' Sp2n : Fix a maximal k-torus T of SO.q/ and let V 0 be the T -equivariant complement to V ? in V . As above, let W D V 0 ˚ ` and naturally identify L WD SO.qjW / with a k-subgroup of SO.q/ containing the maximal k-torus T . The restriction of  to SO.qjW / is the natural exceptional isogeny from a group of adjoint type Bn onto a simply connected group of type Cn in characteristic 2, so  carries SO.q/ onto the semisimple group Sp.B q /. Hence  must kill the smooth connected unipotent normal k-subgroup Ru;k .SO.q/0 /. Therefore, to show that this k-unipotent radical is trivial it would suffice to check that .ker /.k/ is trivial. If g 2 .ker /.k/ then g.v/ v 2 V ? for all v 2 V , so q.v/ D q.g.v// D q..g.v/

v/ C v/ D q.g.v/

v/ C q.v/

and thus g.v/ v 2 ker.qjV ? / D f0g for all v 2 V . This says g D 1, as desired. We have shown that SO.q/0 is pseudo-reductive, so H WD D.SO.q/0 / is pseudo-semisimple. The isogeny jL W L ! Sp.B q / DW L with unipotent kernel maps T isomorphically onto a maximal torus T of L. But clearly ZL .T / D T , so ZSO.q/ .T / is a smooth k-subgroup of SO.q/ whose image in L is the isomorphic copy T of T . Hence, ZSO.q/ .T /.k/ D T .k/ since .ker /.k/ D 1, so ZSO.q/ .T / D T . In particular, all Cartan k-subgroups of SO.q/0 and H are tori, and the schematic center ZSO.q/ is contained in T . Thus, ZSO.q/ D 1 since the connected semisimple subgroup SO.qjW / ' SO2nC1 has trivial center. Let f W SO.q/0  L be the restriction of  to SO.q/0 . Since f jT W T ! T is an isomorphism, .ker fk /0red is unipotent. Thus, by [CGP, Rem. 2.3.6], the sets of roots ˆ WD ˆ.SO.q/0 ; T / and ˆ.L; T / coincide up to positive rational multipliers on the roots, so the Weyl groups coincide and if  2 X .T / is “regular” in the sense that ha; i ¤ 0 for all a 2 ˆ.SO.q/0 ; T / then the cocharacter  WD f ı  2 X .T / is likewise regular for .L; T /. For such , it follows

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Constructions with regular degenerate quadratic forms

that B WD PSO.q/0 ./ is a minimal pseudo-parabolic k-subgroup of SO.q/0 that meets H in a minimal pseudo-parabolic k-subgroup PH ./, and the image of each under f is a common Borel k-subgroup B of L. Since H is visibly normal in SO.q/, for any g 2 SO.q/.k/ we see that gT g 1 is also a maximal k-torus in H and hence is an H.k/-conjugate of T . To show that g 2 H.k/ (thereby proving that SO.q/ is (connected and) pseudosemisimple), we may replace g with a left H.k/-translate so that g normalizes T . The equality of Weyl groups W .H; T / D W .SO.q/0 ; T / D W .ˆ/ further allows us to replace g with another H.k/-translate that also normalizes B. Hence, f .g/-conjugation on L preserves .B; T /, so f .g/ 2 T .k/. But f W T ! T is an isomorphism, so by replacing g with a suitable left T .k/-translate we can arrange that g 2 .ker f /.k/  .ker /.k/ D 1. This completes the proof that SO.q/ is pseudo-semisimple. The comparison of root systems and Weyl groups for .SO.q/; T / and .L; T / via f implies that ˆ WD ˆ.SO.q/; T / is irreducible (as its Weyl group acts irreducibly on X.T /Q ), so SO.q/ is absolutely pseudo-simple with rank n. By the classification of irreducible root systems, the containment ˆ.L; T /  ˆ then forces ˆ to be either Bn or BCn , and in the former case L is a Levi k-subgroup of SO.q/ by (the proof of) [CGP, Thm. 3.4.6]. To rule out the possibility that ˆ is non-reduced, it suffices to show that for a short root b 2 ˆ.L; T / the corresponding root group Ub of SO.q/ has Lie algebra supporting only the T -weight b (and not also 2b). Note that 2b is a long root in ˆ.L; T /, so if 2b 2 ˆ.SO.q/; T / then the nontrivial smooth connected 2b-root group U2b of SO.q/ must map onto the 1-dimensional 2b-root group U 2b  L under f . By equivariance of f W U2b ! U 2b with respect to T ' T , this quotient map between root groups corresponds to a linear map between vector spaces, so it has smooth kernel that must be trivial (as .ker f /.k/ is trivial). Hence, the quotient f W Ub  U 2b splits as a direct product, thereby providing a nontrivial smooth (connected) k-subgroup of ker f , a contradiction. This shows that ˆ D ˆ.L; T /. Finally, we allow k to be arbitrary (of characteristic 2) and will show that SO.q/ admits a pseudo-split ks =k-form. Letting V 0 be a linear complement to V ? in V , .V; q/ is the orthogonal sum of the restrictions .V 0 ; q 0 / and .V ? ; q ? / of q to V 0 and V ? respectively. Since dim V 0 D 2n is even, there is a split ks =k-form .V 0 ; q0 / of .V 0 ; q 0 /, so for the orthogonal sum .V ; q/ of .V 0 ; q0 / and .V ? ; q ? / we see that SO.q/ is a pseudo-split ks =k-form of SO.q/.  We continue to assume that 0 < dim V ? < dim V and that .V; q/ is regular. The k-group SO.q/ is non-reductive whenever q is degenerate, which is to say

113

7.1 Regular degenerate quadratic forms

whenever dim V ? > 1, due to the following result. Proposition 7.1.3. The dimensions of the root groups of SO.q/ks are dim V ? for short roots and 1 for long roots, with the convention that if n D 1 then the roots are short, so SO.q/ is not reductive whenever dim V ? > 1. The minimal field of definition over k for the geometric unipotent radical of SO.q/ is the subfield K  k 1=2 generated over k by the square roots .q.v/=q.v 0 //1=2 for v; v 0 2 V ? f0g. In particular, ŒK W k > dim V ? . If SO.q/ is pseudo-split of type A1 D B1 then it is isomorphic to PHV ? ;K=k . Proof. By the regularity of q, for any nonzero v0 2 V ? the k-linear map V ? ! k 1=2 defined by v 7! .q.v/=q.v0 //1=2 is injective, so ŒK W k > dim V ? . Since the formation of K is compatible with separable extension on k, and an equality between purely inseparable extensions of k holds if and only if it does so after a separable extension of k, we may assume that the group SO.q/ is pseudo-split. We fix a split maximal k-torus T in this group and will use notation (such as L, L, and T ) introduced in the proof of Proposition 7.1.2. If n > 1 then the long roots in ˆ WD ˆ.SO.q/; T / are the short roots in ˆ.L; T /. By equivariance with respect to the isomorphism T ' T induced by f W SO.q/  L, the associated quotient map between corresponding root groups of SO.q/ and L for such roots corresponds to a linear map between vector spaces and hence must be an isomorphism (as .ker f /.k/ is trivial). Thus, if n > 1 then the long root groups of SO.q/ are 1-dimensional. We next show (for any n > 1) that the root group for a short root b has dimension d 2n D dim V ? for all n > 1. Since k D ks , we can choose a k-basis fe1 ; : : : ; ed g of V and k-isomophism GLn1 ' T so that two properties hold: q D x1 x2 C


C x2n

2 2 1 x2n C c2nC1 x2nC1 C


C cd xd

(7.1.3)

with fci g linearly independent over k 2 , and any point t D .t1 ; : : : ; tn / 2 GLn1 D T acts on the k-vector space V D k d via t:.x1 ; : : : ; xd / D .t1 x1 ; t1 1 x2 ; : : : ; tn x2n

1 ; tn

1

x2n ; x2nC1 ; : : : ; xd /

with b.t1 ; : : : ; tn / WD t1 1 . The subtorus S WD .ker b/0red consists of those t 2 T such that t1 D 1. Clearly the b-root group Ub lies in D.ZSO.q/ .S// DW H . For V0 D ke1 ˚ ke2 ˚ V ? and q0 D qjV0 , the natural k-subgroup SO.q0 /  SO.q/ is centralized by S and so lies inside H . But all points of ZSO.q/ .S/ preserve the weight spaces fkei g2 4. The explicit form of q identifies K with k. c =c ; : : : ; c =c 4 3 d 3 /. p ? ? We k-linearly embed V into K via v 7! q.v/=q.e3 /, so V is thereby a ksubspace of K containing 1 that generates K as a k-algebra. This subspace defines the k-subgroup PHV ? ;K=k  RK=k .PGL2 / whose diagonal k-torus is a Cartan k-subgroup since K 2  k (see Definition 3.1.2 and Proposition 3.1.4).

7.1 Regular degenerate quadratic forms

115

For the K-isogeny … W PGL2 ! SL2 it is easy to check via consideration of open cells that RK=k .…/ carries PHV ? ;K=k into the natural k-subgroup SL2  RK=k .SL2 / via a k-homomorphism that is injective on k-points (since q W V ? ! k is injective). Using the identification of each T -root group in SO.q/ with V ? (computed above for one root, and done likewise for the opposite root with the roles of e1 and e2 swapped), we get an evident k-scheme isomorphism  from an open cell  of .SO.q/; T / to that of .PHV ? ;K=k ; D/ over the natural map from each side into L D SL2 . The map  is a rational homomorphism because this can be checked on k-points in a dense open subset of    (using that .ker …/.K/ D 1), so  uniquely extends to an isomorphism of k-groups. Thus, the minimal field of definition over k for Ru .SO.q/k /  SO.q/k coincides with that of PHV ? ;K=k . By Proposition 3.1.4 this field is (uniquely) k-isomorphic to kŒV ?  D K. (In the initial setup with any n > 1, this proof shows that if k D ks then for any short b 2 ˆ.SO.q/; T / we have SO.q/b ' PHjv .V ? /;K=k where 0 p jv0 .v/ D q.v/=q.v0 / for any v0 2 V ? f0g.)  Remark 7.1.4. The description of K in Proposition 7.1.3 shows that the kernel V WD ker.qK jV ? /  VK? is a hyperplane, and the SO.q/K -action on VK is trivial K

on V (it is even trivial on VK? , by Proposition 7.1.2). Thus, the quadratic form qK WD qK mod V W VK =V ! K

(7.1.4)

is well-defined and non-degenerate on the space VK =V of K-dimension 2n C 1, and there is an evident natural map h W SO.q/K ! SO.qK /. In the proof of Proposition 7.1.2 we built Levi k-subgroups L  SO.q/ of the form SO.qjW / where W D V 0 ˚ ` for any linear complement V 0 to V ? in V and line ` in V ? . The restriction of h to LK is an isomorphism since WK ! VK =V is an ss isomorphism, so h identifies SO.q/K with SO.qK /. Proposition 7.1.5. Let T be a maximal k-torus of SO.q/. There is a unique T -equivariant linear complement V0 to V ? inside V . For every line `  V ? the restriction of q to the .2n C 1/-dimensional subspace W WD V0 ˚ ` is nondegenerate, the k-subgroup scheme L of points g 2 SO.q/ such that g.W / D W is a Levi k-subgroup isomorphic to SO.qjW / via g 7! gjW , and every Levi ksubgroup of SO.q/ containing T arises in this manner for a unique `. If n > 2 then there is a unique connected semisimple subgroup of SO.q/ of type Dn containing T , and it is SO.q0 / for the non-degenerate q0 WD qjV0 .with SO.q0 / ,! SO.q/ defined via isometries of V D V0 ˚ V ? that preserve V0 /.

116

Constructions with regular degenerate quadratic forms

Proof. The uniqueness assertions permit us to assume k D ks , so T is split and we may decompose V as a direct sum of T -weight spaces. Since all maximal k-tori in G WD SO.q/ are SO.q/.k/-conjugate (as k D ks ), for the rest of the proof it suffices to treat a single T . We fix a k-linear complement V0 to V ? in V , so q0 WD qjV0 is non-degenerate of rank 2n and .V; q/ is the orthogonal sum of q0 and qjV ? . Since k D ks , we may choose a k-isomorphism q0 ' x1 x2 C


C x2n 1 x2n , so we obtain a maximal k-torus GLn1  G preserving V0 via the action .t1 ; : : : ; tn /:.x1 ; : : : ; x2n / D .t1 x1 ; t1 1 x2 ; : : : ; tn x2n

1 ; tn

1

x2n /

(7.1.5)

on V0 and the trivial action on V ? . We may (and do) take T to be this torus. The T -weights occurring on V0 are visibly nontrivial and have 1-dimensional weight spaces, so V0 is the unique T -equivariant linear complement to V ? in ˙1 V . The 2n characters ˙ of T are exactly the short roots in i W .t1 ; : : : ; tn / 7! ti ss ˆ.G; T / D ˆ.GK ; TK / D ˆ.SO.qK /; TK /. Consider a line ` in V ? . The injectivity of qjV ? implies that qj` is nonzero, so W D V0 ˚ ` equipped with the quadratic form qjW is the orthogonal sum of qj` and q0 . Hence, .W; qjW / is non-degenerate of dimension 2n C 1, and in the proof of Proposition 7.1.2 we showed that the inclusion SO.qjW / ,! G defined by imposing the trivial action on V ? is a Levi k-subgroup. This is precisely the k-subgroup scheme of G defined by the combined conditions of preservation of W and triviality on V ? because the orthogonal group scheme O.qjW / coincides with SO.qjW /  2 and triviality on ` D W \ V ? characterizes SO.qjW / inside O.qjW /. The subspace W  V (and hence the line ` D W \ V ? ) is uniquely determined by the maximal k-torus T  G and the Levi k-subgroup SO.qjW /  G containing T because of the elementary fact that W is the SO.qjW /-span of V0 (see the explicit description of unipotent orthogonal automorphisms arising from short root groups for the case n D 1 in the proof of Proposition 7.1.3). Since V ? is preserved by the action of G on V , to show that every Levi k-subgroup L of G containing T arises by the above construction for some (necessarily unique) line in V ? we recall from [CGP, Thm. 3.4.6] that any such L is uniquely determined by its T -root groups inside those of G for roots belonging to a fixed basis  of the common root system ˆ.L; T / D ˆ.G; T /. The long root groups of .G; T / are 1-dimensional, so they are contained in L. We may and do choose  so that its unique short root b is the character 1 of T D GLn1 . Equip the b-root group Ub  G with its unique T -equivariant linear structure. The Levi k-subgroup L  G containing T is uniquely determined by its b-root group that in turn corresponds to a line `0 inside Ub .k/. The k-vector group

7.1 Regular degenerate quadratic forms

117

Ub  GL.V / was identified with V ? in the proof of Proposition 7.1.3: to each v 2 V ? we associate the orthogonal automorphism of V that pointwise fixes both V ? and every ej for j ¤ 1, and carries e1 to e1 q.v/e2 C v. The b-root group of .L; T / corresponds to those v lying in the line `0 , but the same calculations applied to the non-degenerate qjV0 ˚`0 show that Levi k-subgroup L0 of G associated to .T; `0 / via the above construction has the same b-root group inside Ub . Hence, L D L0 . Now assume n > 2. The group SO.q0 / is naturally identified with a connected semisimple k-subgroup H  SO.q/ of type Dn containing T as a maximal k-torus. As any connected semisimple subgroup of G of type Dn containing T is generated by its root groups, which in turn are precisely the 1-dimensional long root groups of .SO.q/; T /, the uniqueness of H follows.  Assume 1 < dim V ? < dim V . It can happen that the non-reductive absolutely pseudo-simple k-group SO.q/ is k-anisotropic (i.e., does not contain GL1 ss as a k-subgroup), even when the K-group SO.q/K D SO.qK / (see Remark 7.1.4) is K-split. This is interesting because for k of any characteristic and an absolutely pseudo-simple standard pseudo-reductive k-group G with minimal field of definition K=k for Ru .Gk /  Gk , the k-rank of G coincides with the K-rank of the connected semisimple GKred (seen by inspection of the standard construction). To make such .V; q/, first observe that anisotropicity of .V; q/ implies that the pseudo-reductive group SO.q/ is k-anisotropic. Indeed, more generally: Proposition 7.1.6. Let .V; q/ be a regular quadratic space over a field k. There exists an orthogonal sum of r hyperbolic planes in .V; q/ if and only if the kgroup scheme O.q/ contains a split k-torus of dimension r. In particular, q has an isotropic vector if and only if O.q/ contains a nontrivial split k-torus. Proof. The case of non-degenerate .V; q/ is classical (by consideration of weight vectors). Thus, we may assume .V; q/ is degenerate, so by regularity necessarily char.k/ D 2. For any pairwise orthogonal hyperbolic planes H1 ; : : : ; Hr in V , clearly V D H1 ?


? Hr ? V 0 for the orthogonal complement V 0 of ? Hi , so there is an evident split r-torus inside O.q/. Now suppose O.q/ contains T D GLr1 as a k-subgroup. We seek a pairwise orthogonal collection of r hyperbolic planes inside V , so we may assume r > 0. Since T acts on V through a k-subgroup inclusion of T into GL.V /, the weights for T on V generate X.T /Q . Hence, we can find r linearly independent nontrivial weights 1 ; : : : ; r for T on V .

118

Constructions with regular degenerate quadratic forms

If  2 X.T / is nontrivial then any v in the -weight space V ./ satisfies q.v/ D q.t:v/ D q..t/v/ D .t/2 q.v/ for all t 2 T , forcing q.v/ D 0. Likewise, V ./ and V .0 / are Bq -orthogonal for all nontrivial 0 ¤  and each is Bq -orthogonal to V T . Since the 2-linear qjV ? has trivial kernel due to regularity, we see that V ?  V T . For nonzero vi 2 V .i / (so vi 62 V ? ) we have Bq .vi ;
/ ¤ 0, so there exists vi0 2 V . i / such that Bq .vi ; vi0 / D 1. Thus, vi and vi0 span a hyperbolic plane Hi , and the Hi ’s are pairwise orthogonal.  We next build examples of non-degenerate anisotropic .V0 ; q0 / over some k with char.k/ D 2 and dim.V0 / D 2n > 2 such that SO.q0 / splits over a finite extension K=k for which squaring q W K ! K lands inside k and q0 ˚ cq W V0 ˚ K ! k is k-anisotropic for some c 2 k  . For .V; q/ WD .V0 ; q0 / ? .K; cq/, the k-group SO.q/ is then anisotropic and the minimal field of definition over k for Ru .SO.q/k / is K=k (as we see via the explicit description in Proposition 7.1.3 of that minimal field extension of k in terms of square roots of ratios of nonzero values of qjV ? ); our examples will have the additional property that for n even, SO.qK / is K-split. The method is inspired by a construction of Richard Weiss that in turn is modelled on [TW, (14.19)–(14.23)]. Example 7.1.7. Let  be a field of characteristic 2 and let k D .t1 ; : : : ; tnC1 / be the rational function field over  in n C 1 variables with any n > 1. Let F be a quadratic extension of k such that for 1 6 i 6 n C 1 the ti -adic valuation on k has ramification index 1 and residual degree 2 in F . Among such F are the extensions  0 .t1 ; : : : ; tnC1 / for a quadratic extension  0 of  (if such  0 exist) P or the splitting field of the separable polynomial x 2 x j tj 2 kŒx (which is irreducible over k since x 2 x t is irreducible over L.t/ for any field L). Define the anisotropic binary quadratic form  D NF =k W F ! k over k. Let V0 denote the 2n-dimensional k-vector space F ˚n , and let qi W V0 ! k be the composition of  with the i th projection V0 ! F . Squaring on K WD p p . t1 ; : : : ; tn ; tnC1 / is a quadratic form q W K ! k valued in the subfield 2 k 0 WD .t1 ; : : : ; tn ; tnC1 / of k. Consider the quadratic form qD

X

ti qi C tnC1 q W V D V0 ˚ K ! k

16i 6n

over k, so if F=k is separable then SO.q/ is absolutely pseudo-simple and SO.q/F is pseudo-split (with root system Bn ). We shall first prove (for every

7.1 Regular degenerate quadratic forms

119

n > 1) that q is anisotropic, and then that for suitable separable F=k as above and even n > 2 the connected semisimple K-group SO.qK / is split. Since q is valued in k 0 , to prove the anisotropicity it suffices to show by induction on n that for any c1 ; : : : ; cn 2 F and  2 k 0 such that n X

ti .ci / C tnC1  D 0;

(7.1.7)

i D1

necessarily c1 D


D cn D 0 and  D 0. For all 1 6 i 6 n C 1 and all c 2 F  , ord ti ..c// is even since the quadratic extension F=k has ramification index 1 and residual degree 2 over the ti -adic discrete valuation on k. Likewise, ord tnC1 ./ is even when  ¤ 0 since  2 k 0 . Define Ak D Œt1 ; : : : ; tnC1   k, so the fraction field of Ak is k. Let AF denote the module-finite integral closure of Ak in F=k, so AF has fraction field F and dimension n C 1. Since AF is a normal noetherian domain, it coincides with the intersection of its local rings at height-1 primes. By hypothesis the ti -adic valuation vi on k has a unique lift wi on F and ti is a uniformizer for wi , so ti is a unit at all height-1 primes of AF aside from wi . Hence, ti AF is the prime ideal of elements of AF with positive wi -valuation, so AF =ti AF is a domain whose fraction field is the residue field at wi and the natural map Ak =ti Ak ! AF =ti AF is injective. Consider a solution to (7.1.7). We want to show that all ci and  vanish. Scaling through by the square of a suitable nonzero element of Ak allows us to 2 arrange that ci 2 AF for all i and  2 Œt1 ; : : : ; tn ; tnC1  DW Ak 0 . Since the local ring .AF /wi at the height-1 prime ti AF coincides with the localization of AF at the height-1 prime ti Ak for Ak , we see that the induced injective map of discrete valuation rings .Ak /vi ! .AF /wi is module-finite and hence free of rank 2. In particular, it makes sense to compute norms relative to this ring extension, so for any c 2 AF the norm .c/ 2 Ak relative to F=k has mod-ti Ak reduction that coincides with i .c mod ti AF / where i is the norm relative to the induced quadratic extension of residue fields at vi and wi . Such compatibility of norms will enable us to carry out induction on n. P If  ¤ 0 then niD1 ti .ci / 2 k  and its tnC1 -adic valuation is equal to 1 C ord tnC1 ./ 2 1 C 2Z (see (7.1.7)). Each ti .ci / that is nonzero must have even tnC1 -adic valuation, so there would have to be at least two nonzero ci ’s if  ¤ 0. In particular, this rules out the case n D 1, so we may assume n > 1 and that the case of n 1 is settled. For 1 6 i 6 n, since Ak 0 =ti Ak 0 ! Ak =ti Ak is injective by inspection, reducing (7.1.7) modulo ti permits us to apply induction with n 1 to conclude that ti divides both  in Ak 0 and cj in AF for every j ¤ i .

120

Constructions with regular degenerate quadratic forms

Q Hence,  is divisible by  WD niD1 ti in Ak 0 and ci is divisible by i D =ti in AF for all 1 6 i 6 n because the normality of AF reduces i -divisibility to tj -divisibility for each j ¤ i separately (as tj is a unit at all height-1 primes of AF apart from wj ). Substituting the resulting identities  D   for   2 Ak 0 and ci D i ci for ci 2 AF into (7.1.7) and canceling a common factor of  throughout yields n X .=ti /.ci / C tnC1   D 0: i D1

Reducing modulo ti kills all terms in the sum apart from the i th, so Y . tj /
i .ci mod ti / D tnC1 .  mod ti / 16j 6n;j ¤i

in the residue field Frac.Ak =ti Ak / at vi (with i the norm into this from the residue field at wi ). If both sides are nonzero then the left side has even tnC1 adic valuation whereas the right side has odd tnC1 -adic valuation. This is absurd, so both sides vanish for all i . We conclude that ci D  ci0 for some ci0 2 AF and that   is divisible by t1 ; : : : ; tn in Ak 0 , so  D  2  0 for some  0 2 Ak 0 . Substituting these into (7.1.7) and cancelling the common factor of  2 throughout brings us to another solution .c10 ; : : : ; cn0 ;  0 / of (7.1.7) with ci0 2 AF and  0 2 Ak 0 . We can keep iterating the process to conclude that  and every ci are infinitely divisible by  D t1


tn , so  D 0 and ci D 0 for all i . This completes the proof of k-anisotropicity of q. Now assume F=k is separable, so KF WD K ˝k F is a field and the quadratic form qK W .KF /˚n ˚ K ! K as in (7.1.4) is an orthogonal sum of tnC1 x 2 and n copies of the non-degenerate rank-2 quadratic space .KF ; NKF =K /. In general, an orthogonal sum of two copies of a non-degenerate rank-2 quadratic space .U; h/ over any field L of characteristic 2 is a split quadratic space. Indeed, by scaling we can assume .U; h/ D .L ˚ L; x 2 C xy C ay 2 / for some a 2 L, and if fe1 ; e2 g and fe10 ; e20 g are the associated bases of two orthogonal copies of U then fe1 C e10 ; a.e1 C e10 / C e2 g; fe2 C e20 ; e2 C e20 C e10 g are bases of null vectors for orthogonal hyperbolic planes spanning the orthogonal sum U ? U . Thus, if n is even then whenever F=k is separable the quadratic form qK is the orthogonal sum of tnC1 x 2 and n hyperbolic planes, so SO.qK / is K-split. (If n is odd and F=k is separable then typically SO.qK / has K-rank n 1. To be precise, assume in addition that the quadratic residue field extension

7.2 Conformal isometries

121

for F=k at tnC1 is separable, or more generally is distinct from that of K=k, as we can certainly arrange. The preceding method and Witt cancellation reduce our task to checking that tnC1 viewed in K is not a norm from KF , as holds since KF =K has residual degree 2 at tnC1 .) Example 7.1.8. Consider G D SO.q/ for finite-dimensional regular quadratic spaces .V; q/ over k such that 0 < dim V ? < dim V . p For any nonzero v0 2 V ? we obtain a k-linear injection jv0 W V ? ! K via v 7! p q.v/=q.v0 /, and for any nonzero v1 2 V ? clearly jv1 D jv0 , where  D q.v0 /=q.v1 / 2 K. Hence, the k-subspace jv0 .V ? /  K is independent of v0 up to K  -scaling, so the k-subalgebra F D f 2 K j 
jv0 .V ? /  jv0 .V ? /g of K is independent of v0 . This is called the root field of .V ? ; qjV ? /. Observe that V ? thereby acquires a structure of F -vector space over its k-linear structure, and this is independent of v0 ; it is characterized by the property q.v/ D 2 q.v/ for  2 F and v 2 V ? . The formation of F commutes with separable extension on k, and ŒF W k is a power of 2, so if dimk V ? is odd then F D k. In the non-degenerate case F D k, but in the degenerate case F may be larger than k (e.g., for V ? D K with qjV ? the squaring map K ! k we have F D K). It is obvious that the (long) root field of G is k. In general we claim that F coincides with the short root field of G. It suffices to check this equality of purely inseparable extensions of k after scalar extension to ks . In the proof of Proposition 7.1.3 it has been shown that if k D ks then for any short root b 2 ˆ.G; T / we have Gb ' PHjv .V ? /;K=k for some v0 2 V ? f0g. 0

Definition 7.1.9. For regular .V; q/ satisfying 0 < dim V ? < dim V , the spin group Spin.q/ is the universal smooth k-tame central extension of SO.q/. The k-group SO.q/ in Definition 7.1.9 is of minimal type since ZSO.q/ D 1, so by Proposition 5.3.3 the k-group Spin.q/ is of minimal type. This recovers the usual spin group in the non-degenerate case (i.e., when dim V ? D 1).

7.2 Conformal isometries We next turn our attention to ks =k-forms of SO.q/ for finite-dimensional regular quadratic spaces .V; q/ over k such that 0 < dim V ? < dim V ; our main interest will be in the degenerate case (i.e., dim V ? > 1). A conformal isometry between quadratic spaces over a field is a linear isomorphism that respects the quadratic

122

Constructions with regular degenerate quadratic forms

forms up to a nonzero scaling factor (which is uniquely determined when the quadratic forms are nonzero). Let CIsom.q 0 ; q/ be the set of conformal isometries onto .V; q/ from a finitedimensional regular quadratic space .V 0 ; q 0 / satisfying 0 < dim V 0 ? < dim V 0 . Let CO.q/  GL.V / be the smooth k-subgroup that is the Galois descent of the Zariski closure of CIsom.qks ; qks / inside GL.Vks /. The (possibly non-smooth) scheme of conformal isometries is the closed subgroup scheme of GL.V / assigning to any k-algebra A the group of pairs .L; / consisting of an A-linear automorphism L of VA and a unit  2 A such that qA ı L D 
qA ; its maximal smooth k-subgroup is CO.q/ by [CGP, Lemma C.4.1], so if E=k is an arbitrary separable extension field then CO.q/.E/ is the group of conformal isometries of .V; q/E onto itself. Remark 7.2.1. If q is degenerate then the k-group CO.q/ that is smooth by definition generally does not contain the group scheme O.q/; see Example 7.2.3. Let F be the root field of .V ? ; qjV ? / as in Example 7.1.8, so it is also the short root field of SO.q/ and V ? has a canonical structure of F -vector space over its given k-linear structure (with q.v/ D 2 q.v/ for v 2 V ? and  2 F ). When G WD SO.q/ contains a split maximal k-torus T , we can build a copy of RF =k .GL1 / inside CO.q/ centralizing T , as follows. Consider the unique T equivariant decomposition V D V0 ˚V ? as in Proposition 7.1.5, and decompose V0 into a direct sum of T -weight lines `˙1 ; : : : ; `˙n for pairs of opposite weights (the short roots in ˆ.G; T /). For any  2 F  we have 2 2 k  and define Œ 2 CO.q/ via ŒjV ? D , Œj`i D 2 for i D 1; : : : ; n, and Œj`i D 1 for i D 1; : : : ; n. This construction defines a k-subgroup inclusion RF =k .GL1 / ,! CO.q/

(7.2.1.1)

resting on a choice within each pair of opposite nontrivial T -weights on V . The set of such choices is permuted transitively under the action of NSO.q/ .T /.k/ on T through its Weyl group quotient W .ˆ.G; T //, so the composite map q W RF =k .GL1 / ! CO.q/=SO.q/

(7.2.1.2)

is intrinsic to the k-split T . It follows from the SO.q/.ks /-conjugacy of maximal ks -tori in SO.q/ks that (7.2.1.2) is independent of T and so may be constructed in general over k via ks =k-descent (not depending on an initial choice of maximal k-torus). Using k-split T , the restriction of (7.2.1.1) to the maximal k-torus GL1 

7.2 Conformal isometries

123

RF =k .GL1 / carries  to the product of -multiplication and .; 1=; : : : ; ; 1=; id/ 2 T  GL.V0 /  GL.V ? /;

(7.2.1.3)

so if F D k (as when q is non-degenerate) then (7.2.1.1) provides nothing in CO.q/ beyond what is already obtained from T and the central GL1 . In general RF =k .GL1 / \ T D 1 schematically inside GL.V /, so since ZSO.q/ .T / D T it follows that the k-homomorphism RF =k .GL1 / n SO.q/ ! CO.q/ provided by the split k-torus T has trivial kernel (equivalently, ker q D 1). The k-subgroup RF =k .GL1 /  T of CO.q/ contains the central GL1 due to (7.2.1.3). The k-subgroup SO.q/  GL.V / is insensitive to k  -scaling of q (and likewise after any separable extension on k), so there is an evident action of CO.q/=GL1 on SO.q/ and the natural map CIsom.q 0 ; q/ ! Isomk .SO.q 0 /; SO.q//

(7.2.1.4)

is equivariant with respect to CO.q/.k/ ! CO.q/.k/=k  . Proposition 7.2.2. Consider .V; q/ and .V 0 ; q 0 / as above .with no pseudo-split hypothesis on SO.q//. (i) The map .7:2:1:4/ is the quotient by the k  -action on CIsom.q 0 ; q/. In particular, SO.q 0 / ' SO.q/ if and only if .V 0 ; q 0 / is conformal to .V; q/, and CO.q/.k/=k  ' Autk .SO.q//. (ii) The map q in .7:2:1:2/ is an isomorphism, so CO.q/ is connected. In particular, SO.q/.k/ ' Autk .SO.q// if and only if F D k, in which case the ks =k-forms of SO.q/ are the k-groups SO.q 0 / for .V 0 ; q 0 / that becomes isometric to .V; q/ over ks . Moreover, in general SO.q 0 / is a ks =k-form of SO.q/ if and only if dim V 0 =V 0 ? D 2n and q 0 jV 0 ? becomes conformal to qjV ? over ks . Proof. The first assertion in (i) clearly implies the rest of (i), and both sides of (7.2.1.4) are compatible with Galois descent. By Hilbert’s Theorem 90, to prove (i) it suffices to consider the case k D ks . Granting (ii) when k D ks , let us deduce (ii) in general. The general k-isomorphism assertion in (ii) is immediate via Galois descent, so if F D k then the central GL1 in CO.q/ is a complement to SO.q/ (due to how this central GL1 lies inside RF =k .GL1 /  T via q ), so in such cases the ks =k-forms of SO.q/ are classified up to k-isomorphism by the

124

Constructions with regular degenerate quadratic forms

cohomology set H1 .ks =k; Autks .SO.qks /// D H1 .ks =k; CO.q/.ks /=ks / D H1 .ks =k; CO.q/.ks // since H1 .ks =k; ks / D 1. Hence, if F D k then this cohomology set classifies conformal isometry classes .V 0 ; q 0 / that become conformal to .V; q/ over ks (forcing dim V 0 =V 0 ? D dim V =V ? ), and the form of SO.q/ associated to such a .V 0 ; q 0 / is SO.q 0 /. Likewise, if F D k then Autks .SO.qks // D SO.q/.ks / D O.q/.ks /, so in such cases the ks =k-forms of SO.q/ are classified by the set H1 .ks =k; O.q/.ks // of isometry classes of .V 0 ; q 0 / over k that are isometric to .V; q/ over ks . Explicitly, if F D k then the ks =k-form of SO.q/ classified by .V 0 ; q 0 / is the k-group SO.q 0 /. Clearly if .V 0 ; q 0 / and .V; q/ become conformal over ks then the same holds for q 0 jV 0 ? and qjV ? . Conversely, suppose these defect spaces are conformal over ks and dim V 0 =V 0 ? D 2n. Each of .V; q/ and .V 0 ; q 0 / is isometric to the orthogonal sum of its defect space and a non-degenerate quadratic space of dimension 2n, yet there is only one isometry class of 2n-dimensional non-degenerate quadratic spaces over ks , so .V; q/ and .V 0 ; q 0 / become conformal over ks . This completes the reduction of the proof of (ii) to the case k D ks . For the remainder of this proof we may and do assume k D ks . We shall first prove that if SO.q 0 / ' G WD SO.q/ then .V 0 ; q 0 / is conformal to .V; q/. The isomorphism of root systems and equality of dimensions of short root groups imply that dim V 0 =V 0 ? D dim V =V ? D 2n and dim V ? D dim V 0 ? (see Proposition 7.1.3). We have just seen that .V 0 ; q 0 / is conformal to .V; q/ if and only if q 0 jV 0 ? is conformal to qjV ? . We will construct a conformal isometry of the latter type via root groups. Choose a maximal k-torus T  G and a short root b 2 ˆ.G; T /, with Ub  G the associated root group. Let K=k be the minimal field of definition for the geometric unipotent radical of G. By Proposition 7.1.2 we may choose a Levi red k-subgroup L  G containing T , so LK ! G 0 WD GK is an isomorphism; 0 0 explicitly, by Proposition 7.1.5, L D SO.q / with q WD qjV 0 ˚` for the unique T -stable linear complement V 0 to V ? in V and a unique line `  V ? . The choice of L provides a k-structure on G 0 compatible with the natural one on its maximal K-torus T 0 WD TK . For qK as in (7.1.4), the canonical K-isogeny 0 G 0 D SO.qK / ! Sp.B qK / DW G has unipotent infinitesimal kernel (it is the classical isogeny SO2nC1 ! Sp2n in characteristic 2). This identifies T 0 with a 0 0 0 0 split maximal K-torus T  G , and the quotient map .G 0 ; T 0 / ! .G ; T / over

125

7.2 Conformal isometries

K descends to the classical unipotent k-isogeny L D SO.q 0 / ! Sp.B q 0 / D L carrying T isomorphically onto its image T . Under this isogeny, the long roots 0 0 in ˆ.G ; T / D ˆ.L; T / are twice the short roots in ˆ.G; T / D ˆ.G 0 ; T 0 / D ˆ.L; T / via the identification X.T / D X.T 0 /. Let b D 2b 2 ˆ.L; T /, so the associated root group Ub of .L; T / is kisomorphic to Ga (isomorphism unique up to k  -scaling) and provides a k0 0 0 structure on the b-root group of .G ; T /. The quotient map GK  G carries Ub .K/ into Ub .K/ and carries Ub .k/ into Ub .k/ ' Ga .k/ D k. The description of the orthogonal automorphism Œv in the proof of Proposition 7.1.3 provides a T -equivariant isomorphism between Ub and the vector group associated to V ? (equipped with T -action through b). Combining this with Remark 7.1.4, we see that the composite map Ub .k/ ! Ub .k/ D k is a k  -multiple of qjV ? because on VK? WD K ˝k V ? we have 1˝v 

p q.v/=q.v0 / ˝ v0 mod V

for any v 2 V ? and a fixed nonzero v0 2 V ? , where V is the K-hyperplane ker.qK jV ? /  VK? . Therefore, .V 0 ? ; q 0 jV 0 ? / is conformal to .V ? ; qjV ? /, and K hence .V 0 ; q 0 / is conformal to .V; q/. The preceding argument proves that the target of (7.2.1.4) is non-empty if and only if the source of (7.2.1.4) is non-empty, so (i) is reduced to showing that the natural map hq W CO.q/.k/=k  ! Autk .G/ is bijective (with k D ks ). For non-degenerate quadratic spaces with odd dimension > 3 this is a special case of the Isomorphism Theorem for connected semisimple groups (and is due to Dieudonné away from characteristic 2). In the non-degenerate odd-dimensional case, a scheme-theoretic proof that hq is bijective in all characteristics is given in [C2, Lemma C.3.12], and a characteristicfree proof via classical algebraic geometry rather than via schemes is given in [KMRT, VI, 26.12, 26.15, 26.17]. This special case will be applied over K in our general proof below that hq is bijective. As a preliminary step in the general case (with k D ks ), we check that ker hq D 1; i.e., any conformal isometry f of .V; q/ centralizing G inside GL.V / is a scalar. For a maximal torus T  G, let V0 be the unique T -equivariant linear complement to V ? in V , so we may identify q0 WD qjV0 with x1 x2 C


C x2n

1 x2n

making T D GLn1 act on V0 with weight space ke2i

1

for t 7! ti and weight

126

Constructions with regular degenerate quadratic forms

space ke2i for t 7! ti 1 . Let  2 k  be the scalar for which q ı f D q (so Bq ı .f  f / D Bq ). Since f centralizes T , each ej is an eigenvector for f with eigenvalue j satisfying 2i 1 2i D  for all i . Likewise, f carries V ? D V T into itself preserving qjV ? up to multiplication by . For every line `  V ? , the Levi k-subgroup L` WD SO.qjV0 ˚` /  G containing T is centralized by f , so f preserves L` :V0 D V0 ˚ `. Hence, f preserves V ? \ .V0 ˚ `/ D `, so the linear automorphism f jV ? preserves every line and thus is scaling by some c 2 k  . But qjV ? ı f D qjV ? , so  D c 2 . Replacing f with .1=c/f thereby reduces us to the case  D 1, so f arises from T .k/  G.k/. But T is arbitrary in G, so the triviality of ZG (or of the center of any Levi k-subgroup of G containing T ) then forces f D 1; i.e., ker hq D 1. Now we address the surjectivity of hq . Since ZG D 1, G.k/ is a normal subgroup of Autk .G/ via conjugation; this subgroup is obviously contained inside the image of CO.q/.k/=k  under hq . For a given f 2 Autk .G/, to determine if f lies in the image of hq it is harmless to replace f with a G.k/-translate. Thus, we may assume that f preserves a chosen maximal k-torus T of G and a minimal pseudo-parabolic k-subgroup B of G containing T , so f centralizes T (as the diagram of ˆ.G; T / has no nontrivial automorphism). ss Consider the induced automorphism f K of SO.qK / D GK . By the settled case of non-degenerate quadratic spaces of odd dimension 2n C 1, f K arises ss from some t 2 T .K/. Let  be the basis of ˆ.G; T / D ˆ.GK ; TK / associated  to B, so the natural map T ! GL1 is an isomorphism (as we may check over K, or over k using that ZG D 1). This identifies t with an ordered n-tuple .t1 ; : : : ; tn / 2 .K  /n where ti D ai .t/ for the vertices ai 2  denoted according to the Dynkin diagram of .G; B; T /:

a1



a2








an



1

an Db

+3 

Here, b denotes the unique short root in  (where we use the convention for n D 1 that the unique root in  is short). Writing Ua and Ua0 to denote the respective a-root groups of .G; T / and ss 0 .G WD GK ; TK / for any a 2 , we have Ua .k/  Ua0 .K/ since G is pseudoreductive. For long a, the quotient map GK ! G 0 D SO.qK / carries the 1dimensional .Ua /K isomorphically onto Ua0 , so Ua .k/ is a k-line in the K-line Ua0 .K/. Hence, if a is long then the K-linear t -action on Ua0 .K/ preserves this k-line, so the scalar a.t / 2 K  lies in k  for such a.

127

7.2 Conformal isometries

For the unique short root b 2 , multiplication on the K-line Ub0 .K/ by b.t / 2 K  must preserve the image of the additive injection j W V ? D Ub .k/ ! Ub0 .K/. If we use a nonzero v0 2 V ? as a K-basis of Ub0 .K/ then the map j p is identified with v 7! q.v/=q.v0 / (due p to the definition of the isomorphism ? V ' Ub .k/ and the congruence v  q.v/=q.v0 /
v0 mod V with V as in (7.1.4)). Hence, b.t / lies in the short root field F , so we can translate f against a k-point of the k-subgroup RF =k .GL1 /  CO.q/ defined as in (7.2.1.1) to reduce to the case b.t / D 1 without affecting the property that f centralizes T . But then Q f K 2 a2> k   T .k/, so f arises from T .k/ by [CGP, Prop. 1.2.2]. This proves (i), and via (7.2.1.3) the same reasoning shows that CO.q/ is generated by G and the k-subgroup RF =k .GL1 / provided by (7.2.1.1). In other words, q is an isomorphism.  Example 7.2.3. Suppose F D k, so the isomorphism property of q implies that the inclusion GL1  SO.q/  CO.q/ is an equality. We claim that if q is degenerate then the closed subgroup scheme O.q/  GL.V / is not contained in the smooth k-subgroup CO.q/  GL.V /. Suppose such a containment holds. Since SO.q/  CO.q/ and O.q/ meets the central GL1 in 2 , we obtain that O.q/ D 2  SO.q/. Hence, the identification of Vk?s with each of the 2n short root spaces in Lie.SO.q//ks for the action of a maximal ks -torus in SO.q/ks implies that dim Lie.O.q// D 1 C dim SO.q/ D 1 C dim SO2nC1 C 2n.dim V ? D 1 C n.2n C 1/ C 2n.ı D n.2n

1/

1/

1/ C .2nı C 1/

where ı WD dim V ? . However, by direct calculation we see that the Lie subalgebra Lie.O.q//  gl.V / D End.V / consists of those endomorphisms  that satisfy Bq .v; .v// D 0 for all v 2 V , which is to say that the bilinear form Bq .v; .w// on V factors through an alternating form on V =V ? . In particular, there is a short exact sequence 0 ! Hom.V; V ? / ! Lie.O.q// ! ^2 .V =V ? / ! 0; so dim Lie.O.q// D n.2n 1/ C .2n C ı/ı. Hence, if F D k then O.q/ is not contained in CO.q/ whenever ı ¤ 1. The case ı D 1 is the non-degenerate case, so if .V; q/ is degenerate and SO.q/ has short root field equal to k then O.q/ 6 CO.q/. (Note that the short root field of an absolutely pseudo-simple basic exotic k-group is a nontrivial purely inseparable

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extension of k, unlike for SO.q/ when F D k and q is degenerate.) In contrast with the adjoint semisimple case, the above SO.q/-construction with n > 2 does not account for all centerless absolutely pseudo-simple k-groups of type Bn whose Cartan k-subgroups are tori. This is relevant even over nonarchimedean local fields k of characteristic 2: Example 7.2.4. Let k be a non-archimedean local field of characteristic 2. For n > 2, the proof of [CGP, Prop. 7.3.6] provides basic exotic absolutely pseudosimple k-groups G of type Bn such that G has k-rank bn=2c. Thus, the groups G=ZG are absolutely pseudo-simple of type Bn with trivial center (Proposition 5.3.1(ii)) and their Cartan k-subgroups are tori (by inspection). But any SO.q/ as above has k-rank at least n 1 due to the existence of a Levi k-subgroup and the isotropicity of non-degenerate quadratic spaces of dimension > 4 over k. (See [EKM, Lemma 36.8] for an elementary proof of such isotropicity in dimension > 2eC1 over fields F of characteristic 2 such that ŒF W F 2  D 2e .) The following proposition provides sufficient conditions for an absolutely pseudo-simple k-group to be isomorphic to SO.q/ for a regular quadratic form q; it is a partial converse to Proposition 7.1.2. Proposition 7.2.5. For n > 1, consider absolutely pseudo-simple k-groups G with root system Bn over ks such that the Cartan k-subgroups are tori and ZG D 1. If G is pseudo-split then it is isomorphic to SO.q/ for a regular quadratic space .V; q/ of Witt index n with nonzero V ? of codimension 2n in V . If instead G merely admits a pseudo-split ks =k-form but its short root field is equal to k then it is isomorphic to SO.q 0 / for a regular quadratic space .V 0 ; q 0 / with nonzero V 0 ? of codimension 2n in V 0 such that .V 0 ? ; q 0 jV 0 ? / has root field k in the sense of Example 7:1:8. Proof. As the root system over ks is reduced and the center is trivial, it follows from the centrality of the kernel (of iG ) in Proposition 2.3.4 that all such groups G are of minimal type. To prove the first assertion, assume G is pseudo-split and let K=k be the minimal field of definition for its geometric unipotent radical and let T be a fixed split maximal k-torus in G. Note that iG maps G isomorphically onto its ss image in RK=k .GK / since G is of minimal type and its root system is reduced. As G has trivial center and is generated by its root groups, we see that ˆ WD ss ˆ.G; T / .D ˆ.GK ; TK / under the natural identification of the character groups

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129

ss X.T / and X.TK /) spans X.T /. This implies that GK is of adjoint type. We fix a basis  of the root system ˆ. Let Gc be the absolutely pseudo-simple k-subgroup of G generated by the ˙c-root groups for any c 2 ˆ. In view of the construction of these subgroups as derived groups of centralizers of codimension-1 tori, the hypothesis that the Cartan subgroups of G are tori is inherited by each Gc . Likewise, each Gc is of minimal type. Also, recall from 3.2.2 that the natural maps ss ss hc W .Gc /K ! .GK /c

between split connected semisimple K-groups of rank 1 are isomorphisms. ss Since GK has root system Bn (understood to have only short roots when n D 1) and its center is trivial, for any long root a 2  (if n > 1) and the unique ss short root b 2  we conclude via the isomorphisms ha and hb that .Ga /K is ss simply connected (so it is isomorphic to SL2 ) and .Gb /K is adjoint (so it is isomorphic to PGL2 ). As the Cartan subgroups of the pseudo-split absolutely pseudo-simple k-group Ga of minimal type are tori, and the root system of Ga is A1 , Proposition 3.1.8 implies (using the description of Cartan subgroups provided by Proposition 3.1.4) that Ga ' SL2 for any long a 2 . Hence, by Proposition 3.2.5 (and the trivial equality Gb D G when n D 1), the minimal field of definition for the geometric unipotent radical of Gb is K. Now using Proposition 3.1.8 again we see that Gb ' PHV0 ;K=k for a nonzero kK 2 -subspace V0 of K that contains 1 and generates K as a k-algebra. In particular, ZGb D 1. The Cartan subgroups of Gb are (1-dimensional) tori, so the description of Cartan subgroups in Proposition 3.1.4 implies that K 2  k. Now we can define a k-valued anisotropic quadratic form on V0 by q0 W v 7! v 2 , so Proposition 7.1.3 implies Gb ' SO.H ? q0 / for a hyperbolic plane H . This settles the case n D 1. In general, let q be the orthogonal sum of n hyperbolic planes and the anisotropic .V0 ; q0 /, so if n > 2 then the Isomorphism Theorem (i.e., Theorem 6.1.1) and the description of Gb imply G ' SO.q/. This proves the first assertion. To prove the second assertion, suppose instead that G admits a pseudo-split ks =k-form G and has short root field equal to k. Then G inherits the hypotheses from G, so G ' SO.q/ for some q as above. Since the short root field of G is k, by Proposition 7.2.2 applied to SO.q/ we have G ' SO.q 0 / for a regular quadratic space .V 0 ; q 0 / as desired. (The end of Example 7.1.8 relates the short root field of SO.q 0 / to that of .V 0 ? ; q 0 jV 0 ? /.)  Remark 7.2.6. In the second assertion of Proposition 7.2.5, the hypothesis that G admits a pseudo-split ks =k-form cannot be dropped: this property is a

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consequence of the conclusion by Proposition 7.1.2, and examples satisfying all of the other hypotheses but lacking a pseudo-split ks =k-form are given by the k-groups G=ZG for G of type Bn in Example C.4.1 with many k admitting a quadratic Galois extension and satisfying Œk W k 2  > 8.

7.3 Severi–Brauer varieties For imperfect fields k of characteristic 2, an exhaustive description of all pseudosplit absolutely pseudo-simple k-groups of type-B with trivial center and tori as Cartan k-subgroups is provided by the first assertion in Proposition 7.2.5; this applies in particular over ks . To give an exhaustive description without a pseudo-split hypothesis (and without assuming that there exists a pseudo-split ks =k-form) requires a description of k-descents of SO.q 0 / for degenerate regular .V 0 ; q 0 / with V 0 ? ¤ V 0 over finite Galois extensions k 0 =k. Since isomorphisms among such groups SO.q 0 / generally arise from conformal isometries of quadratic spaces rather than from isometries of quadratic spaces (Proposition 7.2.2), such k 0 =k-descents generally do not have the form SO.q/ for regular .V; q/ over k. (The same phenomenon is seen in the classical semisimple case for non-degenerate quadratic spaces with even dimension > 4.) Remark 7.2.6 provides such descents with no pseudo-split ks =k-form (and with short root field equal to k) for many k admitting a quadratic Galois extension and satisfying Œk W k 2  > 8. We now prepare to describe all descents in terms of automorphisms of certain quadrics in Severi–Brauer varieties over k. 7.3.1. Recall that a Severi–Brauer variety of dimension N > 0 over a field F is a smooth proper F -scheme X such that XF ' PN , or equivalently XFs ' PN Fs . F Since the latter isomorphism is well-defined up to the action of PGLN C1 .Fs /, for a geometrically reduced hypersurface D  X we can speak of its degree ı > 1: this means the degree of DFs as a hypersurface in XFs ' PN Fs (or equivalently, the index of the subgroup of Pic.XFs / generated by the invertible ideal sheaf of D in OX ). For instance, when ı D 2 we call D a quadric in X. Every Severi– Brauer variety X over F is projective, and by descent theory the automorphism functor AutX=F is represented by an F -form AutX=F of PGLN C1 . For any F algebra A, an A-automorphism f of XA that carries DA into itself restricts to an automorphism of DA [EGA, IV4 , 17.9.6], so the subfunctor Aut.X;D/=F W A

ff 2 AutA .XA / j fA .DA /  DA g

of AutX=F is a subgroup functor.

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Since D is geometrically reduced, or equivalently reduced and generically smooth, the subset D.Fs /  DFs is schematically dense (in the sense of [EGA, IV3 , 11.10.2]). Thus, Aut.X;D/=F is represented by a closed subgroup scheme Aut.X;D/=F  AutX=F ; namely by the Galois descent of the closed subscheme of AutXFs =Fs defined by the intersection of the closed conditions “f .d / 2 DFs ” for all d 2 D.Fs /. Remark 7.3.2. The case of most interest to us is imperfect F of characteristic 2 with XFs D P.V  / WD Proj.Sym.V  // and DFs D .q D 0/ for a regular quadratic form q W V ! Fs such that V ? is a nonzero proper subspace of V . Since V ? has even codimension in V , necessarily dim V > 3 and when dim V 6 4 we can identify DF ,! XF with .x 2 C yz D 0/  Pdim V 1 . These are instances of the general context above with D a geometrically integral quadric (so N > 2), subject to the further condition that when N D 3 it is non-smooth at a unique geometric point. In such cases, we claim that the restriction map Aut.X;D/=F ! AutD=F to the automorphism scheme of D is an isomorphism, so Aut.X;D/=F is intrinsic to D (and AutD=F is affine). This is not needed below, but for the interested reader we now sketch the proof. We may assume F is algebraically closed, and we claim that Pic.D/ ' Z. The case dim D D 1 is obvious (as D is an integral plane conic, hence smooth), the case dim D D 2 is a classical fact about the quadric cone in P3 , and the case dim D > 3 is a consequence of the global Lefschetz theorem [SGA2, XII, Cor. 3.6] (which gives that Z D Pic.X/ ' Pic.D/ in such cases). The Picard scheme PicD=F is constant since H1 .D; OD / D 0, so the action of AutD=F on PicD=F D Z is trivial (as it must preserve the unique ample generator). Thus, this action extends to a projective representation of AutD=F on €.D; L / for any line bundle L on the integral hypersurface D. The projective embedding j W D ,! X D PN is given by the space of global sections of L D j  .OX .1// since D has degree > 1, so the desired isomorphism property follows. Proposition 7.3.3. Let X be a Severi–Brauer variety over k with dim X > 0, and let D  X be a geometrically reduced quadric that is regular .equivalently, smooth/ at all ks -points. If D is smooth then assume dim X is even. Let GX;D be the maximal smooth k-subgroup scheme Autsm .X;D/=k of Aut.X;D/=k . (i) The k-group GX;D is connected. Its derived group D.GX;D / is absolutely pseudo-simple of type B with trivial center, and its Cartan k-subgroups are tori.

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(ii) If .X 0 ; D 0 / is a second such pair then Isomk ..X 0 ; D 0 /; .X; D// ! Isomk .D.GX 0 ;D 0 /; D.GX;D // is bijective. In particular, the k-isomorphism class of any of .X; D/, GX;D , or D.GX;D / determines the others. (iii) The k-groups D.GX;D / for .X; D/ with X.k/ ¤ ; are precisely SO.q/ for regular .V; q/ with 0 < dim V ? < dim V , and in such cases GX;D D D.GX;D / if and only if .V ? ; qjV ? / has root field k in the sense of Example 7:1:8. sm By Remark 7.3.2, we have GX;D D AutD=k ; this will not be used.

Proof. First we consider (i) and (ii), so we may and do assume k D ks . Thus, we identify X with the projective space P.V  / D Proj.Sym.V  // covariantly attached to a k-vector space V of finite dimension at least 2, and D is thereby the zero scheme of a nonzero quadratic form q 2 Sym2 .V  /. The regularity of D at its k-points is exactly the regularity of q (i.e., the injectivity of the 2-linear qjV ? ). The generic smoothness expresses that qk is reduced, which is to say that q is not a sum of squares; i.e., V ? ¤ V . Since smoothness of the quadric corresponds to non-degeneracy of the quadratic form, and dim.V =V ? / is even, non-smoothness of the quadric expresses exactly the condition that dim V ? > 1. If instead the quadric is smooth then .V; q/ is non-degenerate and by hypothesis dim X is even and positive in such cases, so dim V is odd and > 3 (forcing V ? to be a line in V ). The smooth part of Aut.X;D/=k is the Zariski closure in PGL.V / of the group of projective linear automorphisms preserving .q D 0/. These arise from exactly the linear automorphisms f of V that preserve q up to a k  -scaling factor, which is to say that f is a conformal automorphism of .V; q/. Hence, GX;D is the ksubgroup CO.q/=GL1 ,! PGL.V / that is connected (by Proposition 7.2.2(ii)). Since SO.q/ has trivial center, the natural maps SO.q/ ! PGL.V /; SO.q/ ! CO.q/=GL1 have trivial kernel. But CO.q/=SO.q/ is commutative since (7.2.1.2) is an isomorphism (Proposition 7.2.2(ii)), so the perfect SO.q/ coincides with D.GX;D /. This proves (i), and (ii) is exactly Proposition 7.2.2(ii). Consider (iii). If X.k/ ¤ ; then we may identify X with a projective space over k and hence D with a quadric. Thus, the preceding considerations work over k to show that D.GX;D / D SO.q/ for .V; q/ as in (iii). Conversely, every

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such SO.q/ has the desired form by defining X D P.V  / and D D .q D 0/ since dim X D dim V 1 is even and positive when D is smooth (because V ? ¤ 0 by hypothesis and smoothness of D expresses non-degeneracy for q). This proves the first part of (iii), and for the equivalence at the end we may assume k D ks . Thus, GX;D D CO.q/=GL1 . This has derived group SO.q/ and maximal commutative affine quotient RF =k .GL1 /=GL1 , where F is the root field of .V ? ; qjV ? /, so perfectness of GX;D is indeed the condition F D k.  Remark 7.3.4. For any .X; D/ as above, X is classified up to k-isomorphism by an element ŒX  2 Br.k/ called the Brauer invariant of D.GX;D /. By Proposition 7.3.3(iii), ŒX  D 1 if and only if D.GX;D / ' SO.q/ for a regular .V; q/ with 0 < dim V ? < dim V . We claim that ŒX is 2-torsion, and more generally that a Severi–Brauer variety X containing a geometrically reduced hypersurface D of degree d (over ks ) corresponds to a d -torsion class in Br.k/. This is proved via an argument of Artin [Ar, 5.2] that we now review. Let V be a k-vector space of dimension N C1 > 3 such that X is a k-form of the projective space P.V  / WD Proj.Sym.V  //; this projective space classifies lines in V (equivalently, hyperplanes in V  ) and has a left action by PGL.V / under which the action of L 2 PGL.V / carries the hypersurface .h D 0/ for nonzero h 2 Symd .V  / to the zero scheme of .Symd .L // 1 .h/. For every r > 1, the natural representation GL.V / ! GL..Symr .V  // / induces a corresponding k-homomorphism fr W PGL.V / ! PGL..Symr .V  // /: The fr -pushforward of the 1-cocycle descent datum for X valued in PGL.V / is the 1-cocycle descent datum valued in PGL..Symr .V  // / for a Severi–Brauer variety X.r/ equipped with an inclusion X ,! X.r/ encoding a descent of the r-fold Segre map. But X.d / contains a degree-1 hypersurface H corresponding to a descent of Dks  Xks , so the ideal sheaf IH of H in OX.d / is a k-descent of O. 1/ on Pk0 s , where P 0 D P..Symd .V  /// (so IH is invertible). By computing over ks , we see that L WD IH 1 is generated by its global sections and that the natural map X.d / ! Proj.Sym.€.X.d /; L /// D P.€.X.d /; L // (assigning to each x 2 X.d / the hyperplane of global sections of L that vanish

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at x) is an isomorphism; i.e., X.d / is a trivial Severi–Brauer variety. Hence, the map of pointed sets H1 .fd / W H1 .k; PGL.V // ! H1 .k; PGL..Symd .V  // // kills ŒX . For any r > 1 we have a commutative diagram of central extensions of affine algebraic k-groups 1

/ GL1 tr

1



/ GL1

/ GL.V /

/ PGL.V /





/ GL..Symr .V  // /

/1

fr

/ PGL..Symr .V  // /

/1

from which it follows that H1 .fr / is compatible with multiplication by r on H2 .k; GL1 / D Br.k/ via the connecting map ı W H1 .k; PGL.V // ! H2 .k; GL1 /: Thus, ŒX  2 Br.k/Œd . Proposition 7.3.5. Let G D D.G.X;D/ / for .X; D/ as in Proposition 7:3:3. Then G contains a Levi k-subgroup if and only if G ' SO.q/ for a regular .V; q/ over k with 0 < dim V ? < dim V . Proof. The implication “(” is immediate from Proposition 7.1.5. For the converse, suppose G has a Levi k-subgroup L. Let T  L be a maximal k-torus. By Proposition 7.3.3(iii) over ks , we have Gks ' SO.q/ for a regular quadratic space .V ; q/ over ks with 0 < dim V ? < dim V . Concretely, viewing the quadratic form q W V ! ks as a nonzero element of Sym2 .V  /, the pair .X; D/ is a ks =kdescent of .P.V  /; .q D 0// via a descent datum valued in conformal isometries between .V ; q/ and its ks =k-twists. Letting V0 be the unique Tks -equivariant complement to V ? , Proposition 7.1.5 gives that Lks D SO.qjV0 ˚` / for a unique ks -line ` in V ? . In view of the uniqueness of `, the conformal isometries of .V ; q/ encoding the ks =kdescent datum on .P.V  /; .q D 0// must preserve the ks -line ` in V because L and T are k-subgroups of the ks =k-descent G of SO.q/. Hence, the closed subscheme P.` /  P.V  / is preserved by the ks =k-descent datum that defines X. But P.` / D Spec.ks / is a ks -point of P.V  / D Xks , so its ks =k-descent to a closed subscheme of X must be a k-point of X. Thus, by Proposition 7.3.3(iii) it follows that G ' SO.q/ for some regular .V; q/ with 0 < dim V ? < dim V . 

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Now we finally obtain the result we have been after in this section: Theorem 7.3.6. The absolutely pseudo-simple k-groups G such that the Cartan k-subgroups are tori, ZG D 1, and the root system over ks is Bn for some n > 1 are precisely the k-groups D.GX;D / for .X; D/ as in Proposition 7:3:3. Moreover, the pair .X; D/ is uniquely functorial with respect to isomorphisms in the associated k-group, and for G WD D.GX;D / the following three conditions are equivalent: X.k/ ¤ ;, G ' SO.q/ for a regular .V; q/ over k such that 0 < dim V ? < dim V , and G admits a Levi k-subgroup. Proof. Proposition 7.3.3(i) gives that the k-groups D.GX;D / satisfy the initial list of properties. For a general k-group G satisfying those properties, we shall prove that it has the form D.GX;D / for some such pair .X; D/; once that is shown, Propositions 7.3.3 and 7.3.5 give everything else. We may choose a finite Galois extension k 0 =k such that the k 0 -group Gk 0 is pseudo-split, so by the first assertion in Proposition 7.2.5 we have Gk 0 ' SO.q 0 / for a regular quadratic space .V 0 ; q 0 / over k 0 such that 0 < dim V 0 ? < dim V 0 and dim.V 0 =V 0 ? / D 2n. By Proposition 7.3.3(iii), Gk 0 ' D.GX 0 ;D 0 / for some pair .X 0 ; D 0 / over k 0 (with X 0 .k 0 / ¤ ;). This k 0 -isomorphism transfers the k 0 =k-descent datum on Gk 0 over to such descent datum on the k 0 group D.GX 0 ;D 0 /, and by Proposition 7.3.3(ii) this arises from a uniquely determined k 0 =k-descent datum on .X 0 ; D 0 /. By effectivity of Galois descent for (quasi-)projective schemes over fields, we obtain .X; D/ over k equipped with a k 0 -isomorphism .Xk 0 ; Dk 0 / ' .X 0 ; D 0 / such that the resulting composite k 0 isomorphism f 0 W D.GX;D /k 0 D D.GXk0 ;Dk0 / ' D.GX 0 ;D 0 / ' Gk 0 is compatible with the natural k 0 =k-descent datum on each side. By Galois descent, f 0 descends to a k-isomorphism D.GX;D / ' G.  Proposition 7.3.7. Let k be a field. Non-reductive pseudo-reductive k-groups G whose Cartan k-subgroups are tori exist if and only if k is imperfect of characteristic 2, in which case such groups are precisely H  Rk 0 =k .G 0 / for a connected reductive k-group H , nonzero finite étale k-algebra k 0 , and smooth affine k 0 -group G 0 whose fiber Gi0 over each factor field ki0 of k 0 is a group D.GXi0 ;Di0 / as in Theorem 7:3:6 for which the quadric Di0 is not ki0 -smooth. That such G only exist over imperfect fields of characteristic 2 is [CGP, Thm. 11.1.1], but our proof of the above more precise result is simpler.

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Proof. We first observe that when k is imperfect of characteristic 2, such a triple .H; k 0 =k; G 0 / describing G is unique up to unique isomorphism if it exists. Indeed, to prove this we may (by Galois descent) assume k D ks , and then it suffices to show that if H; H are connected reductive k-groups and fHi gi 2I ; fHj gj 2J are non-empty finite collections of non-reductive pseudosimple k-groups with trivial center then any k-isomorphism Y Y f WH Hi ' H  Hj i

j

arises uniquely from a k-isomorphism ' W H ' H, a bijection  W I ' J , and k-isomorphisms 'i W Hi ' H .i / for i 2 I . The groups Hi are precisely the non-reductive pseudo-simple normal k-subgroups of the left side, and H is the Q centralizer of i Hi in the left side (since ZHi is trivial for all i ). This intrinsic Q characterization of .H; fHi g/ in terms of the k-group H  Hi has an evident analogue on the right side, so the (unique) description of f is immediate. By Galois descent, now we may assume k D ks and that k is imperfect with p D char.k/. Let T  G be a maximal k-torus, so T D ZG .T / and hence G D T
D.G/. By [CGP, Lemma 1.2.5(ii)], T is an almost direct product of the maximal central k-torus Z and the maximal k-torus T \D.G/ of D.G/, so G D Z
D.G/. The collection fGi gi 2I of pseudo-simple normal k-subgroups of G Q is finite and non-empty, the Gi ’s pairwise commute, and Gi ! D.G/ is a surjection with central kernel (see [CGP, Prop. 3.1.8]). The Cartan k-subgroups of any smooth connected normal k-subgroup of G are tori, and G is non-reductive if and only if some Gi is non-reductive, so we may (and will) assume that G is pseudo-simple (and non-reductive). By Theorem 7.3.6, it suffices to show that char.k/ D 2, ZG is trivial, and the root system is of type B. Consider the k-subgroup CG D .ker iG /T as in §2.3. By centrality, CG is contained in ZG .T / D T , yet it is unipotent since ker iG is unipotent, so CG D 1. Thus, G is of minimal type. Hence, its universal smooth e is also of minimal type, by Proposition 5.3.3. k-tame central extension G e has trivial center by [CGP, Assume G has a non-reduced root system, so G e Prop. 9.4.9] and hence G D G. The possibilities for such G are given in [CGP, Thm. 9.8.6], and an explicit description of the Cartan k-subgroups of G is given in [CGP, (9.7.6)] when n ¤ 2 and in [CGP, (9.8.2)] when n D 2 (see [CGP, Thm. 9.8.1(2), Prop. 9.8.4(1)] for this determination of Cartan k-subgroups). By inspection, these k-subgroups are not tori (because dim V0 > 2 for V0 as defined there). Hence, the root system ˆ WD ˆ.G; T / of G is reduced. Let K=k be the minimal field of definition for Ru .Gk /  Gk , so K ¤ k

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137

since G is not reductive. By Proposition 3.2.6, the extension K=k attached to G e Since G e is of minimal type with a reduced coincides with the analogue for G. e0 D G eK =Ru;K .G eK /, so root system, ker iG e is trivial by Proposition 2.3.4. Let G 0 e e naturally G  RK=k .G / and ZG e  RK=k .ZG e0 /. Assume that ˆ is not of type B, _ ec maps isomorphically onto its so cK .GL1 / \ ZG D 1 for every c 2 ˆ. Thus, G e0 e image Gc in the central quotient G of G for all c 2 ˆ, so the Cartan k-subgroup ec is GL1 . But each ker i e is trivial, so the possibilities for each G ec are of each G Gc given by Lemma 3.2.3 (in the “SL2 -case”) in terms of Kc =k (and auxiliary data in characteristic 2). It follows that Kc D k for all c. This contradicts Proposition 3.2.5 since K ¤ k, so ˆ is of type B. Consider the rank-1 case; i.e., ˆ D B1 . By Proposition 3.1.8, if char.k/ ¤ 2 then G D RK=k .G 0 / with G 0 equal to SL2 or PGL2 , but then its Cartan ksubgroups are not tori since K ¤ k. Suppose char.k/ D 2, so we just need to show ZG is trivial. If ZG is nontrivial then Proposition 3.1.8 implies G ' HV;K=k for a nonzero kK 2 -subspace V  K satisfying khV i D K (so dimk V >  1). The Cartan k-subgroup VK=k has dimension larger than 1, so it is not a ktorus. The B1 -case is settled. _ Suppose ˆ D Bn with n > 2, and consider long a 2 ˆ, so aK .GL1 / \ ZG e0 e is trivial. The proof that ˆ is of type B shows that Ga ' SL2 . If p ¤ 2 then e0 / by Theorem 3.4.1(i), so G ea0 / D RK=k .SL2 /, a e ' RK=k .G ea ' RK=k .G G K contradiction. Thus, p D 2 (so we just need to show ZG is trivial) and in the notation of Theorem 3.4.1(iii) we have shown V> D k (even if n D 2). Hence, e is classified by a nonzero k-subspace V  K satisfying khV i D K. G e are GLn1 1 V  ; the factor By Proposition 4.3.3, Cartan k-subgroups of G K=k  VK=k is associated to the unique short root b in a basis of ˆ. Since ZG e  _  RK=k .ZG e0 / D RK=k .bK .2 //, clearly ZG e D VK=k Œ2. The central quotient Gb e eb  Gb is either an of Gb D HV;K=k is of minimal type (as G is), so the map G isomorphism or projection onto PHV;K=k , and the latter case is the quotient by e or G D G=Z e e . Since dimk V > 2 (as khV i D K ¤ k), ZG eb D ZG e . Thus, G D G G  e cannot occur. Hence, G D G=Z e e , so so VK=k is not a torus, the case G D G G ZG is trivial by Proposition 5.3.1(ii). 

8 Constructions when ˆ has a double bond

In 8:1–8:3 we assume char.k/ D 2.

8.1 Additional constructions for type B Inspired by Proposition 7.3.3, we are led to the following construction that goes beyond SO.q/’s and will be seen to provide the right generalization of the basic exotic construction for type-Bn when n ¤ 2. (The case n D 2 admits additional constructions; see §8.3.) Definition 8.1.1. A type-B adjoint generalized basic exotic k-group is a kgroup G of the form D.Autsm .X;D/=k / for .X; D/ as in Proposition 7:3:3 with non-smooth D, subject to the addition requirement in the rank-1 case that G has root field equal to k. A type-B generalized basic exotic k-group is the universal smooth k-tame central extension of a type-B adjoint generalized basic exotic k-group. The root-field hypothesis in the rank-1 adjoint type-B case is automatic in the higher-rank case (as we may verify over ks ). Moreover, by Theorem 7.3.6, for any separable extension k 0 =k the k-group G is in one of the above two classes of k-groups if and only if Gk 0 is in the analogous class of k 0 -groups. The type-B adjoint generalized basic exotic ks -groups are precisely SO.q/ for regular .V; q/ over ks satisfying 1 < dim V ? < dim V , subject to the requirement for rank 1 that .V ? ; qjV ? / has root field ks in the sense of Example 7.1.8. This extra condition for rank 1 serves two purposes: it prevents the intervention of Weil restriction (since PHV ? ;K=ks D D.RF =ks .PHV ? ;K=F //, where F is the root field), and it ensures that the Cartan subgroups are tori (automatic in the higher-rank cases) due to the relation “kK 2  F ” in (3.3.8).

139

8.1 Additional constructions for type B

By inspection of SO.q/’s over ks , all k-groups in Definition 8.1.1 are absolutely pseudo-simple of minimal type, non-standard (by consideration of dimensions of root spaces over ks ), have root system over ks of type B, and have trivial center in the adjoint case. In particular, if G is a type-B generalized basic exotic k-group then G=ZG is a type-B adjoint generalized basic exotic k-group and moreover G has root field equal to k (inherited from G=ZG by Remark 3.3.3). 8.1.2. Let G be a type-B generalized basic exotic k-group with minimal field of definition K=k for its geometric unipotent radical, so K 2  k since the root field is k. To justify the terminology in Definition 8.1.1, we now explain how G lies inside a canonically associated basic exotic k-group G with the same associated invariant K=k and same root system over ks . Since .G=ZG /ks ' SO.q/ for .V; q/ over ks as above (so ks hV ? i D Ks and V ? is a nonzero proper ks -subspace of Ks ), we expect that Gks corresponds to enlarging V ? to Ks . However, we want to describe G directly over k and to make its construction functorial (with respect to isomorphisms) in G. ss The connected semisimple K-group G 0 WD GK is simply connected and absolutely simple of type Bn . As G is of minimal type, the kernel of iG W G ! RK=k .G 0 / has trivial intersection with any Cartan k-subgroup of G, yet (by Proposition 2.3.4) ker iG is central since Gks has a reduced root system, so ker iG D 1. Via iG , G embeds into a type-B basic exotic k-group when n > 2: 0

Proposition 8.1.3. For G and .K=k; G 0 / as above, let  W G 0 ! G be the very special K-isogeny and define f D RK=k ./. The image G D f .G/ is a Levi 0 k-subgroup of RK=k .G /, and if G has ks -rank n > 2 then G WD f 1 .G/ is a basic exotic pseudo-simple k-group of type Bn containing G. Thus, any type-B generalized basic exotic k-group with ks -rank n > 2 is contained in a functorially associated basic exotic k-group of type Bn . 0

Proof. Once it is shown that G is a Levi k-subgroup of RK=k .G /, it is immediate from the definitions that G is a basic exotic pseudo-semisimple k-group if 0 the ks -rank of G is > 2. To verify that G is a Levi k-subgroup of RK=k .G / we may assume k D ks , so G contains a split maximal k-torus T and there exists a Levi k-subgroup L of G containing T [CGP, Thm. 3.4.6]. Since LK ! G 0 is an isomorphism,  is the scalar extension to K of the very special isogeny L ! L. In particular, via the inclusion L ,! RK=k .LK / D 0 RK=k .G / we have f .L/ D L, and L is a Levi k- subgroup of RK=k .LK / [CGP, Cor. A.5.16]. By Proposition 2.1.2(i), G is pseudo-semisimple because it lies between L and RK=k .LK /.

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It now suffices to check that the inclusion L  G of smooth connected kgroups is an equality. By considering the product structure on an open cell for .G; T /, it suffices to check that the inclusion T WD f .T /  f .ZG .T // is an equality and that f .Uc / is 1-dimensional for all c 2 ˆ WD ˆ.G; T /. The direct product structure on ZG .T / (Proposition 4.3.3) further reduces the task to showing that f .Gc / is 3-dimensional for all c. (We shall follow the convention that in the rank-1 case the roots in ˆ are short and the roots in ˆ are long.) The finite-index inclusion X.T /  X.T / identifies each long root a 2 ˆ D 0 0 ˆ.G 0 ; T 0 / with a short root of ˆ WD ˆ.G ; T / D ˆ.G; T / and identifies each short root b 2 ˆ with .1=2/b for a long root b 2 ˆ. In this way,  carries Ga0 0 0 isomorphically onto G a for a 2 ˆ> and carries Gb0 onto G b via a Frobenius isogeny for b 2 ˆ< . From the description of G we see that Ga D SL2 for a 2 ˆ> , so we just have to check that the inclusion Lb  f .Gb / is an equality for b 2 ˆ< . 0 But K 2  k, so the restriction f W RK=k .Gb / ! RK=k .G b / has image equal to Lb for dimension reasons.  We would like to describe maximal k-tori and pseudo-parabolic k-subgroups of type-B generalized basic exotic k-groups in terms of the diagram RK=k .G 0 / 

G

f

/ RK=k .G 0 /

As motivation, for a basic exotic k-group G arising from a triple .k 0 =k; G 0 ; G/, [CGP, Prop. 11.1.3] provides a natural bijective correspondence between the set of maximal k-tori T in G and the set of pairs .T ; T 0 / consisting of a maximal k-torus T  G and a maximal k 0 -torus T 0  G 0 such that the very special k 0 0 isogeny G 0 ! G D G k 0 carries T 0 onto T k 0 . Likewise, in [CGP, Prop. 11.4.6] a similar bijection is established for pseudo-parabolic k-subgroups P , as well as an equivalence between the conditions that T  P and that T 0  P 0 . For a type-B generalized basic exotic k-group G with ks -rank n > 2, we generally cannot establish an analogous dictionary for the sets of maximal k-tori of G (in terms of the associated triple .K=k; G 0 ; G/). The difficulty is that since perfect smooth connected affine k-groups are generated by their maximal k-tori [CGP, Prop. A.2.11], the basic exotic k-group G D f 1 .G/ that is strictly larger than G always contains maximal k-tori that are not inside G. Consequently, we establish the following dictionary for pseudo-parabolic k-subgroups:

8.1 Additional constructions for type B

141

Proposition 8.1.4. Let G be a type-B generalized basic exotic k-group with ks rank n > 2 and associated invariants .K=k; G 0 ; G/. If P is a pseudo-parabolic k-subgroup of G then P 0 WD im.PK ! G 0 / is a parabolic K-subgroup of G 0 and G \ RK=k .P 0 / D P . Moreover, if T  G is a maximal k-torus then P 7! P 0 is a bijection between the set of pseudo-parabolic k-subgroups of G containing T and the set of parabolic K-subgroups of G 0 containing the associated T 0 . In particular, a parabolic K-subgroup P 0 of G 0 arises in this way from a pseudo-parabolic k-subgroup P of G if and only if P 0 contains a maximal K-torus of G 0 arising from a maximal k-torus of G. Proof. Given P , we can choose a 1-parameter k-subgroup  W GL1 ! P such that P D PG ./. Thus, the image P 0 of PK under the surjective map  W GK  ss GK DW G 0 is PG 0 .0 / for the composition 0 D  ı K [CGP, Cor. 2.1.9]. To establish that the inclusion P  G \ RK=k .P 0 / is an equality, we first note that RK=k .P 0 / D PRK=k .G 0 / ./ by [CGP, Prop. 2.1.13], where  is valued in the k-subgroup G of RK=k .G 0 /. Hence, G \ RK=k .P 0 / D PG ./ D P as desired. It remains to show for a maximal k-torus T of G that every parabolic Ksubgroup P 0 of G 0 containing the associated T 0 arises from a pseudo-parabolic k-subgroup P of G containing T . Such a P is unique if it exists (since it must equal G \ RK=k .P 0 /), so for existence we may assume k D ks . By [CGP, Prop. 3.5.4], there exists a unique pseudo-parabolic k-subgroup P of G containing T such that Pk has image in G ss D G 0 coinciding with the parabolic k k subgroup P 0 containing T 0 . Thus, the parabolic K-subgroups im.PK ! G 0 / k

k

and P 0 containing T 0 agree after scalar extension to k, so they coincide.



An exceptional rank-2 construction in §8.3 will require an analogue, for type-B generalized basic exotic k-groups, of the classical very special isogeny Spin.q/ ! SO.q/ ! Sp.B q / associated to non-degenerate .V; q/ with dim V ? D 1. Although such an analogue is only needed in rank-2 cases, we shall now define and study this analogue in any rank since that clarifies the construction and entails no extra effort. Let us first address a version of the very special isogeny for k-groups SO.q/ with regular quadratic spaces .V; q/ satisfying 0 < dim V ? < dim V . For such .V; q/, let B q be the symplectic form on V =V ? obtained from Bq , and let q W SO.q/ ! Sp.B q /

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be the natural map; this is surjective since its restriction to a Levi ks -subgroup is the unipotent isogeny from an absolutely simple ks -group of adjoint type B to an absolutely simple and simply connected ks -group of type C. (In Proposition 8.1.8(i) we will see that dim ker q D 2n.dim V ? 1/.) For c 2 k  we have SO.cq/ D SO.q/ inside GL.V / and Sp.B cq / D Sp.B q / inside GL.V =V ? / (since B cq D cB q ); in this way we have q D cq . Thus, q is functorial with respect to conformal isometries in .V; q/, so by Proposition 7.2.2(i) the k-homomorphism q is intrinsic to the k-group G D SO.q/ in the sense that it is independent of the choice of .V; q/ and isomorphism G ' SO.q/. The following definition is therefore well-posed: Definition 8.1.5. Let G be a type-B generalized basic exotic k-group. The very special quotient G  G is the composition of G ! G=ZG and the quotient map u W G=ZG ! G that descends the ks -homomorphism q W SO.q/  Sp.B q / for the conformal isometry class of .V; q/ over ks such that .G=ZG /ks ' SO.q/. In particular, G is absolutely simple and simply connected of type C. Remark 8.1.6. Very special quotient maps  W G ! G always have connected kernel. Indeed, we may assume k D ks , so the problem is to show that the quotient map G=.ker /0 ! G ' Sp2n is an isomorphism. This latter map has finite étale kernel E D .ker /=.ker /0 , so G=.ker /0 is a smooth connected central extension of Sp2n by the finite étale E. But Sp2n has no such nontrivial central extension since it is simply connected, so E is trivial. Lemma 8.1.7. Let G be a pseudo-split absolutely pseudo-simple group over a field k of characteristic 2, and for a split maximal k-torus T assume the root system ˆ D ˆ.G; T / is of type Bn with n > 2. If b; b 0 2 ˆ are linearly independent short roots then the root groups Ub and Ub 0 commute inside G. This lemma expresses a well-known degeneration of Chevalley commutation relations for B2 in characteristic 2 [Hum, 33.4(b)], but we give a proof avoiding the use of this degeneration. Proof. By [CGP, Lemma 1.2.1], to prove the triviality of .Ub ; Ub 0 / it suffices to prove the triviality of the image of .Ub ; Ub 0 /k in G red . Since ˆ is reduced, it folk

lows from [CGP, Rem. 2.3.6, Thm. 2.3.10] that the quotient map Gk ! G red cark ries root groups onto root groups identifying the root systems. Hence, it suffices to treat the case of semisimple G over k D k. For the classical purely inseparable isogeny $ W G D SO2nC1 ! Sp2n in characteristic 2, the long roots of the target are identified with twice the short roots of the source and $ carries each short root group onto the corresponding long root group [CGP, Prop. 7.1.5(2)].

8.1 Additional constructions for type B

143

Thus, it suffices to prove the commutation of root groups attached to a pair of linearly independent long roots a; a0 for a connected semisimple k-group H of type Cn . But in the root system Cn the sum of two linearly independent long roots is never a root (seen by inspection, or in other ways), so we are done.  The very special quotient map  W G ! G associated to a type-B generalized basic exotic group G satisfies properties analogous to very special isogenies between simply connected semisimple groups as in [CGP, Prop. 7.1.5] (such as swapping long and short root groups). This is primarily an assertion about the intermediate map u W G=ZG ! G that is analogous to the (unipotent) isogeny Q W SO.Q/ ! Sp.B Q / for non-degenerate odd-dimensional .W; Q/: Proposition 8.1.8. Let G be a type-B generalized basic exotic k-group and let T be a maximal k-torus in G=ZG , with n D dim T . (i) The k-group ker u is commutative with no nontrivial ks -points, and if G=ZG D SO.q/ for a regular .V; q/ then ker q is a ks =k-form of .ker q ? /2n for q ? W V ? ! Ga defined by the 2-linear restriction q ? W V ? ! k of q .so dim.ker q / D 2n.dim V ? 1//. (ii) The map u carries T isomorphically onto a maximal k-torus T  G. (iii) Assume T is split .so G=ZG ' SO.q/ for .V; q/ as in (i)/, and define ˆ D ˆ.G=ZG ; T / and ˆ D ˆ.G; T /. Let ˆ< ; ˆ>  ˆ be the respective subsets of short and long roots, with both roots understood to be short when n D 1, and similarly define ˆ< ; ˆ>  ˆ with both roots understood to be long when n D 1. The induced isomorphism X.T / ' X.T / identifies ˆ< with ˆ> and identifies ˆ> with 2
ˆ< . If a root a 2 ˆ> coincides with a root a 2 ˆ< then u W Ua ! U a is an isomorphism, and if a root b 2 ˆ< coincides with b=2 for a root b 2 ˆ> then u W Ub ! U b is identified with q ? . Proof. We may assume G=ZG D SO.q/ for some .V; q/ as in (i), so u D q . Let `  V ? a line, and L  SO.q/ the associated Levi k-subgroup containing T as in Proposition 7.1.5. By design, for the unique T -equivariant complement V0 of V ? we have L D SO.qjW / for W WD V0 ˚ ` of dimension 2n C 1. In particular, qjW is non-degenerate with defect line W ? equal to ` D W \ V ? . The resulting equality W=W ? D V =V ? is compatible with the symplectic forms arising from qjW and q respectively, so q jL is the classical (unipoeD tent) isogeny and its composition with the simply connected central cover L e Spin.qjW / ! L is the very special isogeny for L [CGP, Rem. 7.1.6]. In particular, q carries T isomorphically onto its image T , and for split T the map q

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has the asserted behavior between root systems (via the natural identification ˆ.L; T / D ˆ) as well as from long root groups onto short root groups (since the long root groups of .SO.q/; T / coincide with those of .L; T /). Since Tks is ks -split, for the analysis of ker q and the description of q jUa W Ua ! U 2a for a 2 ˆ< we may and do assume T is k-split. The kernel ker q is connected by Remark 8.1.6. Since T is its own centralizer in SO.q/, it follows from consideration of compatible open cells with respect to q that ker q is a direct product of 2n copies of ker q ? provided that (1) the k-groups ker q jUb for b 2 ˆ< pairwise commute, and (2) q jUb W Ub ! U 2b is identified with q ? for all such b. For linearly independent b; b 0 2 ˆ< the associated root groups Ub and Ub 0 commute by Lemma 8.1.7, so our problem is reduced to the study of q on Gb WD hUb ; U b i. The k-subgroup Gb coincides with D.ZSO.q/ .S// for the codimension-1 subtorus S WD .ker b/0red  T , and the computations in the proof of Proposition 7.1.3 identify Gb with SO.qjWb / where Wb denotes the span Vb ˚ V b ˚ V ? of V ? and the ˙b-weight lines V˙b for T acting on V . The restriction qb D qjWb is regular with defect space of codimension 2, and q jGb is thereby identified with the composition of qb and the natural identification of Sp.B qb / with the copy of SL2 inside Sp.B q / generated by the T -root groups for ˙2b. Hence, we may replace .V; q/ with .Wb ; qb / for each such b, thereby reducing our problem to the case n D 1. The restriction of q to the T -equivariant complement of V ? in V may be identified with .k 2 ; x1 x2 /, and we may k-linearly embed V ? into K via p v 7! q.v/=q.v0 / for a nonzero v0 2 V ? ; this embedding identifies q ? with the restriction to V ? of the 2-linear squaring map K ! k. In this way, as we saw in the proof of Proposition 7.1.3, the map q is identified with the natural map PHV ? ;K=k ! SL2 induced by applying RK=k to the natural unipotent K-isogeny … W PGL2 ! SL2 . The restriction of … between each pair of corresponding root groups (relative to the diagonal tori) is identified with the squaring endomorphism of Ga over K, and its kernels ˛2 in the standard root groups over K commute inside PGL2 by inspection. Hence, the restriction q of RK=k .…/ to the k-subgroup PHV ? ;K=k  RK=k .PGL2 / has the asserted form between compatible root groups and its kernel admits the desired description. 

8.2 Constructions for type C We shall next build a large class of absolutely pseudo-simple k-groups of type C via fiber products using type-B generalized basic exotic groups. To motivate

8.2 Constructions for type C

145

this, recall that the construction of basic exotic k-groups G of type Cn for n > 2 is given by fiber products G

/ RK=k .G 0 /





L

j

f

/ RK=k .G 0 /

where K=k is a nontrivial finite extension satisfying K 2  k and f D RK=k ./ 0 for the very special isogeny  W G 0 ! G from absolutely simple type Cn onto 0 absolutely simple type Bn , with L a Levi k-subgroup of RK=k .G / (equivalently, 0 L is a k-descent of G ) such that L  im.f /. If G 0 contains a split maximal 0 K-torus T 0 and we define T D .T 0 / then  carries long T 0 -root groups iso0 morphically onto short T -root groups and it carries short T 0 -root groups onto 0 long T -root groups via a Frobenius isogeny. In particular, f carries long root groups isomorphically onto short root groups. 0 If L contains the split maximal k-torus T  RK=k .T / then f carries short root groups RK=k .Ga / into the 1-dimensional long T -root groups of L since K 2  k. Hence, the idea is that we try to fatten L into a type-B generalized 0 basic exotic k-subgroup G  RK=k .G / by enlarging the short root groups of L to become the k-subgroup V >  RK=k .Ga / associated to a k-subspace V>  K strictly containing k. Indeed, by inspection of open cells we expect that f 1 .G/ should be pseudo-semisimple of minimal type and that if k D ks then the root system of f 1 .G/ is type Cn with short root groups equal to RK=k .Ga / (as K 2  k) and long root groups equal to V > . If k D ks then this should account for the non-standard possibilities of type Cn in Theorem 3.4.1(iii) when n > 3. (In the special case V> D k, which is to say G of type B is semisimple rather than basic exotic, we recover basic exotic k-groups of type C.) In view of Proposition 3.3.6, to avoid the intervention of nontrivial Weil restrictions we want f 1 .G/ to have root field k, and that is the same as the long root field, so we should only consider those G of type B whose short root field is k. (For pseudo-split G, this says f 2 K j 
V>  V> g D k.) The presence of two vector spaces V and V> in Theorem 3.4.1(iii) for the case n D 2 shows that such a fiber product idea cannot be exhaustive for type B2 D C2 over ks . Although C2 D B2 , when the above idea is specialized to the case n D 2 it does not generally coincide with the type-B generalized basic exotic construction for n D 2 if Œk W k 2  > 2. Thus, to avoid confusion, the reader should regard the notion of “type-C generalized (basic) exotic” defined below (without reference

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Constructions when ˆ has a double bond

to rank) as always distinct from “type-B generalized (basic) exotic”. 8.2.1. We now turn to the actual construction. Let K=k be a nontrivial finite extension satisfying K 2  k, and let G 0 be a connected semisimple K-group that is absolutely simple and simply connected of type Cn with n > 2 (i.e., G 0 is a K0 0 form of Sp2n ). Let  W G 0 ! G be the very special isogeny, so G is a K-form of Spin2nC1 . Let G be either a simply connected and absolutely simple connected semisimple k-group of type Bn or a type-B generalized basic exotic k-group with ks -rank n. Such a G is of minimal type, and the minimal field of definition K 0 =k for its geometric unipotent radical is a (possibly trivial) finite extension of k contained inside k 1=2 . Consider such G for which K 0  K over k and there ss 0 is given a K-isomorphism GK ' G ; we can encode this K-isomorphism via a k-homomorphism 0 j W G ! RK=k .G / ss

0

ss

by composing G D iG with RK 0 =k .GK 0 / ,! RK=k .GK / ' RK=k .G /. (The case of semisimple G is included above so that the “type-C generalized basic exotic” construction below includes as a special case the type-C “basic exotic” construction in §2.2.) Since G ks has a reduced root system, ker iG is central (Proposition 2.3.4). But G is of minimal type, so ker iG is trivial; i.e., j is a closed immersion. We will only be interested in those .G; j / for which j.G/ lies inside the image of 0

f D RK=k ./ W RK=k .G 0 / ! RK=k .G /: Proposition 8.2.2. Let .K=k; G 0 / and .G; j / be as above .so K 0  K/ with j.G/  im.f /, and define G to be the fiber product G

/ RK=k .G 0 /





G

j

f

/ RK=k .G 0 /

.so G D f .G//. The k-group G is absolutely pseudo-simple of minimal type with G ss simply connected of type Cn , K=k is the minimal field of definition k for Ru .Gk /  Gk , and the inclusion G ,! RK=k .G 0 / is identified with iG .so the corresponding K-homomorphism GK ! G 0 is surjective and identifies G 0 ss with GK /. In particular, the data .K=k; K 0 =k; G 0 ; G; j / satisfying K 0  K is uniquely functorial with respect to k-isomorphisms in G.

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8.2 Constructions for type C

Proof. We assume (as we may) after replacing k with ks that k D ks . The quotient scheme G=.ker f / D G is connected, so to prove that G is connected it suffices to show that ker f is connected. As a K-scheme (ignoring the K-group structure), the infinitesimal group scheme ker  is a direct product of copies of the K-scheme Spec.KŒx=.x 2 // D ˛2 D 2 . Thus, the k-scheme ker f D RK=k .ker / is a direct product of copies of the k-scheme RK=k .˛2 / that is a quadric in the affine space RK=k .Ga / (since K 2  k). This quadric hypersurface is geometrically irreducible since the underlying reduced scheme of its k-fiber is a hyperplane, so ker f is connected and thus so is G. Since k is separably closed, by [CGP, Thm. 3.4.6] there exists a Levi kss subgroup L in the pseudo-simple k-group G, so LK ! GK is an isomorphism. 0 ss 0 The inclusion j W G ,! RK=k .G / corresponds to a K-isomorphism GK ' G , 0 so the composite inclusion L ,! RK=k .G / corresponds to a K-isomorphism 0 0 LK ' G . Thus, L is a Levi k-subgroup of RK=k .G /. Clearly L  im.f /, so the triple .K=k; G 0 ; L/ is exactly the setup for a basic exotic k-group of type Cn (see [CGP, Prop. 7.2.7, Prop. 7.3.1]). The schematic preimage f 1 .L/ is therefore a basic exotic k-group of type Cn . Let L be a Levi k-subgroup of f 1 .L/ (see [CGP, Thm. 3.4.6]), so L is also a Levi k-subgroup of RK=k .G 0 / (since the natural map f 1 .L/K ! G 0 is a Kdescent of the maximal geometric reductive quotient [CGP, Prop. 7.2.7(2)]). The connected k-subgroup scheme G of RK=k .G 0 / contains the Levi k-subgroup L of RK=k .G 0 /, so if G is smooth then by Proposition 2.1.2(i) the k-group G is pseudo-reductive with L a Levi k-subgroup of G. Assume G is smooth, so Lk ! G red D G 0 is an isomorphism. Thus, G red is k k k semisimple, simply connected, and absolutely simple of type Cn (so G has an irreducible root system and therefore is pseudo-simple if it is pseudo-semisimple). By Proposition 2.1.2(i), GK \ Ru;K .RK=k .G 0 /K / is a K-descent of Ru .Gk /, so it coincides with Ru;K .GK / and the natural map GK ! G 0 (corresponding to the given inclusion G ,! RK=k .G 0 /) is the quotient modulo Ru;K .GK /. Hence, K=k is a field of definition Ru .Gk /. In fact, K=k is the minimal such field because it is the minimal field of definition for the geometric unipotent radical of the basic exotic f 1 .L/ and f

1

.L/k \ Ru .Gk / D f

1

.L/k \ Ru .RK=k .G 0 /k / D Ru .f

1

.L/k /:

It follows that the inclusion G ,! RK=k .G 0 / is iG , and the consequent triviality of ker iG implies that G is of minimal type. It remains to prove that the connected k-group G is smooth and perfect.

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Constructions when ˆ has a double bond

For this purpose we shall study compatible open cells of RK=k .G 0 /, RK=k .L/, and G built as follows. Pick a maximal k-torus T of f 1 .L/, so T is also a maximal k-torus of RK=k .G 0 /. Letting T be the maximal k-torus image of T in L D f .f 1 .L//, we may naturally identify TK and T K with respective 0 maximal K-tori of G 0 and G . Define the root systems ˆ D ˆ.RK=k .G 0 /; T / D ˆ.G 0 ; TK /; 0

0

ˆ D ˆ.G; T / D ˆ.RK=k .G /; T / D ˆ.G ; T K /; so the inclusion X.T /  X.T / identifies ˆ< with ˆ> and identifies ˆ> with 0 2
ˆ< , and the map  W G 0 ! G carries long root groups isomorphically onto short root groups and carries short root groups onto long root groups via a Frobenius isogeny. For c 2 ˆ, let c 2 ˆ be the corresponding root (i.e., c D c if c is long and c D 2c if c is short). Choose compatible positive systems of roots ˆC and ˆ spectively, so we get compatible open cells

C

in ˆ and ˆ re-

0

  G 0 ;  D ./  G : By [CGP, Prop. 2.1.8(3)], L \ RK=k ./ and G \ RK=k ./ are the corresponding open cells of L and G respectively. Since ker  is infinitesimal, the inclusion of open subschemes    1 ./ is an equality. Thus, G \ RK=k ./ D f

1

.G \ RK=k .//:

By expressing  W  !  as a direct product of maps between maximal tori and root groups (arranged in whatever compatible order we wish among the positive roots, and among their negatives), we get an analogous expression for f W RK=k ./ ! RK=k ./ as direct product of maps. Since the open cell C

G \ RK=k ./ associated to .G; T ; ˆ / is likewise such a direct product, the f -preimage of G \ RK=k ./ may be computed within each factor separately. Hence, G is smooth if and only if (i) the restriction f W RK=k .TK / ! RK=k .T K / has smooth preimage of C WD ZG .T /, and (ii) the restriction f W RK=k .Uc / ! RK=k .U c / has smooth preimage for the c-root group of .G; T /. Note that (i) is _ _ a direct product of maps RK=k .cK .GL1 // ! RK=k .cK .GL1 // for roots c in the C

basis  of ˆC (with the associated c’s constituting the basis  of ˆ ), and C is given by a compatible direct product due to Proposition 4.3.3 (applied to G). If c is long then Uc ! U c is an isomorphism and if also c 2 > then the

8.2 Constructions for type C

149

c-factor of (i) is an isomorphism, so (i) and (ii) are clear for long roots of ˆ (and associated coroots). For short c 2 ˆ we have c D 2c 2 ˆ> and the 1-dimensional c-root group of .G; T / coincides with that of .L; T /, so its f -preimage meets RK=k .Uc / in the c-root group of f 1 .L/. If also c 2 < then c D 2c 2 > and the c-factor of (i) is the squaring endomorphism of RK=k .GL1 / whose image is GL1  RK=k .GL1 / (since K 2  k). The c-factor of C is c _ .GL1 /, and its _ preimage in the c-factor RK=k .cK .GL1 // of RK=k .TK / is the entire c-factor. Thus, G is smooth and therefore pseudo-reductive. 0 This argument proves: Ga D G a inside RK=k .Ga0 / D RK=k .G a / for a 2 ˆ> , Gb D RK=k .Gb0 / for b 2 ˆ< , and ZG .T / is generated by a Cartan k-subgroup of Ga0 for the unique a0 2 > and a Cartan k-subgroup of the pseudo-simple subgroup f 1 .L/  G, so ZG .T /  D.G/. But G D D.G/
ZG .T /, so G D D.G/; i.e., G is perfect.  Definition 8.2.3. A type-C generalized basic exotic group is a k-group G as in Proposition 8.2.2 (so G is absolutely pseudo-simple of minimal type with G ss k simply connected of type Cn , n > 2) such that the root field of G (or equivalently, the short root field of G) is k. By Proposition 8.2.2, if k 0 =k is a separable extension then a k-group G is type-C generalized basic exotic if and only if Gk 0 is such a k 0 -group. 8.2.4. It is instructive to compare the root groups for pseudo-split “type-B” and “type-C” generalized basic exotic groups. Let G be a group of either type with rank n > 2, and let K=k the minimal field of definition for the geometric unipotent radical, so K ¤ k and K 2  k. The type-B construction has long root groups of dimension 1 and short root groups V for a nonzero k-subspace V  K satisfying khV i D K (so 1 < dimk V 6 ŒK W k); the case V D K coincides with “basic exotic of type Bn ”. The proof of Proposition 8.2.2 shows that (as expected from the motivation given near the beginning of §8.2) the type-C construction has as its short root groups RK=k .Ga / and as its long root groups either Ga (when G is semisimple) or the k-group V > (when G is a type-B generalized basic exotic k-group) with associated nontrivial field extension khV> i=k, where V>  khV> i  K and V> has root field k (so 1 < dimk .V> / < ŒK W k). The type-C case with semisimple G is the basic exotic construction of type C, so the new cases of interest are when G is a type-B generalized basic exotic k-group with ks -rank n > 2. When k D ks , the generalized basic exotic groups that are not basic exotic rest on a choice of a nonzero k-subspace of K whose k-dimension is strictly

150

Constructions when ˆ has a double bond

between 1 and ŒK W k. The type-B generalized basic exotic construction of rank n > 2 is obtained from the basic exotic construction of type Bn relative to K=k by shrinking short root groups from RK=k .Ga / to V for V satisfying khV i D K, whereas the type-Cn generalized basic exotic construction for n > 2 is obtained from a basic exotic construction of type Cn relative to K=k by enlarging long root groups from Ga to V > where the nonzero proper k-subspace V>  K has root field k. By Theorem 3.4.1(iii), the isomorphism class of each such k-group depends only on the root system over ks and on the K  -homothety class of the indicated k-subspace V> of K. Such k-groups that are not basic exotic can only exist when ŒK W k > 2, and since K  k 1=2 this can only occur if Œk W k 2  > 2. For n D 2 over k D ks with ŒK W k > 2 (so Œk W k 2  > 2), we therefore have two entirely different classes of new absolutely pseudo-simple k-groups G for which G ss is simply connected of type B2 D C2 : the “type-B” case has long root k groups of dimension 1 and short root groups of dimension strictly between 1 and ŒK W k whereas the “type-C” case has short root groups of dimension ŒK W k > 1 and long root groups of dimension strictly between 1 and ŒK W k. Here is the classification of pseudo-split possibilities for the generalized basic exotic constructions beyond rank 1; it is established by the preceding considerations along with Theorem 3.4.1(iii) and Proposition 4.3.3. Proposition 8.2.5. Fix n > 2 and let K=k be a nontrivial finite extension satisfying K 2  k. Consider pseudo-split type-B or type-C generalized basic exotic k-groups G of rank n that have K=k as the minimal field of definition for Ru .Gk /  Gk . Let T be a split maximal k-torus in G, and ˆ D ˆ.G; T /. The set of k-isomorphism classes of such G in the “type-B” case is in bijective correspondence with the set of K  -homothety classes of nonzero ksubspaces V  K satisfying khV i D K by assigning to G the homothety class of V for which Gb ' HV;K=k for short roots b 2 ˆ. Moreover, if  is a basis of ˆ then naturally Y _  ZG .T / D . a_ .GL1 //  .RK=k .bK //.VK=k / a2>

for the unique b 2 < . The set of k-isomorphism classes of such G in the “type-C” case is in bijective correspondence with the set of K  -homothety classes of nonzero ksubspaces V>  K with root field k such that Ga ' HV> ;K=k for long a 2 ˆ .so Ga ' HV> ;K> =k for K> WD khV> i/. Moreover, if  is a basis of ˆ then

151

8.2 Constructions for type C

naturally Y

_  ZG .T / D .RK=k .aK //..V> /K=k /

RK=k .GL1 /

b2
. In Proposition 8.2.5, the k-group G is basic exotic precisely when V D K in the “type-B” case and precisely when V> is a k-line in the “type-C” case. Note that we allow khV> i to be a proper subfield of K over k for the “type-C” case (it is precisely the subfield K> in such cases). In the spirit of Proposition 8.1.4, here is a description of the pseudo-parabolic k-subgroups of a type-C generalized basic exotic k-group G in terms of an associated type-B generalized basic exotic k-group G: Proposition 8.2.6. Let G be a type-C generalized basic exotic k-group with ks rank n > 2 and associated triple .K=k; G 0 ; G/ as in Proposition 8:2:2. Define 0 f D RK=k ./ for the very special K-isogeny  W G 0 ! G . For any pseudoparabolic k-subgroup P of G, define P D f .P /  f .G/ D G; P 0 D im.PK ! G 0 / 0

ss

0

.so P WD .P 0 / is the image of P K ,! GK  GK ' G /. (i) The k-subgroup P  G is pseudo-parabolic, the K-subgroup P 0  G 0 is pseudo-parabolic, and P D G \ RK=k .P 0 /. Moreover, the natural map G=P ! .G=P / R

K=k .G

0

0

=P /

RK=k .G 0 =P 0 /

is an isomorphism. (ii) For any pseudo-parabolic k-subgroup Q  G and pseudo-parabolic K0 subgroup Q0  G 0 such that .Q0 / D im.QK ! G /, there is a unique pseudo-parabolic k-subgroup Q  G giving rise to Q0 and Q. (iii) Let T  G be a maximal k-torus, and let T 0 WD im.TK ,! G 0 /. Then P  T if and only if P 0  T 0 , and every pseudo-parabolic K-subgroup of G 0 containing T 0 comes from a unique such P . 0

Since G 0 and G are connected reductive, we remind the reader that over a field F the pseudo-parabolic F -subgroups of a connected reductive F -group are precisely its parabolic F -subgroups [CGP, Prop. 2.2.9].

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Constructions when ˆ has a double bond

Proof. The proof of (iii) and the proof of (i) apart from the assertions concerning G=P proceed exactly as in the proof of Proposition 8.1.4. In particular, the uniqueness in (ii) is proved. It remains to establish the description of G=P in (i) and to prove the existence of Q in (ii). For both assertions we may assume k D ks , and the arguments will rest on calculations with the Bruhat decomposition for the pairs 0

0

.G 0 .K/; P 0 .K//; .G .K/; P .K//; .G.k/; P .k// similar to the basic exotic case in the proof of [CGP, Prop. 11.4.6]. However, 0 although G 0 and G are reductive, now G is not reductive (in contrast with the basic exotic case). This causes two complications: (1) the proof of the Bruhat decomposition for G.k/ relative to a choice of pseudo-parabolic k-subgroup of G lies deeper than in the reductive case, 0 (2) the map GK ! G corresponding to the tautological inclusion j W G ,! 0 ss RK=k .G / is the quotient map GK  GK rather than an isomorphism.

Q0

Consider the existence part of (ii): given a pseudo-parabolic K-subgroup  G 0 and a pseudo-parabolic k-subgroup Q  G such that ss

0

.Q0 / D im.QK ! GK ' G /; we seek a (necessarily unique) pseudo-parabolic k-subgroup Q  G giving rise to Q0 and Q. As in the basic exotic case in [CGP, Prop. 11.4.6], we first pick a ss maximal k-torus T  G and let T 0 denote the maximal K-torus TK ,! GK D G 0, so G 0 .K/-conjugacy of maximal K-tori in G 0 provides g 0 2 G 0 .K/ such that the pseudo-parabolic K-subgroup P 0 WD g 0 Q0 g 0 1 of G 0 contains T 0 . Hence, by (iii) there is a unique pseudo-parabolic k-subgroup P  G containing T such ss that P 0 is the image of PK ! GK ' G 0. Let P denote the pseudo-parabolic image of P under the quotient map G  G. It is harmless to change g 0 by left P 0 .K/-multiplication and also by right G.k/-multiplication (since for g 2 G.k/ there is no harm in replacing Q0 and Q with their respective conjugates against the images of g in G 0 .K/ and G.k/ respectively). Thus, it suffices to show that necessarily g 0 2 P 0 .K/G.k/. For the pseudo-parabolic k-subgroups P and Q in G, the scalar extensions ss 0 0 0 P K and QK in GK have respective images in GK D G equal to P and Q . 0 These images are G .K/-conjugate by design, so P and Q are G.k/-conjugate due to:

8.2 Constructions for type C

153

Lemma 8.2.7. Let H be a pseudo-reductive group over a field F , and consider a pair of pseudo-parabolic F -subgroups P; Q  H and the respective parabolic images P 0 ; Q0  H 0 DW H red of PF and QF . If P 0 is H 0 .F /-conjugate to Q0 F then P is H.F /-conjugate to Q. This is a strengthening of [CGP, Cor. C.2.6] in which a separability condition has been removed (but otherwise the proof is essentially the same). Proof. By the H.F /-conjugacy of minimal pseudo-parabolic F -subgroups of H [CGP, Thm. C.2.5], we may replace Q with an H.F /-conjugate so that P and Q contain a common (minimal) pseudo-parabolic F -subgroup. But then P 0 and Q0 are parabolic subgroups of H 0 containing a common parabolic subgroup, and hence containing a common Borel subgroup. By hypothesis P 0 and Q0 are H 0 .F /-conjugate, and in a connected reductive F -group there is no conjugacy among distinct parabolic subgroups containing a common Borel subgroup. Thus, P 0 D Q0 , so P D Q by [CGP, Cor. 3.5.11].  We shall now complete the proof of Proposition 8.2.6. Since P and Q are now known to be G.k/-conjugate, by exactly the same reasoning as in the treatment of the basic exotic case (see the proof of [CGP, Prop. 11.4.6(1)]) it suffices for the existence aspect of (ii) to show that under the map 0

0

G 0 .K/=P 0 .K/ ! G .K/=P .K/ 0

the preimage of the P .K/-coset of any point of G.k/ admits a representative 0 in G.k/. In contrast with the basic exotic case, GK ! G is now merely a surjection (it is the maximal reductive quotient) rather than an isomorphism. 0 Nonetheless, the map G.k/ ! G .K/ is injective since it is induced by the 0 inclusion G ,! RK=k .G /. Choose a positive system of roots ˆC in ˆ WD ˆ.G; T / D ˆ.G 0 ; T 0 /, and let C 0 0 ˆ be the corresponding positive system of roots in ˆ WD ˆ.G; T / D ˆ.G ; T /. Via the equality X.T /Q D X.T /Q and the length-swapping correspondence c 7! c between ˆ and ˆ, the reflections rc and rc coincide. Thus, W WD W .ˆ/ coincides with W .ˆ/ as reflection groups on the common rational character space. Letting  be the basis of ˆ corresponding to ˆC , the parabolic set of roots ˆ.P; T / is labelled by a unique subset J   (with J D ; when P is a pseudo-Borel K-subgroup). 0 Inspection of the behavior of the very special K-isogeny  W G 0 ! G on 0 root groups relative to T 0 and T shows that P and P 0 are also labelled by J via C the natural bijection between  and the basis of ˆ corresponding to ˆ . Thus,

154

Constructions when ˆ has a double bond 0

via the injection G.k/ ,! G .K/ we can conclude by the exact same argument as in the basic exotic case (in the proof of [CGP, Prop. 11.4.6(1)]) provided that the following two properties are proved: (a) the natural map a

U w .k/
nw
P .k/ ! G.k/

w2W J

is bijective, (b) each constituent on the left side of the map in (a) is a direct product set. Here we use standard notation as in the reductive case [Bo, 21.29]: W J is the set of unique shortest-length representatives in W D W .G 0 ; T 0 / of the cosets in W =WJ where WJ is generated by the reflections associated to elements of J , nw 2 NG .T /.k/ is a representative of w, and U w is the k-subgroup of G directly spanned in any order by the root groups of .G; T / associated to the roots C C c 2 ˆ such that w 1 .c/ 62 ˆ . (See [CGP, Cor. 3.3.13(1)] for the existence of the k-subgroup U w .) Using the “open cell” structure associated to pseudo-parabolic subgroups of pseudo-reductive groups in [CGP, Prop. 2.1.8(2),(3)], properties (a) and (b) are straightforward consequences of the Bruhat decomposition for pseudo-reductive groups [CGP, Thm. C.2.8] (applied to .G; P /) exactly as in the reductive case. This completes the proof of (ii), and the proof of the formula for G=P in (i) goes exactly as in the basic exotic case (using calculations with the bijection as in (a) above and its better-known analogue for the pairs .G 0 .K/; P 0 .K// and 0 0 0 .G .K/; P .K// with reductive G 0 and G ). 

8.3 Exceptional construction for rank 2 Let k be imperfect with char.k/ D 2. Among the absolutely pseudo-simple kgroups G of minimal type with a reduced root system over ks such that G ss k is simply connected of type B2 D C2 , Theorem 3.4.1(iii) provides pseudo-split non-standard possibilities beyond the type-B and type-C generalized basic exotic constructions in §8.1–§8.2. These additional G are those for which .Gks /c is non-standard for all roots c (rather than only for short c as in §8.1, or only for long c as in §8.2). We aim to go beyond the pseudo-split case and describe such G in terms of a variant of the fiber product construction in Proposition 8.2.2. The pseudo-split pseudo-simple groups with root system B2 are classified by field-theoretic and linear-algebraic data: the minimal field of definition K=k

8.3 Exceptional construction for rank 2

155

for the geometric unipotent radical (a nontrivial purely inseparable finite extension), the subextension K> =k from Theorem 3.3.8(ii) that contains kK 2 by (3.3.8), and the nonzero kK 2 -subspace V>  K> and nonzero K> -subspace V  K satisfying khV> i D K> and khV i D K (see Theorem 3.3.8(iii)). Since Ga ' HV> ;K> =k for long roots a and Gb ' HV;K=k for short roots b, nonstandardness of Gc for all roots c says exactly V ¤ K and V> ¤ K> (so ŒK W K>  > 2 and ŒK> W kK 2  > 2). Remark 8.3.1. We claim that such pseudo-split G exist with non-standard Gc for all c if and only if Œk W k 2  > 16. Suppose Œk W k 2  > 16, so we can choose K=K> =k inside k 1=2 with ŒK W K>  > 4 and ŒK> W k > 4. Thus, we can choose V> to be a k-hyperplane in K> and V to be a K> -hyperplane in K. Conversely, non-standardness forces ŒK> W kK 2  > 2 and ŒK W K>  > 2, and these degrees are powers of 2, so both degrees are multiples of 4 and ŒK W K 2  > ŒK W kK 2  > 16. It remains to show that Œk W k 2  D ŒK W K 2 , or more generally that if E is any field of characteristic p > 0 then ŒE 0 W E 0 p  D ŒE W E p  for any finite extension E 0 of E. If E 0 =E is separable then E ˝E p E 0 p is a field and E ˝E p E 0 p ! E 0 is a map of fields that is both separable and purely inseparable (hence an equality). Thus, the general case reduces to E 0 D E.˛/ where ˛ p D a 2 E E p . This special case amounts to an elementary computation given in the proof of [Mat, Thm. 26.10] since ŒE W E p  D dimE 1E=E p (by the relationship between p-bases and differential bases [Mat, Thm. 26.5]) and 1E=E p D 1E=Fp . Now consider G as above without a pseudo-split hypothesis, and assume .Gks /c is non-standard for all c and that G has root field k. The next two lemmas show how to construct G in terms of a fiber product analogous to the generalized basic exotic construction in Proposition 8.2.2, after which we will show that such fiber products account for all G in a canonical manner. Let K=k be the minimal field of definition for Ru .Gk /  Gk , and let K> =k be the subextension that is the ks =k-descent of the analogous subextension of Ks =ks associated to Gks in Theorem 3.4.1(iii). Let F> ; F< denote the long and short root fields of G (as in Definition 3.3.7), so kK 2  F>  K>  F< by Galois descent of (3.3.8) applied to Gks . The root field of G is k by hypothesis, but F> is equal to the root field of G by (3.3.2), so F> D k and hence K 2  k. prmt

Lemma 8.3.2. Let G D GF< . (i) The F< -group G is type-B generalized basic exotic of rank 2 but not basic exotic, and the natural map G ! RF< =k .G / has trivial kernel. (ii) Let  W G ! G be the very special quotient as in Definition 8:1:5 .so G is semisimple and simply connected of type C2 D B2 /, and define f D

Constructions when ˆ has a double bond

156

RF< =k ./ and G WD f .G/. The k-group G is type-B generalized basic exotic of rank 2 but not basic exotic, and the natural map G F< ! G is an F< -descent of the maximal geometric reductive quotient of G. (iii) If G is pseudo-split and is classified by the data .K=k; V; V> / in Theorem 3:4:1(iii) then in Proposition 8:2:5 the F< -group G is classified by .K=F< ; V / and the k-group G is classified by .K> =k; V> /. Before we prove this lemma, for the convenience of the reader we display the groups and maps in the form of a commutative diagram / RF =k .G /
/

in which the top map is an inclusion by (i) and the lower-right horizontal map ss is induced by the k-inclusion K>  F< and the identification G F< ' G arising from (ii). Proof. For the proofs of (i) and (ii) it is harmless to extend scalars to ks , so for the proof of the entire lemma we may assume G has a split maximal ktorus T and associated linear algebra data .V; V> / as in Theorem 3.4.1(iii). Let ˆ D ˆ.G; T /. The maximal geometric reductive quotient of G coincides with pred that of G WD GF< and hence with that G , so G ss is simply connected of type k B2 D C2 . Identify T WD TF< with a maximal F< -torus of G, so ˆ.G; T/ D ˆ. For every c 2 ˆ, the quotient map GF<  G restricts to a quotient map …c W .Gc /F<  Gc whose target is pseudo-reductive of minimal type (see Lemma prmt 2.3.10). We claim that …c identifies Gc with .Gc /F< as quotients of .Gc /F< . Since the geometric unipotent radical of Gc is defined over K, by Remark 2.3.12 it suffices to identify Gc (as a quotient of .Gc /F< ) with the image of the natural ss ss map jc W .Gc /F< ! RK=F< ..Gc /K /. But .Gc /K is naturally a K-subgroup of ss GK by 3.2.2, so the image of jc is identified with that of the composite map ss .Gc /F< ,! GF< ! RK=F< .GK /:

The second map in this composition has image G by Remark 2.3.12, and the resulting map .Gc /F< ! G clearly has image Gc , as required. For long roots a 2 ˆ, since K>  F< over k we conclude that prmt

Ga ' .HV> ;K> =k /F< ' SL2

8.3 Exceptional construction for rank 2

157

pred

because .HV> ;K> =k /K> ' SL2 . Likewise, for short roots b 2 ˆ, since F<  K over k and V is an F< -subspace of K we have prmt

Gb ' .HV;K=k /F< ' HV;K=F< via (3.1.6) and (2.3.13). Thus, by Theorem 3.4.1(iii), the F< -group G is classified by the data .K=F< ; V; F< /, so it is a type-B generalized basic exotic F< -group of rank 2 and is not basic exotic since V ¤ K (as Gc is non-standard for short c). This settles the first assertions in (i) and (iii). To establish the second assertion in (i) we note that the quotient map GK  ss factors through the quotient GK of GK D .GF< / ˝F< K since GK has no nontrvial central unipotent K-subgroup scheme. Thus, we can factor iG through the map of interest G ! RF< =k .G /, so this latter map has trivial kernel since ker iG D 1 (by Proposition 2.3.4, since G is of minimal type). ss GK

Turning to (ii), for a Levi k-subgroup L  G containing T (which exists by [CGP, Thm. 3.4.6]) clearly LF< is a Levi F< -subgroup of G containing TF< . The restriction  W LF< ! G is the very special isogeny for LF< (by Definition 8.1.5 and Proposition 7.1.5), so it identifies G with LF< for the very special quotient L of L. Hence, f .L/ D L; this is a Levi k-subgroup of RF< =k .LF< / D RF< =k .G /. It follows from Proposition 2.1.2(i) that G is pseudo-semisimple with L as a Levi k-subgroup and with the same root system as L relative to the split maximal k-torus T WD f .T /. Since L is simultaneously a Levi k-subgroup of RF< =k .G / and G, we have established the final assertion in (ii). Consider the split maximal F< -torus T F< in G and associated root system ˆ D ˆ.L; T / D ˆ.G; T /. By [CGP, Prop. 7.1.5] applied to the very special isogeny L ! L, the isogeny T ! T identifies each long c 2 ˆ with a short c 2 ˆ and identifies each short c 2 ˆ with c=2 for a long c 2 ˆ. Moreover, the definition of  implies that the restriction c W Gc  G c is an F< isomorphism for long c whereas if c is short then c corresponds to the natural map HV;K=F< ! SL2 induced by RK=F< applied to the Frobenius isogeny of .Lc /K ' SL2 . Thus, fc W Gc  G c (induced by RF< =k .c /) is an isomorphism if c is long, but if c is short then fc has 1-dimensional image on the ˙c-root groups since even on RF< =k .RK=F< .SL2 // D RK=k .SL2 / the effect of the Frobenius isogeny of the K-group SL2 carries each root group of RK=k .SL2 / onto a 1-dimensional image (as K 2  k). By our description of G c for roots c 2 ˆ, Theorem 3.4.1(iii) now classifies

158

Constructions when ˆ has a double bond

G via the data .K> =k; V> ; k/, so G is a type-B generalized basic exotic k-group of rank 2 and it is not basic exotic since V> ¤ K> (as Gc is non-standard for long c). Parts (ii) and (iii) are now proved.  Lemma 8.3.3. The inclusion G  f

1 .G/

induced by .8:3:2/ is an equality.

Proof. We can replace k with ks so that k D ks . We first check that f 1 .G/ is smooth by computing where it meets an open cell of RF< =k .G /. Let T be a maximal k-torus in G, and define T D f .T /. Note that ˆ WD ˆ.G; T / D ˆ.RF< =k .G /; T / D ˆ.G ; TF< /, so upon fixing a positive system of roots ˆC  ˆ and using the corresponding positive system of roots for the common root system ˆ of .G; T /, .RF< =k .G /; T /, and .G ; T F< / we get open cells   G and   G such that G \ RF< =k ./ and G \ RF< =k ./ are the corresponding open cells of G and G (by [CGP, Prop. 2.1.8(3)]). Let ˆ> ; ˆ< be the respective subsets of long and short roots in ˆ, and define the subsets ˆ> and ˆ< of ˆ similarly. The restriction f W RF< =k ./ ! RF< =k ./ is a direct product of copies of RF< =k applied to maps between root groups and between coroot subgroups of Cartan subgroups of Gc and G c (with c short precisely when c is long). By Lemma 8.3.2(iii) for G , the maps between corresponding direct factors for long c arise from the identity map of RF< =k .SL2 / and such maps for short c arise from RF< =k .'/ W RF< =k .HV< ;K=F< / ! RF< =k .SL2 / where ' W HV< ;K=F< ! SL2 is the “Frobenius” over F< . The description of RF< =k ./ as a direct product yields a description of the open cell G \ RF< =k ./ of G as well as of the open cell G \ RF< =k ./ of G and the map induced between them by f . By computing separately with root groups for long roots and short roots in ˆ as well as for direct factors of Cartan subgroups in accordance with Proposition 4.3.3, we see (using that K 2  k when working with short roots in ˆ) that f 1 .G/ \ RF< =k ./ D G \ RF< =k ./. This establishes the smoothness of f 1 .G/ and that G D f 1 .G/0 , so G is normal in f 1 .G/. It remains to show f 1 .G/ is connected, and it suffices to check that every k-point 2 f 1 .G/ lies in G. By G.k/-conjugacy of maximal k-tori of G and the equality NG .T /.k/=ZG .T /.k/ D W .ˆ/, we may replace with a G.k/translate so that conjugation by stabilizes T and preserves ˆC . Since the diagram of ˆ has no nontrivial automorphisms, the effect of -conjugation on G fixes  pointwise. Hence, centralizes T . But the map RF< =k .TF< / ! RF< =k .T F< /

159

8.3 Exceptional construction for rank 2

induced by f D RF< =k ./ is identified with the map  VK=k  RF< =k .GL1 / ! RF< =k .GL1 /  RF< =k .GL1 /

given by squaring between the first factors and the identity between the second  factors. Inside the target, ZG .T / is compatibly identified with GL1  .V> /K > =k by Lemma 8.3.2(iii) for G (recall khV> i D K>  F< ). But K 2  k, so 2   VK=k  .V> /K D ZG .T /.  > =k 8.3.4. The two preceding lemmas motivate the following construction. Let G be a type-B generalized basic exotic k-group of rank 2 that is not basic exotic, 2  k and with K> =k the minimal field of definition for Ru .G k /  G k (so K> ŒK> W k > 4). Let F 0 =K> be a finite-degree subfield of k 1=2 and G a type-B generalized basic exotic F 0 -group of rank 2 with short root field F 0 such that G is not basic exotic and the minimal field of definition K=F 0 for Ru .GF 0 /  GF 0 satisfies K 2  k. Note that ŒK W F 0  > 4. Let  W G ! G be the (semisimple) very special quotient over F 0 as in Definition 8:1:5, and define f D RF 0 =k ./. ss

Proposition 8.3.5. Let there be given an F 0 -isomorphism  W G F 0 ' G , and iG

ss

assume that the inclusion j W G ,! RK> =k .GK> / ,! RF 0 =k .G / defined via  lands inside im.f /. The fiber product G







G

/ RF 0 =k .G /

j

f

/ RF 0 =k .G /

.with ker j D 1/ is pseudo-reductive and satisfies the following properties: (i) G is absolutely pseudo-simple with root system B2 D C2 , (ii) G has short root field F 0 and G ss is simply connected, k

prmt

(iii)  corresponds to an isomorphism GF 0 ' G .so G is of minimal type since ker  D 1/, (iv) .Gks /c is non-standard for all roots c relative to a maximal k-torus of G. In particular, since G D .f ı /.G/, the 4-tuple .G; F 0 =K> ; G ; / is uniquely functorial with respect to k-isomorphisms in G.

Constructions when ˆ has a double bond

160

Proof. We may and do assume without loss of generality that k D ks . The main work is to prove that G is smooth. To prove such smoothness we will build a basic exotic k-group H of type B2 inside G, so let us first construct such an H . Since k D ks , by [CGP, Thm. 3.4.6] there exists a Levi F 0 -subgroup L  G (so L ' Spin5 and the natural map LK ! GKred is an isomorphism). A (split) maximal F 0 -torus T of L is also one for G , and ˆ.L ; T / D ˆ.G ; T / since both coincide with ˆ WD ˆ.GKred ; TK ). Hence, by consideration of open cells we see that the schematic center ZL is contained in the center of G , so the containment L \ ZG  ZL is an equality. In particular, L =ZL is identified with a Levi F 0 -subgroup of the type-B adjoint generalized basic exotic k-group G =ZG , so (via Proposition 7.1.5) the restriction of the very special quotient map  W G ! G to L is the very special isogeny L for L . Let fL D RF 0 =k .L /, Let V be a nonzero proper F 0 -subspace of K that classifies G via the “typeB” case of Proposition 8.2.5 (so F 0 hV i D K). For b 2 ˆ< the natural map b W Gb ! G b (with b D 2b 2 ˆ> ) is identified with the natural “Frobenius” quotient map HV;K=F 0 ! SL2 (recall K 2  k), whereas for a 2 ˆ> the natural map a W Ga ! G a (with a D a 2 ˆ< ) is an isomorphism (with source and target isomorphic to SL2 ). Hence, by treating long and short roots in ˆ separately and using that K 2  k when considering short roots, we see that the images f .RF 0 =k .Gc // and f .RF 0 =k .Lc // coincide for all c 2 ˆ. The Cartan F 0 subgroup ZG .T / has a natural product decomposition as in the “type-B” case of Proposition 8.2.5, so RF 0 =k .G / is generated by the groups RF 0 =k .Gc / for c 2 ˆ. This proves that f .RF 0 =k .G // D fL .RF 0 =k .L //, hence G  fL .RF 0 =k .L //. If L denotes a Levi k-subgroup of G (so it is a Levi k-subgroup of RF 0 =k .G /, ss as G F 0 ! G is an isomorphism) then the triple .L ; F 0 =k; L/ is data as required in the construction of a basic exotic k-group of type B2 . Thus, the schemetheoretic preimage H WD fL 1 .L/  G is a basic exotic k-group of type B2 . In particular, this preimage is smooth (even absolutely pseudo-simple) over k and f .H / D L. Let T  H be a (split) maximal k-torus, so for rank reasons T is maximal in RF 0 =k .G / and hence TF 0 is a maximal F 0 -torus in G . Likewise the image T D f .T / is a maximal k-torus in f .H / D L and so also in G. Clearly T F 0 is identified with a maximal F 0 -torus of G . Let  be a basis of ˆ.H; T / D ˆ.L ; TF 0 / D ˆ.G ; TF 0 / D ˆ.RF 0 =k .G /; T /. The basis  defines a positive system of roots ˆC for the common root system ˆ of .RF 0 =k .G /; T / and C

.G ; TF 0 /. Let ˆ be the corresponding positive system of roots for .G; T /, .RF 0 =k .G /; T /, and .G; T F 0 /. For the corresponding open cells   G and

161

8.3 Exceptional construction for rank 2

 D ./  G D G F 0 , clearly RF 0 =k ./ and RF 0 =k ./ are respective open cells of RF 0 =k .G / and RF 0 =k .G /, and G \ RF 0 =k ./ is the associated open cell of G. The smoothness of G WD f to the smoothness of X Df

1

1 .G/

is equivalent

.G \ RF 0 =k .// \ RF 0 =k ./:

The compatible product decompositions of all of these open cells (including product decompositions of Cartan factors via Proposition 8.2.5) decomposes X into a direct product of analogous constructions indexed by the elements of ˆ and _ . Thus, the smoothness of X can be studied separately on each factor of the direct product. For roots in ˆ> and the coroot a_ for the long root a 2 , the corresponding factors are identified with those of G a ' HV> ;K> =k since Ga ! G a is an isomorphism. For roots in ˆ< and the coroot b _ for the short root b 2 , the corresponding factors are identified with those of RF 0 =k .Gb / ' HV;K=k since the inclusion Lb  b .RF 0 =k .Gb // is an equality for dimension reasons (as K 2  k). This completes the proof that G is smooth. Next we prove that the smooth connected G 0 is pseudo-reductive. Note that Ru;k .G 0 /  ker f since G 0 ! G is surjective with pseudo-reductive target. But ker f D RF 0 =k .ker /, so it suffices to show that ker  has no nontrivial F 0 points. That .ker /=ZG does not have nontrivial F 0 -points follows from Proposition 8.1.8(i), and ZG .F 0 / is trivial since ZG  RK=F 0 .ZGKss / D RK=F 0 .2 /. Rank considerations and pseudo-semisimplicity of G D f .G/ D f .G 0 / imply that the maximal tori of G lie inside D.G 0 /. ss Since .G; F 0 =K> ; G / and  W G F 0 ' G are the data arising from applying Lemma 8.3.2 to D.G 0 /, via the compatible open cells considered above we see that D.G 0 / satisfies all of the desired properties (i)–(iv) for G (using the inclusion iG W G ,! RK=F 0 .G 0 / and E D K in Remark 2.3.12 for (iii)). Hence, Lemma 8.3.3 gives that D.G 0 / D f 1 .G/, so the inclusion D.G 0 /  G inside f 1 .G/ is an equality.  Definition 8.3.6. A k-group G is (rank-2) basic exceptional if it arises as in Proposition 8.3.5 (so G is absolutely pseudo-simple of rank 2) with root field k. By Remark 8.3.1 over ks , such k-groups exist if and only if Œk W k 2  > 16. The following analogue of Proposition 8.2.5 is now immediate from Theorem 3.4.1(iii), Proposition 4.3.3, and Lemmas 8.3.2 and 8.3.3.

Constructions when ˆ has a double bond

162

Corollary 8.3.7. Let k be imperfect with char.k/ D 2 such that Œk W k 2  > 16, and let K=K> =k be finite subextensions of k 1=2 =k with ŒK W K>  > 4, ŒK> W k > 4. A pseudo-split basic exceptional k-group G with associated invariants K and K> is classified up to isomorphism by a pair .V< ; V> / consisting of a K  homothety class of nonzero proper K> -subspaces V<  K such that khV< i D K  -homothety class of nonzero proper k-subspaces V  K satisfying and a K> > > khV> i D K> and f 2 K> j 
V>  V> g D k, and all such pairs arise. For a   split maximal k-torus T in such a G, we have ZG .T / ' .V> /K  .V< /K=k . > =k We finish our discussion of the rank-2 basic exceptional construction with an analogue of Proposition 8.2.6: Proposition 8.3.8. Let G be a basic exceptional k-group with associated 4-tuple .G; F< =K> ; G ;  /. Let  W G ! G be the very special quotient over F< , and let f D RF< =k ./. For any pseudo-parabolic k-subgroup P  G, define P D f .P /  G; P 0 D im.PF< ! G / 0

ss



.so P WD .P 0 / is the image of P F< ,! G F<  G F< ' G /. Assertions (i), (ii), (iii) of Proposition 8:2:6 hold in the present situation 0 upon replacing .G 0 ; G ; K/ with .G ; G ; F< /. Proof. We may carry over the proof of Proposition 8.2.6 essentially without change except that we work with the Cartesian diagram as in Proposition 8.3.5 and now G is not reductive (so the Bruhat decomposition for .G 0 .K/; P 0 .K// in the proof of Proposition 8.2.6 is replaced with the pseudo-reductive Bruhat decomposition for .G .F< /; P 0 .F< // provided by [CGP, Thm. C.2.8], and we ss replace the K-isomorphism GK ' G 0 from the proof of Proposition 8.2.6 with prmt the F< -isomorphism GF< ' G ). Proposition 8.1.8 provides the required properties of the very special quotient G ! G on root groups, akin to very special isogenies in the semisimple case. 

8.4 Generalized exotic groups Over imperfect fields of characteristic 2, the generalized basic exotic groups of types B and C, as well as the rank-2 basic exceptional groups, underlie a construction beyond the standard case that is exhaustive under a locally minimal type hypothesis. To explain this, we need to introduce auxiliary Weil restrictions: Definition 8.4.1. Let k be an imperfect field of characteristic p 2 f2; 3g. A generalized exotic k-group is any G ' D.Rk 0 =k .G 0 // for a nonzero finite reduced

8.4 Generalized exotic groups

163

k-algebra k 0 and k 0 -group G 0 whose fiber over each factor field of k 0 is typeG2 basic exotic when p D 3 or is either type-F4 basic exotic, type-B or type-C generalized basic exotic, or rank-2 basic exceptional when p D 2. Any such k-group is pseudo-semisimple and is clearly of minimal type with a reduced root system over ks . By Proposition 3.4.4 it is also clear that such G are never standard. If fki0 g is the set of factor fields of k 0 and Gi0 denotes the ki0 -fiber of G 0 then applying [CGP, Prop. A.4.8] over k gives that G ss ' k

YY .Gi0 ˝ki0 ;i k/ss i

(8.4.1)

i

where i varies through the set of k-embeddings of ki0 into k. In particular, G ss k is simply connected. Remark 8.4.2. By (8.4.1), G as in Definition 8.4.1 is absolutely pseudo-simple if and only if k 0 is a purely inseparable field extension of k (see [CGP, Lemma 3.1.2]). In such cases with p D 2 and absolutely pseudo-simple G with root system of type B or C, the next result implies that G is generalized basic exotic if and only if k 0 D k; for rank 1 (arising for type-B cases) this requires the root field condition that we included in Definition 8.1.1. It is important that the data .G 0 ; k 0 =k/ used in the construction of a generalized exotic k-group G is functorial with respect to isomorphisms in G: Proposition 8.4.3. Let G and H be generalized exotic k-groups arising from respective pairs .G 0 ; k 0 =k/ and .H 0 ; `0 =k/. Any k-isomorphism  W G ' H has the form D.R˛=k .'// for a unique pair .'; ˛/ consisting of a k-algebra isomorphism ˛ W k 0 ' `0 and a group isomorphism ' W G 0 ' H 0 over ˛. Proof. In view of the uniqueness assertion, by Galois descent we may and do Q assume k D ks . Writing k 0 D i 2I ki0 for finite extensions ki0 =k (which must be Q purely inseparable) and Gi0 for the ki0 -fiber of G 0 , we have G D D.Rki0 =k .Gi0 //. Each of the factors in this direct product is absolutely pseudo-simple by Remark 8.4.2, so these factors are the minimal nontrivial smooth connected normal ksubgroups of G due to [CGP, Prop. 3.1.6]. Hence, if f`j0 ; Hj0 gj 2J is the analogous such data for H , any isomorphism G ' H arises from a bijection  W I ' J and a collection of k-isomorphisms D.Rki0 =k .Gi0 // ' D.R`0 .i / =k .H0.i / //: In this way we are reduced to the special case where k 0 and `0 are fields.

164

Constructions when ˆ has a double bond

By definition of generalized basic exotic groups, the root field of G 0 is k 0 . Thus, by Proposition 3.1.8 and (3.1.6), the (rank-1) minimal-type k-subgroup of G generated by a pair of opposite long root groups has root field equal to k 0 . In particular, the long root field of G is k 0 . Likewise, the long root field of H is `0 . Hence, the existence of a k-isomorphism G ' H implies that there is a k-isomorphism ˛ W k 0 ' `0 between the long root fields (and ˛ is unique since k D ks ). Now identifying `0 with k 0 via ˛, the inclusions G ,! Rk 0 =k .G 0 / and H ,! Rk 0 =k .H 0 / correspond to k 0 -homomorphisms Gk 0 ! G 0 and Hk 0 ! H 0 that are maximal among pseudo-reductive quotients of minimal type over k 0 by Proposition 2.3.13 (since k 0 =k is purely inseparable). Thus, k 0 induces a k 0 -isomorphism ' W G 0 ' H 0 , and it is clear that this is the unique ' which does the job.  By Galois descent, we obtain: Corollary 8.4.4. A k-group G is generalized exotic if and only if Gks is generalized exotic. For an imperfect field k of characteristic p 2 f2; 3g, among all pseudosemisimple k-groups of minimal type with a reduced root system over ks there is a simple intrinsic characterization of the generalized exotic k-groups: Theorem 8.4.5. Let G be a nontrivial pseudo-semisimple k-group of minimal type with a reduced root system over ks . Then G is generalized exotic if and only if G satisfies the following two properties: (i) G ss is simply connected, k (ii) the minimal normal pseudo-simple ks -subgroups of Gks are non-standard. Note that non-standardness of a pseudo-reductive k-group is insensitive to scalar extension to ks [CGP, Cor. 5.2.3]. Proof. By Corollary 8.4.4, we can (and we will) assume that k D ks . The implication “)” follows from the preceding discussion. For the converse, we may assume k D ks and first reduce to considering pseudo-simple G. Let fGj g be the collection of minimal nontrivial perfect smooth connected normal k-subgroups of G, so the Gj ’s pairwise commute and multiplication Q defines a k-homomorphism  W Gj ! G that is surjective with central kernel [CGP, Prop. 3.1.8]. For a maximal k-torus T of G the associated k-subtori Q Tj D T \ Gj  Gj are maximal [CGP, Cor. A.2.7], and the map Tj ! T is ` Q an isogeny that identifies ˆ.G; T / with ˆ.Gj ; Tj / inside X.T /Q D X.Tj /Q [CGP, Prop. 3.2.10]. Thus, each Gj has a reduced root system.

165

8.4 Generalized exotic groups

Since G is of minimal type, so are its normal k-subgroups Gj (Lemma 2.3.10). The central quotient  restricts to an isomorphism between corresponding root groups, and it restricts to an isomorphism between Cartan k-subgroups due to Proposition 4.3.3, so  restricts to an isomorphism between open cells and hence is an isomorphism (as for any birational homomorphism between smooth connected k-groups). In particular, each .Gj /ss is a direct factor of G ss k k and hence is simply connected. Consequently, it suffices to treat each Gj separately, so we may and do assume G is (absolutely) pseudo-simple over k D ks . The hypotheses on G say that it is non-standard of minimal type with a reduced root system of rank n > 1 and G ss is simply connected. If F  K is k

prmt

the root field of G then Proposition 3.3.6 implies G D D.RF =k .GF // where prmt GF over F satisfies the analogous hypotheses as G does over k. Hence, we prmt may replace G and k with GF and F respectively so that G has root field k. Let K=k denote the minimal field of definition over k for Ru .Gk /  Gk , so 0 G WD GK =Ru;K .GK / is a simply connected and absolutely simple connected semisimple K-group. Since G is of minimal type, by Proposition 2.3.4 the natural map iG W G ! RK=k .G 0 / has trivial kernel. Since G is not standard, by Theorem 3.4.2 the reduced and irreducible root system ˆ of rank n > 1 must be G2 if p D 3 and either Bn (n > 1), Cn (n > 1), or F4 if p D 2. If it is G2 with p D 3 or F4 with p D 2 then G is exotic by Proposition 3.4.3, so we may and do assume p D 2 and ˆ is either Bn or Cn with n > 1. The rank-1 case is settled by Proposition 3.1.8, so we may assume n > 2. Hence, the possibilities for G up to isomorphism are classified by Theorem 3.4.1(iii) for types B and C with L simply connected (having root system ˆ). Consider type Bn with n > 3, so G with root field k is classified by the subfield K> D k containing kK 2 (so K 2  k) and the K  -homothety class of a nonzero k-subspace V  K satisfying khV i D K. Since K 2  k and K ¤ k, the same field-theoretic and linear-algebraic data arises from a type-B generalized basic exotic k-group, so G must be that group by Proposition 8.2.5. For type Cn with n > 3, G with root field k is classified by the subfield K>   -homothety class of a nonzero kK 2 -subspace K containing kK 2 and the K> V>  K> such that f 2 K> j 
V>  V> g D k (so K 2  k) and khV> i D K> . By Proposition 8.2.5, the invariants .K=k; V> / arise from a type-C generalized basic exotic k-group. Finally, consider type B2 D C2 , so G with root field k is classified up to k-isomorphism by a triple of invariants .K> =k; V; V> / as in Theorem 3.4.1(iii)

166

Constructions when ˆ has a double bond

with V> satisfying f 2 K> j 
V>  V> g D k. In particular, since V> is a kK 2 subspace of K> , necessarily K 2  k. If V D K then the type-C generalized basic exotic construction for .K=k; V> / recovers G. If dimk .V> / D 1 then the type-B generalized basic exotic construction for .K=k; V / recovers G. Thus, we may assume V ¤ K and dimk .V> / > 1. Let F D f 2 K j 
V  V g. Since F hV i D K, the hypothesis V ¤ K rules out the possibility ŒK W F  6 2, so ŒK W F  > 4 (since K 2  k  F ). Likewise, since khV> i D K> with k the root field of V> , the hypothesis dimk .V> / > 1 ensures that ŒK> W k > 4 and V> ¤ K> . Thus, G is as in Corollary 8.3.7. 

8.5 Structure of ZG and ZG;C The structure of the center of a basic exotic k-group is given in [CGP, Cor. 7.2.5]. We now give an analogue in the generalized basic exotic and basic exceptional cases for types B and C (types G2 and F4 are settled basic exotic cases). Let k be imperfect with char.k/ D 2, let G be a generalized basic exotic or rank-2 basic exceptional k-group, and let T  G be a maximal k-torus. Assume ˆ WD ˆ.Gks ; Tks / equals Bn or Cn with n > 1. Let K=k be the minimal field ss , so K 2  k and iG W G ! of definition of Ru .Gk /  Gk and let G 0 WD GK RK=k .G 0 / has trivial kernel. Proposition 8.5.1. Using notation as above, ZG D G \ RK=k .ZG 0 /. In particular, G=ZG  RK=k .G 0 =ZG 0 /. More explicitly: (i) If ˆ D Cn with even n > 2 then ZG D RK=k .ZG 0 /. (ii) Assume T is split and ˆ equals Bn with n > 1 or Cn with odd n > 2. Choose c 2 ˆ> for type Cn and c 2 ˆ< for type Bn .including n D 2/. For a nonzero k-subspace V  K such that Gc ' HV;K=k , the unique K  isomorphism ZG 0 ' 2 identifies ZG with VK=k \ RK=k .2 / D VK=k Œ2 inside RK=k .GL1 /. Proof. We may assume k D ks , so G is pseudo-split. Define T 0 D TK  G 0 and let  be a basis of ˆ D ˆ.G; T / D ˆ.G 0 ; T 0 /. Since G is a k-subgroup of RK=k .G 0 /, it is obvious that G \ RK=k .ZG 0 /  ZG . To prove the reverse inclusion, note that the map GK ! G 0 associated to the inclusion G ,! RK=k .G 0 / is a quotient map, so it carries .ZG /K into ZG 0 . Hence, ZG  RK=k .ZG 0 /, so ZG  G \ RK=k .ZG 0 /. Consider G with root system Cn for n > 3. Suppose n is even, so ZG 0 is contained in the direct product of the coroot groups associated to the short roots in . For each b 2 ˆ< we have Gb D RK=k .Gb0 / by comparing dimensions of

8.5 Structure of ZG and ZG;C

167

root groups (since Gb0 ' SL2 ). Thus, RK=k .b _ .GL1 //  Gb for all b 2 ˆ< , so the description of ZG 0 using the direct product structure with respect to < (as n is even) implies that RK=k .ZG 0 /  G and hence ZG D RK=k .ZG 0 /. Assume ˆ is of type Cn with odd n > 3. Let a be the unique long root in , so the Dynkin diagram is:

a1



a2



an

 ks






1

an Da



Clearly the center ZG D G \ RK=k .ZG 0 / is equal to ZG .T / \ RK=k .ZG 0 /. To compute this intersection, we shall describe ZG .T / inside RK=k .T 0 / D Q _ j RK=k ..aj /K .GL1 //. By Proposition 8.2.5, _  ZG .T / D RK=k .aK /.VK=k /

Y

_ RK=k .bK .GL1 //

b2
; F<  K as in Definition 3:3:7, the map .8:5:3/ is a closed immersion that lands inside RF< =k .TFad< / and fits into a fiber product diagram ZG;C

/ RF =k .T ad / < F
/

where the bottom map is the natural inclusion .so RF> =k .TFad> /  ZG;C /. If T is split and  is a basis of ˆ D ˆ.G; T / with respective subsets of short and long roots denoted < and > then via the isomorphism T 0 ad ' GL 1 Q induced by t 0 mod ZG 0 7! c2 c.t 0 / we have ZG;C D

Y c2
=k .GL1 /

(8.5.4)

c2>

inside RK=k .T 0 ad /. By Theorem 3.3.8, for types F4 and G2 we have F< D K and F> D K> . Proof. The closed immersion property in the basic exotic case (for types B, C, F4 , and G2 ) is part of [CGP, Cor. 8.2.7], and the same argument works in general because (as in basic exotic cases) the inclusion G ,! RK=k .G 0 / corresponds to the maximal reductive quotient map GK  G 0 . For the rest of the argument we may assume k D ks , so T is split. By [CGP, Prop. 7.1.5], the isomorphism 0 0 0 X.T /Q ' X.T 0 /Q induced by the very special isogeny .G 0 ; T 0 / ! .G ; T / 0 0 0 0 identifies ˆ.G ; T /< with ˆ> and identifies ˆ.G ; T /> with pˆ< . Thus, since K p  k and X.T ad / has  as a Z-basis, the right side of (8.5.4) coincides with the proposed fiber product description of ZG;C and so it suffices to prove (8.5.4). The natural action of T 0 ad on G 0 preserves every root group via a scaling action through the isomorphism T 0 ad ' GL 1 and the expansion of elements of ˆ as unique Z-linear combinations of elements of . Since W .ˆ/ is generated by reflections in elements of  and acts transitively on ˆ< and ˆ> , the root groups U˙c of .G; T / for c 2  generate G. Thus, since the action of RK=k .T 0 ad / on RK=k .G 0 / preserves all T -root groups for RK=k .G 0 /, the smooth connected right side of (8.5.4) is precisely the k-subgroup scheme of points of RK=k .T 0 ad / whose action on RK=k .G 0 / preserves the k-subgroup G. It follows that the right side of (8.5.4) is the image of ZG;C ,! RF< =k .TFad< /. 

9 Generalization of the standard construction

9.1 Generalized standard groups The notion of “generalized standard” pseudo-reductive group over a field k is defined in [CGP, Def. 10.1.9] subject to the requirement Œk W k 2  6 2 when char.k/ D 2, and in characteristic 2 such k-groups have a reduced root system over ks by construction. (This coincides with the notion of standard pseudoreductive group if char.k/ ¤ 2; 3.) To remove the restriction on Œk W k 2  and permit non-reducedness of the root system when char.k/ D 2, the definition we shall give in Definition 9.1.7 will involve root fields (due to their role in Proposition 3.3.6, Definitions 8.2.3 and 8.3.6, and Proposition 9.1.3 below). The notion of root field has only been defined for absolutely pseudo-simple G whose root system over ks is reduced, so our first step is to remove that restriction. Recall that when Gks has a reduced root system, the long and short root fields F> and F< in Definition 3.3.7 satisfy F>  F< , and F> is equal to the root field of G. Thus, in the non-reduced case we will focus on the longest (equivalently, divisible) roots when defining the root field of G. Let G be an absolutely pseudo-simple k-group of minimal type for which Gks has a non-reduced root system. Fix a maximal k-torus T  G, and let a be a divisible root in ˆ.Gks ; Tks /. Since a is not multipliable, the absolutely pseudo-simple ks -subgroup .Gks /a generated by the ˙a-root groups has as its roots only ˙a; i.e., its root system is A1 (rather than BC1 ). By Lemma 2.3.10, .Gks /a is of minimal type since Gks is of minimal type. Since G ss is simply k connected (see [CGP, Thm. 2.3.10]), we have ..Gks /a /ss ' SL2 by 3.2.2. Hence, k .Gks /a is given by the construction in Proposition 3.1.8 over ks . Example 9.1.1. Let K=k be a nontrivial purely inseparable finite extension. Pseudo-simple pseudo-split k-groups G of minimal type with root system BCn (n > 1) and with minimal field of definition K=k for Ru .Gk /  Gk are classi-

172

Generalization of the standard construction

fied in [CGP, Thm. 9.8.6] in terms of linear algebra data .V .2/ ; V 0 / when n ¤ 2 and .V .2/ ; V 0 ; V 00 / when n D 2, with V 0 a nonzero proper kK 2 -subspace of K. The description of G in terms of such data implies that for a split maximal ktorus T  G and a divisible root a 2 ˆ.G; T /, we have Ga ' HV 0 ;K=k . Hence, the common root field F of such Ga ’s is a proper subfield of K containing kK 2 . When n D 2, V 00 is a nonzero F -subspace of K. The linear algebra data classifying G makes sense relative to the extension K=F instead of K=k. Letting GK=F denote the analogous BCn -construction made over F via the same linear algebra data, we obtain a non-canonical isomorphism from [CGP, Prop. 9.8.13] (using the pair .F; k/ in place of .k; k0 /): G ' D.RF =k .GK=F //:

(9.1.1)

A canonical formulation of (9.1.1) will be provided in Proposition 9.1.3, but first we use Example 9.1.1 to make a definition beyond the pseudo-split case: Definition 9.1.2. Let G be an absolutely pseudo-simple k-group of minimal type with a non-reduced root system over ks . The root field F of G is the ks =kdescent of the common root field of .Gks /a for divisible a 2 ˆ.Gks ; T / with T  Gks any maximal ks -torus. Letting K=k be the minimal field of definition for Ru .Gk /  Gk , applying Example 9.1.1 to Gks shows that kK 2  F  K. The root field F as just defined satisfies an analogue of Proposition 3.3.6: Proposition 9.1.3. The maximal pseudo-reductive quotient of minimal type prmt GF of GF is absolutely pseudo-simple with a non-reduced root system over Fs , its geometric unipotent radical has minimal field of definition K=F , and its root field is F . Moreover, the natural map prmt

jG W G ! D.RF =k .GF // is an isomorphism. Proof. Without loss of generality we may (and do) assume that k D ks , so G is pseudo-split and F D Fs . Thus, the non-canonical isomorphism (9.1.1) takes the form of a k-isomorphism f W G ' D.RF =k .H // for an absolutely pseudosimple F -group H of minimal type with a non-reduced root system, minimal field of definition K=F for its geometric unipotent radical, and root field F . But Proposition 2.3.13 implies that the natural map GF ! H arising from f identiprmt fies H with GF , and using the universal property of Weil restriction to proceed in reverse identifies jG with f . Hence, the canonical jG is an isomorphism. 

9.1 Generalized standard groups

173

Remark 9.1.4. Assume G as above is pseudo-split. In contrast with the nonprmt reducedness of the root system for GF (or equivalently for its pseudo-reductive pred central extension GF ), it is a nontrivial condition on a proper subfield E  K prmt over kK 2 that GE has a non-reduced root system. Indeed, if ŒK W kK 2  > 2 then there exists pseudo-split G as above and a proper subextension E of pred K=kK 2 such that the maximal pseudo-reductive quotient GE has a reduced prmt pred root system (and hence likewise for the central quotient GE of GE ); this is shown in [CGP, 9.8.17, 9.8.18]. prmt If ŒK W kK 2  D 2 then necessarily E D kK 2 D F , so GE always has a nonreduced root system in such cases. Since ŒK W kK 2  D 2 whenever Œk W k 2  D 2, prmt the phenomenon of GE having a reduced root system for some proper subfield E  K over kK 2 is only seen when Œk W k 2  > 2. In view of Proposition 9.1.3, for absolutely pseudo-simple k-groups of minimal type with a non-reduced root system over ks the condition that the root field coincides with the ground field (i.e., F D k) is a reasonable one to impose for the purpose of a structure theorem. This is used in: Definition 9.1.5. Let k be a field, k 0 a nonzero finite reduced k-algebra, and G 0 ! Spec.k 0 / a smooth affine group scheme. The pair .G 0 ; k 0 =k/ is primitive if for each factor field ki0 of k 0 the ki0 -fiber Gi0 is one of the following: (1) connected semisimple, absolutely simple, and simply connected group, (2) basic exotic of type F4 or G2 , generalized basic exotic of types B or C, or rank-2 basic exceptional group, (3) minimal type absolutely pseudo-simple group with a non-reduced root system over the separable closure of ki0 and root field equal to ki0 . Groups mentioned in (2) exist only if k is imperfect with characteristic 2 or 3, and groups in (3) exist only if k is imperfect with characteristic 2. Also, for imperfect k, generalized exotic k-groups are precisely k-groups of the form D.Rk 0 =k .G 0 // for primitive pairs .G 0 ; k 0 =k/ such that Gi0 is as in (2) for all i . For every i the ki0 -group Gi0 is of minimal type, and its maximal geometric semisimple quotient is simply connected. The pseudo-split possibilities for Gi0 are parameterized by known invariants: the root system for case (1), the root system along with the minimal field of definition Ki0 =ki0 for the geometric unipotent radical and some additional field-theoretic and linear-algebraic data inside Ki0 as in Theorem 3.4.1 for case (2) (also see Proposition 8.2.5 and Corollary 8.3.7 for types B and C), and similarly in [CGP, Thm. 9.8.6, Prop. 9.8.9] for case (3). If Ci0  Gi0 is a Cartan ki0 -subgroup with maximal ki0 -torus Ti0 then the ki0 group ZGi0 ;Ci0 is commutative and pseudo-reductive (see Proposition 6.1.4) and

174

Generalization of the standard construction

its structure can be expressed in terms of Ti0 : in case (1) it is Ti0 =ZGi0 , in case (2) it is described by Lemma 6.1.3 for rank 1 and by Proposition 8.5.4 for rank > 2, and in case (3) it is described by [CGP, Prop. 9.8.15]. Proposition 9.1.6. Let .G 0 ; k 0 =k/ be a primitive pair as above, and define G D D.Rk 0 =k .G 0 //. The k-group G is pseudo-semisimple of minimal type, G ss is k simply connected, and G is absolutely pseudo-simple if and only if k 0 is a field purely inseparable over k. Moreover, the center ZG is k-tame. Proof. We may assume k D ks (since if k 0 is not a field purely inseparable over k then k 0 ˝k ks has at least two factor fields and hence Gks is not pseudo-simple; note that by Proposition 5.1.2, ZG is k-tame if and only if ZGks D .ZG /ks is Q ks -tame). Write k 0 D ki0 for fields ki0 , and let Gi0 be the ki0 -fiber of G 0 , so G D D.Rk 0 =k .G 0 // D

Y

D.Rki0 =k .Gi0 //:

Q Proposition 2.3.13 gives that G ss D .Gi0 ˝ki0 k/ss , and this is simply connected k by inspection. In particular, G is pseudo-simple if and only if k 0 is a field (see [CGP, Lemma 3.1.2]), and it is of minimal type by Proposition 5.3.5(ii). To prove that ZG is k-tame we may clearly assume that k 0 is a field. Since Rk 0 =k .G 0 /k 0 ! G 0 is surjective [CGP, Prop. A.5.11(1)], so is Gk 0 ! G 0 . Thus, .ZG /k 0 is carried into ZG 0 , so ZG  Rk 0 =k .ZG 0 /. Arguing as immediately below (5.1.1), it therefore suffices to show that ZG 0 is k 0 -tame, so we may assume k 0 D k. If G has a non-reduced root system then ZG is trivial by [CGP, Prop. 9.4.9] (using that G is of minimal type). Likewise, ZG is trivial for types G2 and F4 by [CGP, Cor. 7.2.5(2)]. For the remaining cases with type B or C, Proposition 8.5.1 identifies ZG with a k-subgroup of the Weil restriction to k of finite multiplicative type group over a finite extension of k. Again arguing as immediately below (5.1.1), ZG is k-tame.  Definition 9.1.7. Let k be a field. A pseudo-reductive k-group G is generalized standard if either it is commutative or there exists a 4-tuple .G 0 ; k 0 =k; T 0 ; C / consisting of a nonzero finite reduced k-algebra k 0 , a smooth affine k 0 -group G 0 such that .G 0 ; k 0 =k/ is a primitive pair, a maximal k 0 -torus T 0  G 0 , a commutative pseudo-reductive k-group C , and a factorization diagram 

C ! C ! ZG ;C D Rk 0 =k .ZG 0 ;C 0 /

(9.1.7.1)

9.1 Generalized standard groups

175

with G D D.Rk 0 =k .G 0 //, C 0 D ZG 0 .T 0 /, and C D G \ Rk 0 =k .C 0 / (a Cartan k-subgroup of G ) such that there is a k-isomorphism .G o C /=C ' G;

(9.1.7.2)

where C is anti-diagonally embedded as a central k-subgroup of G o C . (The maximal k-torus T in Rk 0 =k .T 0 / is contained in C , so the Cartan k-subgroup C of G coincides with ZG .T /.) The 4-tuple .G 0 ; k 0 =k; T 0 ; C / equipped with the factorization (9.1.7.1) and isomorphism (9.1.7.2) is called a generalized standard presentation of G. Remark 9.1.8. The equality in (9.1.7.1) is an instance of Proposition 6.1.7, and Definition 9.1.7 recovers the definition of the same terminology in [CGP, Def. 10.1.9] assuming Œk W k 2  6 2 when char.k/ D 2 except for one crucial aspect: when Œk W k 2  D 2 and char.k/ D 2 the definition in [CGP] avoids case (3) of Definition 9.1.5. That avoidance is reasonable when Œk W k 2  D 2 due to a splitting result in [CGP, Thm. 10.2.1(1)], but such splitting generally fails beyond the perfect minimal-type case whenever Œk W k 2  > 2 (see Example 6.1.5). Any k-group G of the form (9.1.7.2) is obviously non-commutative. Let us show that any such G is pseudo-reductive. Since C is a Cartan k-subgroup of G and the formation of Cartan k-subgroups of smooth connected affine k-groups is compatible with quotients, the evident inclusion C ! G makes C a Cartan k-subgroup of the central quotient G in (9.1.7.2). The centrality of C and the evident pseudo-reductivity of G o C therefore imply that G is pseudo-reductive, by [CGP, Prop. 1.4.3]. The pseudo-semisimple D.G/ coincides with the central quotient G =.ker / since C is commutative, so if G D D.G/ then C D .C /. (The k-group ker  is central in G because ı  encodes the action of C on G via conjugation.) Remark 9.1.9. For .G 0 ; k 0 =k/ as in Definition 9.1.7 and G WD D.Rk 0 =k .G 0 //, any pseudo-reductive central quotient G D G =Z is generalized standard. To see this, choose a Cartan k-subgroup C of G , so C WD C =Z is identified with a Cartan k-subgroup of the pseudo-reductive G. Thus, C is pseudo-reductive and the natural action of C on G (arising from conjugation by C and the centrality of Z in G ) provides a factorization diagram C  C ! ZG ;C giving a generalized standard presentation .G 0 ; k 0 =k; T 0 ; C / for G, where T 0 is the (unique) maximal k 0 -torus in G 0 such that C D G \ Rk 0 =k .ZG 0 .T 0 //.

176

Generalization of the standard construction

9.1.10. In the setting of Definition 9.1.7 we claim that the k-group G is determined by the central quotient k-group G =.ker / D D.G/ functorially with respect to isomorphisms in this central quotient, so by Proposition 8.4.3 the triple .G 0 ; k 0 =k; j / incorporating the k-homomorphism j W D.Rk 0 =k .G 0 //  D.G/ ,! G is determined by the k-group G functorially with respect to isomorphisms in G. In [CGP, Prop. 10.1.12 (1)] this is treated via a splitting result for central extensions by affine k-group schemes Z of finite type such that Z 0 .ks / D 1, assuming Œk W k 2  6 2 when char.k/ D 2. (This splitting result rests on [CGP, Prop. 5.1.3, Ex. 5.1.4, Prop. 8.1.2], and to allow G with a non-reduced root system over ks with Œk W k 2  D 2 one can use [CGP, Thm. 10.2.1(1)].) The preceding method does not apply when char.k/ D 2 with Œk W k 2  > 2 because for every such k and n > 1 there exists a pseudo-split absolutely pseudosimple k-group H with root system BCn such that H is not of minimal type, so H is a non-split k-tame central extension of G WD H=CH . Moreover, when Œk W k 2  > 16 analogous examples exist with H=CH generalized basic exotic of type B or C (with any rank n > 1). Appendix B provides many such H , and proves it is necessary that Œk W k 2  > 16 when the root system is reduced. We shall use universal smooth k-tame central extensions to bypass this problem via: eD Lemma 9.1.11. Let .G 0 ; k 0 =k/ and .G00 ; k00 =k/ be primitive pairs. Define G 0 0 e0 D D.Rk 0 =k .G //, and let Z  Z e and Z0  Z e be D.Rk 0 =k .G // and G G G0 0 0 e e0 =Z0 . closed k-subgroup schemes. Define G D G=Z and G0 D G (i) For any separable extension field F=k, every F -isomorphism f W GF ' eF ! .G e0 /F , and fe is .G0 /F uniquely lifts to an F -homomorphism feW G 0 0 0 0 an isomorphism. Moreover, if .G0 ; k0 =k/ D .G ; k =k/ then fe restricts eF if f restricts to to the identity on a smooth connected F -subgroup of G the identity on its image in GF . (ii) Let H be a smooth connected k-group equipped with a left action on G. e The lifted action restricts to the This lifts uniquely to a left action on G. e if H acts trivially trivial action on a smooth connected k-subgroup of G on its image in G. Proof. The special case when G 0 and G00 have semisimple fibers is [CGP, Prop. 5.1.7]. In the present additional generality, note that the initial setup is compatible with separable extension of the ground field and in all cases the fiber of ZG 0 over each factor field ki0 of k 0 is contained in the Weil restriction to ki0 of a finite group scheme over a finite extension field of ki0 (see Proposition 8.5.1 for the generalized basic exotic and rank-2 basic exceptional cases in characteristic 2). Hence, the fibers of ZG 0 do not admit a nontrivial map of pointed schemes from

9.1 Generalized standard groups

177

a smooth connected scheme, so we can argue exactly as in the proof of [CGP, Prop. 5.1.7] to reduce (ii) to (i) and furthermore reduce (i) to the special case F D k D ks . The problem is to uniquely fill in a commutative diagram e G

‹

/G e0





G

' f



0

/ G0

where  and 0 are the respective quotient maps modulo Z and Z0 , and to show e!G e0 across the top of the diagram is an isomorphism. that the map G The central k-subgroups Z and Z0 are k-tame since ZG e and ZG e0 are k-tame (Proposition 9.1.6). Thus, via the isomorphism f , the perfect smooth connected e and G e0 are k-tame central extensions of G. Each has maximal k-groups G geometric reductive quotient that is simply connected, and this property uniquely characterizes (up to unique isomorphism) the universal smooth k-tame central e!G e0 over G and it is extension. Hence, there is a unique k-homomorphism G an isomorphism.  As an application of Lemma 9.1.11, we now prove a rigidity property of generalized standard presentations. This involves the notion of “pseudo-isogeny” that is defined in §A.1: Proposition 9.1.12. Let G be a non-commutative generalized standard k-group arising from a 4-tuple .G 0 ; k 0 =k; T 0 ; C / and the factorization diagram .9:1:7:1/. (i) The natural map j W D.Rk 0 =k .G 0 // ! G arising from the generalized standard presentation is a pseudo-isogeny onto D.G/ with central kernel. (ii) The triple .G 0 ; k 0 =k; j / is unique up to unique isomorphism in the sense that if .G00 ; k00 =k; T00 ; C0 / is part of a generalized standard presentation of G then there is a unique isomorphism of primitive pairs .G 0 ; k 0 =k/ ' .G00 ; k00 =k/ compatible with the quotient maps j and j0 onto D.G/. Proof. Using notation as in Definition 9.1.7, the central quotient presentation .G oC /=C ' G with commutative C and perfect G implies that the natural map j W G ! G has image D.G/. Moreover, the kernel of j W G  D.G/ is central because C is central in G o C . The uniqueness up to unique isomorphism in (ii) is an immediate consequence of Lemma 9.1.11 applied to D.G/ as a central quotient of G WD D.Rk 0 =k .G 0 //.

178

Generalization of the standard construction

To show that j is a pseudo-isogeny onto the image D.G/, the problem is to prove that ker j contains no nontrivial tori. Using the notation from (9.1.7.1), this kernel is identified with ker.C ! C /  ker.C ! ZG ;C /: Since C ! ZG ;C is a pseudo-isogeny by Proposition 6.1.7, we are done.



By Proposition 9.1.12, if G is a generalized standard k-group then every generalized standard presentation of G encodes the same underlying triple .G 0 ; k 0 =k; j / in a manner that is functorial with respect to isomorphisms in G. The following result goes further and makes precise the sense in which a choice of generalized standard presentation of a given G is “the same” as a choice of maximal k-torus of G, or a choice of maximal k 0 -torus of G 0 : Proposition 9.1.13. Let G be a non-commutative generalized standard k-group arising from the 4-tuple .G 0 ; k 0 =k; T 0 ; C / and factorization diagram .9:1:7:1/. Let G D D.Rk 0 =k .G 0 // and C 0 D ZG 0 .T 0 /. (i) The maximal k-torus T in Rk 0 =k .T 0 / is a maximal k-torus in G , and if T is the maximal k-torus of G that is the almost direct product of j.T / and the maximal central k-torus Z of G then C D ZG .T / D ZG .T /. (ii) There is a bijection between the set of maximal k 0 -tori S 0  G 0 and the set of maximal k-tori in G via S 0 7! j.S /
Z where S is the maximal k-torus in Rk 0 =k .S 0 / .necessarily also a maximal k-torus in G /. (iii) Let S be a maximal k-torus in G, S 0 the corresponding maximal k 0 -torus in G 0 , and S the maximal k-torus in Rk 0 =k .S 0 /. The conjugation action of ZG .S / on D.G/ D j.G / uniquely lifts to an action on G that restricts to the identity on S .and hence on ZG .S //. Using the resulting factorization diagram ZG .S / ! ZG .S/ ! ZG ;ZG .S / D Rk 0 =k .ZG 0 ;S 0 / .equality by Proposition 6:1:7/, the natural k-homomorphism .G o ZG .S//=ZG .S / ! G is an isomorphism. Proof. We saw in the proof of Lemma 9.1.11 that the fibers of ZG 0 do not admit a nontrivial map of pointed schemes from a smooth connected scheme, so by using Proposition 6.1.7 as a replacement for [CGP, Lemma 10.1.8] and using

9.2 Structure theorem

179

Lemma 9.1.11 as a replacement for [CGP, Prop. 10.1.12] we can argue exactly as in the special case when G 0 has semisimple fibers (see [CGP, Prop. 4.1.4]).  Corollary 9.1.14. A pseudo-reductive k-group G is generalized standard if and only if D.G/ is generalized standard. Proof. The case of commutative G is trivial, so we may assume G is noncommutative. The analogue for the standardness property is [CGP, Prop. 5.2.1], and by using Lemma 9.1.11(ii) we can easily adapt the same proof to the generalized standard case since (by Proposition 9.1.13) we may use whatever maximal k-torus we wish (in G or in D.G/) to underlie a choice of generalized standard presentation when one exists (for the non-commutative G or D.G/). 

9.2 Structure theorem Over any imperfect field k (with any positive characteristic) there are standard absolutely pseudo-simple k-groups that are not of minimal type [CGP, Ex. 5.3.7]. There is a weaker condition that is satisfied by all pseudo-reductive k-groups G if char.k/ ¤ 2 or if char.k/ D 2 and Œk W k 2  6 2: G is locally of minimal type (see Remark 4.3.2). This condition is also satisfied if char.k/ D 2 and Œk W k 2  6 8 provided that Gks has a reduced root system (see Proposition B.3.1). Theorem 9.2.1. Let G be a pseudo-reductive group over a field k. Then G is generalized standard if and only if it is locally of minimal type. Proof. The implication “)” is easy, as follows. We may assume k D ks . By Corollary 9.1.14, we may replace G with D.G/ so that G is perfect. Thus, G is a central quotient of D.Rk 0 =k .G 0 // for a primitive pair .G 0 ; k 0 =k/. Identifying the root systems of G and G 0 relative to compatible maximal tori, and working with Gc and Gc0 in place of G and G 0 respectively for each c in the root system, reduces us to the case where G has rank 1. Now k 0 is a field, G 0 has rank 1, and G 0 is of minimal type over k 0 with maximal geometric reductive quotient SL2 . By Proposition 5.3.5(ii), the rank-1 pseudo-simple k-group D.Rk 0 =k .G 0 // (that is a central extension of G) is of minimal type over k. Now we assume G is locally of minimal type and prove that it is generalized standard. By Corollary 9.1.14, we may and do assume G is perfect. The e of G is of minimal type (Propouniversal smooth k-tame central extension G sition 5.3.3). It suffices to prove that there exists a primitive pair .G 0 ; k 0 =k/ e ' D.Rk 0 =k .G 0 //, as then it would follow from Remark 9.1.9 that such that G e to arrange that G is also generalized standard. Thus, we may replace G with G

180

Generalization of the standard construction

G is of minimal type with G ss simply connected, and we seek a primitive pair k .G 0 ; k 0 =k/ such that there is a k-isomorphism f W G ' D.Rk 0 =k .G 0 //. By Proposition 8.4.3, .G 0 ; k 0 =k; f / is unique up to unique isomorphism if it exists. Galois descent thereby reduces the proof of existence of .G 0 ; k 0 =k/ to the case k D ks . By [CGP, Prop. 3.1.8], there are finitely many normal pseudosimple k-subgroups Gi of G and they pairwise commute and generate G, with multiplication defining a central quotient map Y W Gi ! G: The groups .Gi /ss are the simple normal subgroups of the connected semisimple k group G ss that is simply connected (see [CGP, Prop. 3.1.6, Prop. A.4.8]), so each k .Gi /ss is simply connected. Normality of Gi in G implies that Gi is of minimal k type (Lemma 2.3.10). If we can handle the pseudo-simple case (applied to Gi for Q every i ) then each ZGi is k-tame (by Proposition 9.1.6), so ZGi is also k-tame. Hence,  would be a k-tame central extension of the perfect smooth connected affine k-group G that is its own universal smooth k-tame central extension, so  is an isomorphism and thus we would obtain the desired description of G. We may therefore assume that G is pseudo-simple. Let T be a maximal k-torus in G. We may assume that G is not standard (as otherwise we are done), so by Theorem 3.4.2 the field k is imperfect with characteristic p 2 f2; 3g and the root system ˆ D ˆ.G; T / is G2 if p D 3 and is Bn (n > 1), Cn (n > 1), F4 , or BCn (n > 1) if p D 2. We have arranged that G is of minimal type, so if ˆ D BCn (n > 1) then G D D.Rk 0 =k .G 0 // for a primitive pair .G 0 ; k 0 =k/ by Proposition 9.1.3. Thus, G is generalized standard in the BCn -cases, so we may assume ˆ is reduced. Now as G is a non-standard absolutely pseudo-simple group of minimal type with a reduced root system over k D ks , and G ss is simply connected, Theorem 8.4.5 implies that there is a k primitive pair .G 0 ; k 0 =k/ (with k 0 a field) such that G ' D.Rk 0 =k .G 0 //. 

Appendix A Pseudo-isogenies

A.1 Main result Observe that over a field k, a surjective k-homomorphism f W G ! G 0 between connected semisimple k-groups is an isogeny if and only if it restricts to an isogeny between maximal k-tori. More generally: Definition A.1.1. Let G and G 0 be perfect smooth connected affine k-groups. A k-homomorphism f W G ! G 0 is a pseudo-isogeny if it is surjective and restricts to an isogeny between maximal k-tori. A natural example of a pseudo-isogeny that is not an isogeny is Rk 0 =k .SLp / ! Rk 0 =k .SLp /=Rk 0 =k .p / D D.Rk 0 =k .PGLp // for a nontrivial purely inseparable finite extension k 0 =k in characteristic p > 0. Our formulation of a “Pseudo-Isogeny Theorem” will use the following convenient shorthand: if G is a pseudo-reductive k-group and S is a split maximal k-torus of G then for a 2 ˆ.G; S/ we define U.a/ to be the root group Ua when a is not divisible and to be the root group Ua=2 when a is divisible. (See [CGP, Def. 2.3.4, 2.3.12–2.3.14] for a broader context.) As usual, Ga denotes hUa ; U a i (see [CGP, Prop. 3.4.1]). For the commutative Cartan k-subgroup C WD ZG .S / and a 2 ˆ, the intersection Ca WD C \ Ga coincides with the Cartan k-subgroup ZGa .Ga \ S/  Ga . If moreover G is pseudo-semisimple then Q the product map a2 Ca ! C is surjective by Proposition 4.3.3(i). Theorem A.1.2 (Pseudo-Isogeny Theorem). Let k be a field, and let .G; S/ and .G 0 ; S 0 / be pseudo-reductive k-groups equipped with split maximal k-tori S and S 0 , with G pseudo-semisimple. Let C D ZG .S/, C 0 D ZG 0 .S 0 /, and let ˆ D ˆ.G; S / and ˆ0 D ˆ.G 0 ; S 0 / be the respective root systems. We fix a basis  of ˆ. For a 2 ˆ let Ca WD C \ Ga be the Cartan k-subgroup ZGa .Ga \ S/ of Ga . 0 0 Let a 7! a0 be a map  ! ˆ0 such that for distinct a; b 2 , U.a 0 / and U. b 0 / 0 commute. For each a 2  let fa W Ga ! G be a k-homomorphism carrying

182

Pseudo-isogenies

0 _ a_ .GL1 / into S 0 such that fa .U˙a / D U.˙a 0 / and fa is b .GL1 /-equivariant with respect to the inclusion b _ .GL1 /  S and the map fb W b _ .GL1 / ! S 0 for all b 2  fag.

(i) For all a 2 , fa .Ca /  C 0 . Q Q (ii) If the k-homomorphism a2 Ca ! C 0 defined by .ca / 7! fa .ca / factors through a k-homomorphism fC W C ! C 0 then there is a unique k-homomorphism f W G ! G 0 extending fa for every a 2 . 0 (iii) Assume the existence of fC . If G 0 is generated by the k-subgroups U.˙a 0/ for a 2  then f is surjective. If the k-subgroup scheme ker fa is central in Ga for all a 2  then ker f is central in G. If G 0 is pseudo-semisimple and the image of  in ˆ0 is a basis then G 0 is 0 generated by the k-subgroups U.˙a 0 / for a 2  due to [CGP, Lemma 3.1.5] (since reflections associated to elements of a basis of ˆ0 generate W .G 0 ; S 0 /). Our proof of the Pseudo-Isogeny Theorem is completely different from the traditional proof of the Isogeny Theorem for connected semisimple groups (see [Spr, Thm. 9.6.5]). It rests on a pseudo-reductive variant of an idea of Steinberg for proving the Isomorphism Theorem in the connected semisimple case by constructing homomorphisms via graphs built as connected semisimple subgroups of a direct product of two connected semisimple groups.

A.2 Proof of Pseudo-Isogeny Theorem 0 0 The hypothesis that U.a 0 / and U. b 0 / commute for distinct a; b 2  implies that b 0 cannot be a positive rational multiple of a0 , so in particular a 7! a0 is an 0 0 0 0 injective map of  into ˆ0 . Let G.a 0 / WD hU.a0 / ; U. a0 / i, so fa .Ga / D G.a0 / . By the functoriality of dynamic constructions with 1-parameter subgroups (e.g., UG ./) given in [CGP, §2.1], fa carries Ua D UGa .a_ / into UG 0 .fa ıa_ /, so on the nonzero Lie.fa .Ua // all weights for the GL1 -action through a0 WD fa ı a_ are positive. But all weights for the S 0 -action on Lie.U.a0 / / are positive rational multiples of a0 , so ha0 ; a0 i > 0 for all a 2 . 0 The codimension-1 subtorus Sa0 0 D .ker a0 /0red in S 0 centralizes G.a 0 / , and 0 the cocharacter a0 W GL1 ! S has image that is an isogeny complement to 0 Sa0 0 since ha0 ; a0 i ¤ 0. The image fa .Ca /  G.a 0 / commutes with the subtori 0 0 a0 .GL1 / and Sa0 that generate S , so fa .Ca /  C 0 . This establishes (i), and for the rest of the argument we shall assume the existence of fC as in (ii). Since fa .Ga /  D.G 0 / and (by [CGP, Lemma 1.2.5(ii),(iii)]) C 0 \D.G 0 / D ZD.G 0 / .S 0 \D.G 0 // with S 0 \D.G 0 / a split maximal k-torus in D.G 0 /, we may

A.2 Proof of Pseudo-Isogeny Theorem

183

replace G 0 with D.G 0 / so that G 0 is also pseudo-semisimple (in addition to G being pseudo-semisimple, by hypothesis). For a 2  the pseudo-split pseudoreductive k-subgroup Ha WD Ga
C has the presentation .Ga o C /=Ca ' Ha ; and by construction fC agrees with fa on Ca , so there exists a (visibly unique) k-homomorphism a W Ha ! G 0 that extends fa and fC provided that fa W Ga ! G 0 is C -equivariant when C acts on G 0 through fC . Q To establish equivariance of fa with respect to the quotient C of b2 Cb , it is equivalent to prove Cb -equivariance for all b 2 . The case b D a is a tautology since fa is a k-homomorphism, so we may assume b 2  fag. Since 0 0 fa .Ga / D G.a 0 / , to construct a it suffices to show that Ga  G.a0 / is Cb equivariant. By hypothesis, equivariance holds for the action of the maximal k-torus Tb WD b _ .GL1 / in Cb . To prove the Cb -equivariance we may assume k D ks , so then it suffices to show that for each z 2 Cb .k/ the map fa W Ga  0 1 f .zxz 1 /f .z/. This is a comparison of two G.a a 0 / agrees with x 7! fb .z/ b 0 surjective homomorphisms Ga ⇒ G.a0 / between pseudo-reductive groups, so by [CGP, Prop. 1.2.2] it is equivalent to check equality for the induced maps between maximal geometric reductive quotients. By 3.2.2, these quotients over k respectively coincide with the subgroups of of the

G ss and G 0 ss generated by the evident opposite root groups. Thus, in view k k Tb -equivariance, we may conclude by noting that the composite maps ss

Ck ! G ss ; Ck0 ! G 0 k k

have images respectively coinciding with the images of the maximal tori Sk and S 0 . This completes the construction of the k-homomorphism a W Ha ! G 0 k extending fa and fC for all a 2 . Since C 0 is commutative, fC must carry S into the unique maximal ktorus S 0 in C 0 . Let fS W S ! S 0 be this restricted map. This map carries the maximal central torus Sa WD .ker a/0red of Ha into the maximal central torus 0 0 Sa0 0 D .ker a0 /0red of G.a 0 /
C since fS extends to the map a whose image con0 tains the group G.a0 / D fa .Ga /. Thus, a0 ı fS kills Sa , so a0 ı fS is a rational multiple of a. But ha0 ı fS ; a_ i D ha0 ; fS ı a_ i D ha0 ; fa ı a_ i D ha0 ; a0 i > 0; so a0 ı fS is a positive rational multiple of a.

184

Pseudo-isogenies

Let G D G  G 0 , and let S  G be the graph of fS , so S is a k-torus in G such that the projection map pr1 W S ! S is an isomorphism. This identifies  with a linearly independent subset of X.S /. The composite map X.S 0 /

X.pr2 /

! X.S / ' X.S/

is X.fS /, so it carries a0 to a positive rational multiple of a. Let Ha  G be the graph of a , and let C  Ha be the graph of fC . Since pr1 W Ha ! Ha is an isomorphism carrying S onto S, Ha is pseudoreductive and its Cartan k-subgroup ZHa .S / is identified with ZHa .S/ D C . More precisely, ZHa .S / D C inside G . The set of nontrivial S -weights on Ha coincides with the set of nontrivial S -weights on Ha via the identification pr1 W S ' S , so this set of weights contains ˙a and is contained inside Z
a. The a-root group Ua of the pair .Ha ; S / is contained inside the direct prod0 uct Ua  fa .Ua /, and by hypothesis fa .U˙a / D U.˙a 0 / . Thus, for any distinct a; b 2 , Ua commutes with U b because Ua commutes with U b and also 0 0 U.a 0 / commutes with U. b 0 / (the latter by hypothesis). We conclude via [CGP, Thm. C.2.29] that the k-groups fHa ga2 generate a pseudo-split pseudo-reductive k-subgroup H of G  G 0 such that: S is a maximal k-split torus of H , ZH .S / D C ,  is a basis of ˆ.H ; S /, and for a 2  the ˙a-root groups for the pair .H ; S / coincide with the ˙a-root groups U˙a for the pair .Ha ; S /; these latter root groups are the graphs of fa on U˙a , and the coroot associated to a is the 1-parameter subgroup .a_ ; fa ı a_ / W GL1 ! S  S  S 0 : Pseudo-semisimplicity of G implies that it is generated by the k-subgroups fGa ga2 , so the projection H ! G is surjective. Hence, D.H / ! G is also surjective, so the map C \ D.H / ! C between Cartan k-subgroups is surjective. However, pr1 W C ! C is an isomorphism by design, so C D C \ D.H /  D.H /. Since H D C
D.H /, it follows that H D D.H /; i.e., H is pseudosemisimple. We claim that pr1 W H ! G is an isomorphism. Since pr1 restricts to isomorphisms C ' C and Ha ' Ha for all a 2 , pr1 is surjective and the normal closed k-subgroup scheme ker pr1  H has trivial intersection with the root group of the pair .H ; S / for every root in . Every NH .S /.k/-orbit of a non-divisible root in ˆ.H ; S / meets  because NH .S /.k/ maps onto the Weyl group of ˆ.H ; S /, so .ker pr1 / \ Ua is trivial for all non-divisible a 2 ˆ.H ; S /. Passing to Lie algebras, the S -action on Lie.ker pr1 / has no

A.3 Relation with the semisimple case

185

nontrivial weight and hence is the trivial action. Thus, by [CGP, Cor. A.8.11], S centralizes .ker pr1 /0 . But Hk is generated by tori since H is perfect, so .ker pr1 /0 is central. The normal ker pr1 is therefore also central since H is perfect (see [CGP, Lemma 5.3.2]), so ker pr1  C . But ker .pr1 jC / D 1, so pr1 is an isomorphism as desired. Composing the inverse of this isomorphism with pr2 W H ! G 0 defines a k-homomorphism f W G ! G 0 . It is clear that f extends fC and fa for all a 2 , so (ii) is proved. 0 The first part of (iii) is obvious since U.˙a 0 / D f .U˙a / for all a 2 . Now assume ker fa is central in Ga for all a 2 ; we aim to show that ker f is central in G. As with the study of ker pr1 in the proof of (ii), since G is perfect it suffices to show that Ua \ ker f is trivial for all a 2 . For such a we know that f jGa D fa , and the centrality hypothesis for ker fa in Ga implies that ker fa is contained in the Cartan k-subgroup ZGa .S \ Ga /. Thus, Ua \ ker fa is trivial as a k-subgroup scheme of Ga . The Pseudo-Isogeny Theorem is proved.

A.3 Relation with the semisimple case The Pseudo-Isogeny Theorem implies the Isogeny Theorem for split connected semisimple groups G and G 0 as stated in [CGP, Thm. A.4.10] for a p-morphism .f; b; q/ W R.G 0 ; S 0 / ! R.G; S/ between root data (where p > 1 is the characteristic exponent of k and q.a/ D p e.a/ for some e.a/ > 0 for each a 2 ˆ.G; S/); note the “contravariant” convention in the source and target for this terminology. In [CGP] we gave a proof of the Isogeny Theorem that reduced to the case of an algebraically closed ground field, which in turn is proved in [Spr, Thm. 9.6.5]. Now we sketch a proof of the Isogeny Theorem directly over k via the Pseudo-Isogeny Theorem without reducing to work over k. For each a 2  we may identify Ga with SL2 or PGL2 by carrying a_ to t 7! diag.t; 1=t / or t 7! diag.t; 1/ respectively and carrying the a-root group onto the upper triangular unipotent subgroup. For the split connected semisimple target group G 0 and a0 2 0 we likewise identify Ga0 0 with SL2 or PGL2 . Consider roots a 2  and a0 2 0 that correspond to each other under the given p-morphism between the root datum. The given p-morphism restricts to a p-morphism from the root datum of .Ga0 0 ; a0 _ .GL1 // to that of .Ga ; a_ .GL1 //, but there is no p-morphism from the root datum of SL2 to that of PGL2 except if p D 2, in which case necessary q.a/ ¤ 1. Thus, if p ¤ 2 then the case Ga ' PGL2 and Ga0 0 ' SL2 cannot occur. If p D 1 then we define fa W Ga ! Ga0 0 to be the identity endomorphism of SL2 or PGL2 or the central quotient map SL2 ! PGL2 , and if p > 2 then we define fa to be the composition of that map with the q.a/-Frobenius endo-

186

Pseudo-isogenies

morphism of the source. If p D 2 then we use the same procedure as if p > 2 provided that we are not in the case Ga ' PGL2 and Ga0 0 ' SL2 . In this latter case we have seen that necessarily q.a/ ¤ 1, so we define fa to be the classical (non-central) unipotent isogeny PGL2 ! SL2 if q.a/ D 2 and its composition with the q.a/=2-Frobenius endomorphism of PGL2 if q.a/ > 2. Note that in all cases with p > 1, the effect of fa between root groups is a k  -multiple of the q.a/-Frobenius endomorphism of Ga . The GL1 -equivariance requirement on fa with respect to b _ for distinct a; b 2  is deduced from the purely formal [Spr, Lemma 9.6.4(i)]. Moreover, the hypothesis concerning existence fC in (ii) and (iii) is immediate via the given p-morphism of root datum (we can take fC to correspond to the given map of character lattices). Hence, the Pseudo-Isogeny Theorem provides the desired isogeny, as well as the desired combinatorial characterization of when it is central, and it reduces uniqueness (up to the .S 0 =ZG 0 /.k/-action on G 0 ) to the rank-1 case. Consideration of open cells for SL2 and PGL2 settles uniqueness for rank 1.

Appendix B Clifford constructions

Throughout this monograph we have seen the utility of passing to pseudoreductive k-groups of minimal type when proving general theorems. However, over every imperfect field there exist standard absolutely pseudo-simple groups that are not of minimal type (see [CGP, Ex. 5.3.7]). The concept “locally of minimal type” introduced in Definition 4.3.1 is more robust: in the pseudosemisimple case it is equivalent to the universal smooth k-tame central extension being of minimal type (Proposition 5.3.3), and in general it characterizes the output of the “generalized standard” construction (Theorem 9.2.1). Hence, it is natural to seek examples of absolutely pseudo-simple k-groups that are not locally of minimal type. Let k be a field of characteristic 2. If Œk W k 2  > 16 then an explicit construction in 4.2.2 produces non-standard pseudo-split absolutely pseudo-simple k-groups G with root system A1 such that G is not locally of minimal type. The construction rests on a commutative Clifford algebra: for a subfield K  k 1=2 of degree 16 over k and 6-dimensional k-subspace V  K as in (4.2.2), such G are built by using the Clifford algebra C associated to the squaring map q W V ! k viewed as a quadratic form over k. (The k-algebra C is a non-reduced finite local commutative k-algebra with residue field K.) Our aim in this appendix is to show that constructions based on commutative Clifford algebras are not limited to the rank-1 case: whenever Œk W k 2  > 16 we use such triples .K=k; V; C / to construct pseudo-split absolutely pseudo-simple k-groups not locally of minimal type, with root system Bn or Cn for any n > 2. These k-groups are built as smooth connected central extensions of pseudo-split generalized basic exotic pseudo-reductive k-groups that are not basic exotic. (The necessity of avoiding the basic exotic case is due to [CGP, Prop. 8.1.2].) In §B.3 we show the bound Œk W k 2  > 16 is optimal: if k is imperfect of characteristic 2 and Œk W k 2  < 16 then every pseudo-reductive k-group with a reduced root system over ks is locally of minimal type. The reducedness hypothesis is essential since if Œk W k 2  > 2 then another technique in §B.4 (not involving Clifford algebras) yields pseudo-split absolutely pseudo-simple kgroups with root system BCn (n > 1) that are not locally of minimal type.

188

Clifford constructions

B.1 Type B Let G be a split connected semisimple k-group that is absolutely simple and simply connected with root system Bn for n > 2, and let T  G be a split maximal k-torus. Let  be a basis of ˆ D ˆ.G ; T /, so _ is a Z-basis of X .T /. For all c 2 ˆ, Tc WD c _ .GL1 / is a maximal k-torus in Gc ' SL2 . Let .K=k; V; C / be as in 4.2.2. Clearly T is a maximal k-torus in RC =k .GC /. For the unique short root b 2 , choose a k-isomorphism Gb ' SL2 carrying the maximal k-torus Tb onto the diagonal k-torus. Define the pseudo-semisimple k-group H WD HV;C =k  RC =k .SL2 / D RC =k ..Gb /C / as in 4.2.2. The commutative k-group C WD ZH .Tb / 

Y

Ta  RC =k .TC /

(B.1.0)

a2 fbg

is pseudo-reductive since H is pseudo-reductive. By definition C normalizes Q H , and H
C D H o a2 fbg Ta , so H
C is pseudo-reductive and contains T with ZH
C .T / D C . We conclude via [CGP, Thm. C.2.29] that the smooth connected k-subgroup G  RC =k .GC / generated by H and fGa ga2 fbg is pseudo-reductive with maximal k-torus T and satisfies the following additional properties: ZG .T / D C , ˆ.G; T / has basis , and for c 2  the ˙c-root groups of .G; T / coincide with those of .H; Tb / if c D b and coincide with those of .Gc ; Tc / if c ¤ b. In particular, by pseudo-semisimplicity of H we see that G is generated by its T root groups, so G is pseudo-semisimple. Moreover, for all c 2  the coroot c _ associated to .G; T / is the same as for .G ; T /, so ˆ.G; T / D ˆ inside X.T /. The minimal field of definition over k for the geometric unipotent radical of Gc is k if c is long and is K if c is short, as we can compute these extensions of k using c 2 . Hence, by Proposition 3.2.5, the minimal field of definition over k for Ru .Gk /  Gk is K. Lemma B.1.1. The pseudo-reductive k-group G is not locally of minimal type. Proof. By construction, Gb is k-isomorphic to HV;C =k , so .Gb /ss ' SL2 . Thus, k according to Proposition 4.1.6, Gb does not admit a central extension that is absolutely pseudo-simple of minimal type.  The natural map RC =k .GC /  RK=k .GK / carries G into a type-B generalized basic exotic k-group G of minimal type as in Proposition 8.2.5 using V  K. Let’s show that for the k-group G that is not locally of minimal type,

B.2 Type C

189

the natural map f W G ! G is surjective with central kernel isomorphic to ˛2 if v D e1 e2 C e3 e4 and isomorphic to Z=.2/ if v D e1 e2 e3 e4 (with v as in 4.2.2). By construction, f restricts to the identity map between the natural copies of T as a maximal k-torus of G and G, so it carries Gc into Gc for all c 2 ˆ. The resulting map fc W Gc ! Gc recovers the identity map on Gc for c 2 > , and since the composite map V ,! C  K is the canonical inclusion it follows that fb is identified with the natural map HV;C =k ! HV;K=k that (as we reviewed in 4.2.2) is a central quotient map having kernel ˛2 if v D e1 e2 C e3 e4 and kernel Z=.2/ if v D e1 e2 e3 e4 . In view of the structure of C D ZG .T / as a direct product, the Pseudo-Isogeny Theorem in §A.1 implies that f is a central quotient map with central kernel equal to ker fb .

B.2 Type C Now let G be a split connected semisimple k-group that is absolutely simple and simply connected with root system Cn (n > 2), and let T be a split maximal k-torus. Let  W G ! G be the very special isogeny. The image T D .T / is a split maximal k-torus in G . Let .K=k; V; C / be as above. Applying the preceding Bn -construction to .G ; T / yields a pseudo-split and absolutely pseudo-simple k-subgroup G  RC =k .G C / with maximal k-torus T  RC =k .T C / such that ˆ WD ˆ.G; T / is identified with ˆ.G ; T / D Bn and if c 2 ˆ< then G c ' HV;C =k (so G is not locally of minimal type). We showed that the image of G under the quotient map RC =k .G C /  RK=k .G K / is a generalized basic exotic k-group G, and that the quotient map f W G  G has central kernel isomorphic to ˛2 or Z=.2/ (depending on V ). We are not going to search for the desired group with root system Cn by looking inside RC =k .GC /, but rather we will build it from G using a fiber product involving Weil restrictions from K rather than from C : Proposition B.2.1. The k-group G defined by the fiber square G

/ RK=k .GK /





G

RK=k .K /

/ RK=k .G K /

is pseudo-split and absolutely pseudo-simple with root system Cn . It is not locally of minimal type.

190

Clifford constructions

Proof. Choose a basis  of the root system ˆ WD ˆ.G ; T / D ˆ.RK=k .GK /; T / D Cn ; and let a be the unique long root in . Since K 2  k, it follows from [CGP, Prop. 7.1.5(1)] that the very special isogeny  W .G ; T / ! .G ; T / carries Ga isomorphically onto G a , so RK=k .K / carries RK=k ..Ga /K / isomorphically onto RK=k ..G a /K /. Thus, the k-subgroup Ga ' HV;K=k of RK=k ..G a /K / is identified with its inverse image in RK=k ..Ga /K / that we define to be Ga . By (the proof of) Proposition 8.2.5, inside RK=k .GK / the k-subgroup Ga and the ksubgroups RK=k ..Gc /K / for c 2 ˆ< generate a type-C generalized basic exotic k-subgroup G  RK=k .GK / containing T . By [CGP, Prop. 7.1.5(2)], RK=k .K /.G/ D G. The kernel ker RK=k .K / D RK=k .ker K / is directly spanned by closed k-subgroup schemes of the form RK=k .2 / and RK=k .˛2 / inside the k-groups RK=k ..Gc /K / for c 2 ˆ< due to the description of Lie.ker K / given in [CGP, Lemma 7.1.2], so ker RK=k .K / is connected (as K 2  k) and is contained inside G. Thus, G D RK=k .K / 1 .G/ and G D G G G, so G ! G is faithfully flat with central kernel ˛2 or Z=.2/ (depending on V ) and G ! G is faithfully flat with connected kernel RK=k .ker K /. The k-group G therefore inherits connectedness from G. An inspection of compatible open cells for the pairs .G; T /; .G; T /; .G; T / shows that the fiber product G is smooth, so G inherits pseudo-reductivity from its isogenous central quotient G. By centrality of the finite kernel of G  G, we see that G is absolutely pseudo-simple with ˆ.G; T / equal to ˆ.G; T / D ˆ D Cn . For any c 2 ˆ> and the corresponding c 2 ˆ< ,  W Gc ! G c is an isomorphism by [CGP, Prop. 7.1.5(1)]. Thus, RK=k .K / carries Gc isomorphically onto G c ' HV;C =k , so G is not locally of minimal type (via the same reasoning as we used for the type-B analogue). 

B.3 Cases with Œk W k 2  6 8 The following result was brought to our attention by Gabber. Proposition B.3.1. Let k be imperfect of characteristic 2 such that Œk W k 2  6 8. If G is a pseudo-reductive k-group whose root system over ks is reduced then G is locally of minimal type. The reducedness hypothesis is necessary; see Examples B.4.1 and B.4.3.

B.3 Cases with Œk W k 2  6 8

191

Proof. We may assume k D ks and G is absolutely pseudo-simple of rank 1 with e so that G ss D root system A1 . By Proposition 5.3.3 we may replace G with G k SL2 . Let K=k be the minimal field of definition for the geometric unipotent radical of G, so the canonical minimal type central quotient G=CG is of the form HV;K=k for a nonzero kK 2 -subspace V  K containing 1 and satisfying kŒV  D K. It suffices to show that no such HV;K=k has a nontrivial central extension by an affine finite type k-group scheme Z with no nontrivial smooth connected k-subgroup (e.g., Z D CG ). We shall use the splitting criterion in [CGP, Prop. 5.1.3]: for the diagonal k  , it suffices to express VK=k rationally torus D  HV;K=k and its centralizer VK=k ˙ in terms of the D-root groups UV of HV;K=k . That is, we seek an ordered  / U ˙ such that the product finite sequence of rational maps hj W VK=k V Q  / hj W VK=k HV;K=k coincides (as a rational map) with the canonical

inclusion. Let F D f 2 K j 
V  V g, so kK 2  F and Example 3.3.5 gives that HV;K=k D D.RF =k .HV;K=F //. Note that ŒF W F 2  D Œk W k 2  (see Remark 8.3.1 for a general such degree identity in any positive characteristic). If dimF .V / 6 2 then V D F ŒV  D kŒV  D K and G D RK=k .SL2 /, in which case hj ’s are provided by the classical formula diag.t; 1=t / D uC .t/u . 1=t/uC .t

1/u .1/uC . 1/

(B.3.1)

(with the standard parameterizations u˙ W Ga ' U ˙ of the standard root groups of SL2 ) expressing a diagonal point of SL2 universally as a finite product of points in the standard root groups. Hence, we may assume dimF .V / > 2, so ŒK W F  > 4. It is a standard fact that if E 0 =E is a finite extension of fields with characteristic p > 0 then ŒE 0 W EE 0 p  6 ŒE W E p , so ŒK W kK 2  6 Œk W k 2  6 8. For the convenience of the reader, we briefly recall the proof. Since 1E=Fp D 1E=E p and 1E 0 =E D 1E 0 =E E 0 p , by the relation between p-bases and differential bases [Mat, Thm. 26.5] it is equivalent to show that dimE 0 1E 0 =E 6 dimE 0 .E 0 ˝E 1E=Fp /: This inequality follows from a formula of Cartier [Mat, Thm. 26.10], which gives that dimE 0 1E 0 =E D dimE 0 ker.E 0 ˝E 1E=Fp ! 1E 0 =Fp /:

192

Clifford constructions

Now we may assume ŒK W F  D 4; 8 with 2 < dimF .V / < ŒK W F . Suppose ŒK W F  D 4, so dimF .V / D 3. Since 1 2 V , clearly V has an F -basis of the form f1; e; e 0 g with ee 0 extending this to an F -basis of K (because F .e/ D F ˚ F e and ŒK W F .e/ D 2). For z; z 0 2 RF Œe=k .Ga / such that z 2 RF Œe=k .GL1 /, the point z Cz 0 e 0 in RK=k .Ga / has the form z.1Cye 0 / for y WD z 0 =z 2 RF Œe=k .Ga /. But 1Cye 0 D ..1=e 0 2 /e 0 Cy/e 0 and the D-root groups of HV;K=k correspond to V D spanF .1; e; e 0 /  K inside the D-root groups of RK=k .SL2 /, so we get the  required rational maps hj via (B.3.1). This also shows that VK=k D RK=k .GL1 / when ŒK W F  D 4. Suppose ŒK W F  D 8, so F D kK 2 . We note that dimkK 2 .V / > 4, for otherwise ŒK W kK 2  D dimkK 2 .kŒV / 6 4. As 1 2 V and kŒV  D K, V must contain a 2-basis fe; e 0 ; e 00 g of K=k. Let V0 be the kK 2 -span of f1; e; e 0 ; e 00 g, so   2 dim.V0 /K=k 6 dim RkK 2 =k ..V0 /K=kK 2 / 6 7ŒkK W k by [CGP, Prop. 9.1.9].   has dimension 7ŒkK 2 W k .so .V0 /K=k D Lemma B.3.2. The k-group .V0 /K=k   RkK 2 =k ..V0 /K=kK 2 // and .V0 /K=k is expressed rationally in terms of the root groups of HV0 ;K=k relative to the diagonal k-torus.

Proof. Consider the map of k-schemes f W RkK 2 =k .GL1 /  RkK 2 =k .Ga /6 ! RK=k .Ga / that carries .c; x0 ; y0 ; y1 ; z0 ; z1 ; z2 / to c.1 C x0 e/.1 C .y0 C y1 e/e 0 /.1 C .z0 C z1 e C z2 e 0 /e 00 /: Let   RkK 2 =k .Ga /6 be the Zariski-dense open locus defined by 1 C x0 e; 1 C .y0 C y1 e/e 0 ; 1 C .z0 C z1 e C z2 e 0 /e 00 2 RK=k .GL1 /;  so f carries U WD RkK 2 =k .GL1 /   into .V0 /K=k because the factors

1 C .y0 C y1 e/e 0 D e 0 .y0 C y1 e C e 0

2


e 0 /;

1 C .z0 C z1 e C z2 e 0 /e 00 D e 00 .z0 C z1 e C z2 e 0 C e 00

2 00

(B.3.2)

e /

lie in e 0 V 0 and e 00 V 0 respectively. Note that dim U D 7ŒkK 2 W k. We claim that f jU is a monomorphism. Granting this, let us see how to  conclude. By monicity applied to dual-number points, f W U ! .V0 /K=k is injective on tangent spaces at k-points. Thus, by smoothness and dimension  considerations, dim.V0 /K=k D 7ŒkK 2 W k and f is étale on U . But an étale

B.3 Cases with Œk W k 2  6 8

193

monomorphism is an open immersion [EGA, IV4 , 17.9.1], so f jU is an open  immersion and hence (via (B.3.1) and (B.3.2)) .V0 /K=k is expressed rationally in terms of root groups of HV0 ;K=k relative to the diagonal k-torus. It therefore remains to prove monicity on U . Note that K D kK 2 Œe ˚ kK 2 Œee 0 ˚ kK 2 Œe; e 0 e 00 . Consider the k-scheme map f 0 W RkK 2 =k .GL1  Ga /  RkK 2 Œe=k .Ga /  RkK 2 Œe;e0 =k .Ga / ! RK=k .Ga / defined by f 0 ..u; x/; y; z/ D u.1Cxe/.1Cye 0 /.1Cze 00 /. Let 0 be the Zariskiopen locus of points .x; y; z/ such that 1 C xe; 1 C ye 0 ; 1 C ze 00 2 RK=k .GL1 /. We can write any point of RK=k .Ga / in the unique form a0 C a1 e C be 0 C ce 00 where a0 ; a1 2 RkK 2 =k .Ga /, b 2 RkK 2 Œe=k .Ga /, and c 2 RkK 2 Œe;e0 =k .Ga /. The combined conditions a0 ; a0 C a1 e; a0 C a1 e C be 0 ; a0 C a1 e C be 0 C ce 00 2 RK=k .GL1 / define a dense open subscheme of RK=k .Ga /, and f 0 carries RkK 2 =k .GL1 /  0 isomorphically onto that open subscheme. Since   0 , it follows that f jU is monic. (This argument adapts in an evident manner to more general towers of field extensions in place of K=kK 2 =k.)  Now we may suppose V ¤ V0 , so dimkK 2 V > 5 and hence we can apply: Lemma B.3.3. Let F be a field of characteristic 2 and K=F an extension of degree 8 with K 2  F . Let V be an F -subspace of K containing F such that dimF .V / > 5. If dimF .V / > 6 then V contains a 6-dimensional F -subspace V 0 such that F is strictly contained in f 2 K j 
V 0  V 0 g, and if dimF .V / D 5 then there is a 2-basis fe; e 0 ; e 00 g of K over F such that V D spanF f1; e; e 0 ; e 00 ; vg where v D ee 0 or v D ee 0 e 00 . If v D ee 0 e 00 then e 00 V is the F -span of f1; "; "0 ; "00 ; ""0 g for the 2-basis " D ee 0 , "0 D e 0 e 00 , "00 D e 00 of K over F . Proof. First assume dimF .V / D 5. Clearly F ŒV  D K, so V certainly contains a 2-basis fe1 ; e2 ; e3 g of K over F . Thus, V has an F -basis of the form f1; e1 ; e2 ; e3 ; vg where v involves some ei ej (i < j ) or e1 e2 e3 . Suppose v does not involve e1 e2 e3 , so we may take it to be v D e1 e2 C ae2 e3 C be1 e3

194

Clifford constructions

for some a; b 2 F that we may assume are not both zero. Hence, we can assume b D 1, so v D e1 .e2 C e3 / C ae2 e3 D .e1 C ae3 /.e2 C e3 / C ae32 : Since e32 2 F , we can use the 2-basis fe1 C ae3 ; e2 C e3 ; e3 g to arrive at the case v D ee 0 . Suppose v involves e1 e2 e3 . By replacing each ej with ej cj for some cj 2 F we can eliminate all ei ej ’s (i < j ) that appear in v, so we arrive at the case v D ee 0 e 00 . It is trivial to check that in this case fee 0 ; e 0 e 00 ; e 00 g is a 2-basis of K over F that presents e 00 V as being in the preceding case. If dimF .V / > 6 then by the preceding considerations we can choose the 2-basis so that V contains ee 0 . Hence, either V contains F .e; e 0 / ˚ F .ee 0 /e 00 or (after change in the 2-basis) V contains F .e; e 0 / ˚ F .e/e 00 . These 6-dimensional F -subspaces V 0 are subspaces over a strictly larger subfield of K than F . 

If dimkK 2 .V / > 6 then Lemma B.3.3 provides a kK 2 -subspace V 0  V of dimension 6 such that F 0 WD f 2 K j 
V 0  V 0 g strictly contains kK 2 , so  ŒK W F 0  6 4. Our earlier arguments imply that V 0K=k D RK=k .GL1 / and that this is expressed rationally in terms of the standard root groups of HV 0 ;K=k  HV;K=k . We may therefore assume dimkK 2 .V / D 5. Since V only matters up to K  -multiple, by Lemma B.3.3 we may replace it with a suitable such multiple so that V is the kK 2 -span of f1; e; e 0 ; e 00 ; v D ee 0 g for some 2-basis fe; e 0 ; e 00 g of K=k (equivalently, of K=kK 2 ). The proof of Lemma B.3.2 provides a dense open 0  RkK 2 =k .Ga /7 and an open immersion f 0 W RkK 2 =k .GL1 /  0 ! RK=k .GL1 /  that lands inside VK=k because f 0 jf1g0 is a product of maps valued in V ,   e 0 V , e 00 V respectively. Thus, VK=k D RK=k .GL1 / and VK=k is rationally expressed in terms of standard root groups of HV;K=k . This completes the proof of Proposition B.3.1. 

Proposition B.3.4. Let k be imperfect of characteristic 2 such that Œk W k 2  6 8. For any generalized exotic pseudo-semisimple k-group G and commutative affine finite type k-group scheme Z that does not contain a nontrivial smooth connected k-subgroup, every central extension of G by Z uniquely splits.

B.3 Cases with Œk W k 2  6 8

195

The interest in this proposition is due to the role of central extensions (1.4.1.2) in [CGP] when char.k/ ¤ 2 or char.k/ D 2 with Œk W k 2  D 2. Proof. We shall apply the splitting criterion in [CGP, Prop. 5.1.3]. It is sufficient to verify this criterion when k D ks (as we now assume). Let T be a (split) maximal k-torus, and  a basis of ˆ.G; T /. In view of the direct product decomposition in Proposition 4.3.3, it suffices to show that for each a 2  and the split maximal k-torus Ta WD T \ Ga in Ga , the Cartan k-subgroup ZGa .Ta / of Ga is expressed rationally in terms of the root groups for .Ga ; Ta /. This property of the minimal type rank-1 pseudo-simple groups Ga is exactly what was established in the proof of Proposition B.3.1 (since Œk W k 2  6 8).  As an application of Proposition B.3.1 and Proposition 3.1.9, we now remove the “minimal type” hypothesis from the maximality property of (2.3.13), subject to some restrictions when char.k/ D 2. This is of interest even in the standard case since over every imperfect field there exist standard absolutely pseudosimple groups that are not of minimal type [CGP, Ex. 5.3.7]. Proposition B.3.5. Let k be an arbitrary field, and if char.k/ D 2 then assume Œk W k 2  6 8. Let K=k be a purely inseparable finite extension and let 0 G 0 be a pseudo-semisimple K-group. Assume the root system of GK is reduced s .automatic when char.k/ ¤ 2/. Via the natural quotient map q W D.RK=k .G 0 //K  G 0 , the induced map pred D.RK=k .G 0 //K  G 0 is an isomorphism. In particular, G 0 is canonically determined by K=k and the k-group D.RK=k .G 0 //. In §B.4 we will see that the reducedness hypothesis on the root system cannot be dropped when char.k/ D 2 and Œk W k 2  > 2. Proof. We claim that for any pseudo-semisimple K-group H 0 such that HK0 s has a reduced root system, if .H 0 ˝K k/ss is simply connected then H 0 is of minimal type. To prove this, by Proposition 4.3.3 it is equivalent to show that H 0 is locally of minimal type. If char.k/ ¤ 2 then by Proposition 3.1.9 (applied over Ks ) every pseudo-reductive K-group is locally of minimal type. The same holds if char.k/ D 2 by Proposition B.3.1 since ŒK W K 2  D Œk W k 2  6 8 (degree equality proved in Remark 8.3.1). Assume .G 0 ˝K k/ss is simply connected, so G 0 is of minimal type. For the pred pseudo-semisimple G 0 WD D.RK=k .G 0 //K and the unique factorization D.RK=k .G 0 //K  G 0  G 0

196

Clifford constructions

of q, we shall prove that G 0  G 0 is an isomorphism. Since G 0  G 0 is a quotient of minimal type, it factors through G 0 =CG 0 ; i.e., G 0 =CG 0 dominates G 0 as pseudoreductive quotients of D.RK=k .G 0 //K of minimal type. The maximality of G 0 as such a quotient follows from Proposition 2.3.13, so G 0 =CG 0 D G 0 . Hence, 0 GK0 s and GK have the same root data, so .G 0 ˝K k/ss is simply connected (as s we are temporarily assuming .G 0 ˝K k/ss is simply connected) and GK0 s has a reduced root system. Thus, G 0 is of minimal type and so G 0 D G 0 as desired. pred

Now consider the general case, and again let G 0 D D.RK=k .G 0 //K . Let f W D.RK=k .G 0 //K ! G 0 be the natural surjective K-homomorphism. (We do not assume that the root system of GK0 s is reduced.) To prove the proposition in general, it suffices to show that f factors uniquely through q. Uniqueness fole0 ! G 0 be the universal smooth Klows from the surjectivity of q. Letting  W G 0 e has the same associated field K=k (see tame central extension, by centrality G Proposition 3.2.6) and Ks induces an isomorphism between root systems and an isomorphism between root groups for corresponding roots over Ks . Hence, e0 has a reduced root system, so the desired result has been settled for G e0 , and G Ks consideration of root groups over Ks shows that D.RK=k .//K is surjective. e0 //K Thus, via f we may view G 0 as a pseudo-reductive quotient of D.RK=k .G 0 e and so (since the original problem is settled for G ) we get a unique factorization e q

'

e0 //K ! G e0 ! G 0 D.RK=k .G of f ı D.RK=k .//K . We claim that ' factors through  and that the resulting map G 0 ! G 0 provides a factorization of f through q. In the commutative diagram e0 G

e 

/ D.RK=k .G e0 //K

e q



/G e0





G0



 / D.RK=k .G 0 //K

q

 / G0

(with canonical inclusionse  and ), the horizontal compositions are the identity maps. Thus, the existence of a factorization of ' through  may be checked after composing it with e q ıe . Since ' ı e q D f ı D.RK=k .//K , the commutativity of the left square does the job. It is clear from the surjectivity of the vertical maps in the diagram that the map G 0 ! G 0 through which ' uniquely factors has composition with q that recovers f . 

B.4 Type BC

197

B.4 Type BC If Œk W k 2  D 2 then every pseudo-reductive k-group is locally of minimal type (see Remark 4.3.2). But when Œk W k 2  > 2, we shall construct pseudo-split pseudo-simple k-groups G with a non-reduced root system such that G is not of minimal type. As G ss is simply connected, by Proposition 4.3.3 such G cannot k be locally of minimal type. The following example in rank 1 was suggested by Gabber. For examples of such G with rank > 1, see Example B.4.3. Example B.4.1. Let k be a field of characteristic 2 such that Œk W k 2  > 2, so we can choose a subfield K  k 1=2 with degree 4 over k. Explicitly, K D k.u1 ; u2 / where u21 D e, u22 D e 0 for fe; e 0 g part of a 2-basis of k=k 2 . Let p p K 0 D k. u1 ; u2 /  K 1=2 , so ŒK 0 W K D 4, and define V  K 0 to be the p p K-span of f1; u1 ; u2 g. Let H D HV;K 0 =K  RK 0 =K .SL2 /. The k-group RK=k .H / has derived group HV;K 0 =k by (3.1.6). For the ksubspace V  K 0 we have K D f 2 K 0 j 
V  V g with ŒK 0 W K D 4 and dimK V D 3, so the arguments for the degree-4 case in the proof of Proposition B.3.1 imply VK0 =k D RK 0 =k .GL1 / with dimension 16. Hence, the inclusion HV;K 0 =k  RK=k .H / inside RK=k .SL2 / is an equality by comparison of Cartan k-subgroups and the associated root groups, so RK=k .H / is pseudo-simple and the minimal field of definition over k for its geometric unipotent radical is khV i D K 0 . For 1 WD diag.1; 1/, let n D sl2 .K/ C V
1 inside Lie.H /  sl2 .K 0 / (so n is the direct sum of sl2 .K/ and any K-plane in V
1 complementary to K
1). This is a p-Lie subalgebra (with p D 2), so by [CGP, Prop. A.7.14, Ex. A.7.16] it is the Lie algebra of a unique K-subgroup scheme N  ker FH=K of the Frobenius kernel of H . Since H is generated by its root groups relative to the diagonal K-torus, by computing the adjoint action of such root groups we see that the K-subspace n  Lie.H / is AdH -stable. Hence, N is normal in H [CGP, Ex. A.7.16], so RK=k .N / is a closed normal k-subgroup scheme of RK=k .H /. Consider the k-group G WD RK=k .H /=RK=k .N /; so Ru .Gk / D Ru;K 0 .GK 0 /k . Clearly dim RK=k .H / D ŒK W k.ŒK 0 W K C 2 dimK .V // D 10ŒK W k; and if we ignore the group scheme structure then N D ˛25 , so RK=k .N / has dimension 5.ŒK W k 1/ since RK=k .˛2 / has dimension ŒK W k 1 (as K 2  k).

198

Clifford constructions

Thus, dim G D 5ŒK W k C 5 D 25. We will show that G is pseudo-semisimple with root system BC1 and that it is not of minimal type. Step 1. Let D be the diagonal maximal k-torus in RK=k .H / D HV;K 0 =k , with image T in G. The k-group ZG .T / is the image of the Cartan k-subgroup RK 0 =k .GL1 / D ZRK=k .H / .D/ of RK=k .H /, so ZG .T / D RK 0 =k .GL1 /=RK=k .N0 / where N0 WD N \ RK 0 =K .GL1 / is the K-subgroup scheme killed by Frobenius for which Lie.N0 / D V  K 0 D Lie.RK 0 =k .GL1 //. Clearly dim RK=k .N0 / D 3.ŒK W k 1/ D 9, so dim ZG .T / D 16 9 D 7. Now we prove ZG .T / is pseudo-reductive. The inclusion N0  RK 0 =K .GL1 /Œ2 D RK 0 =K .2 / yields a map ZG .T /  RK 0 =k .GL1 /=RK 0 =k .2 / D Œ2.RK 0 =k .GL1 //  RK 0 =k .GL1 /; so since RK 0 =k .GL1 / is commutative and pseudo-reductive we see that Ru;k .ZG .T //  RK 0 =k .2 /=RK=k .N0 / D RK=k .RK 0 =K .2 //=RK=k .N0 /: We shall prove this latter quotient has no nontrivial ks -points, so it would follow that Ru;k .ZG .T // is trivial. Since RK 0 =K .2 /=2 D RK 0 =K .GL1 /=GL1 is a K-form of G3a for the fppf topology, its Frobenius kernel is an fppf form of ˛23 and thus is K-isomorphic to ˛23 (because ˛pr has no nontrivial fppf forms over any field of characteristic p, as its automorphism functor is represented by GLr ). The Frobenius kernel M of RK 0 =K .2 / has order 24 since it coincides with the Frobenius kernel of the smooth 4-dimensional RK 0 =K .GL1 /, so M is a commutative extension of ˛23 by 2 . Hence, via the inclusion 2  N0 inside RK 0 =K .2 / we see that M=N0 ' ˛2 . The underlying K-schemes of M and N0 (ignoring their K-group structure) are ˛24 and ˛23 respectively, so RK=k .M / and RK=k .N0 / have respective dimensions 4.ŒK W k 1/ D 12 and 3.ŒK W k 1/ D 9. Hence, the closed immersion RK=k .M /=RK=k .N0 / ,! RK=k .M=N0 / D RK=k .˛2 / is an isomorphism since RK=k .˛2 / is reduced and irreducible of dimension 3 (defined by x 2 C ey 2 C e 0 z 2 C ee 0 w 2 D 0).

B.4 Type BC

199

The k-group RK 0 =k .2 / is defined by the system of four equations xj2 C eyj2 C e 0 zj2 C ee 0 wj2 D cj for 0 6 j 6 3 with c0 D 1 and c1 ; c2 ; c3 D 0, so it is irreducible of dimension 12. Thus, the inclusion RK=k .˛2 / D RK=k .M /=RK=k .N0 / ,! RK 0 =k .2 /=RK=k .N0 / is between irreducible k-groups of dimension 3, so it is an equality of underlying reduced schemes and hence the target has no nontrivial ks -points (since RK=k .˛2 /.ks / D ˛2 .Ks / D 0). This concludes the proof that ZG .T / is pseudoreductive. Step 2. To prove that Ru;k .G/ is trivial, we first note that any T -weight occurring on Lie.Ru;k .G// must be nontrivial since ZG .T / is commutative and pseudo-reductive. Hence, it suffices to rule out the occurrence of nontrivial T -weights on this Lie algebra. The diagonal k-torus D  HV;K 0 =k D RK=k .H / meets RK=k .N / in DŒ2, so T D D=DŒ2 in G. Let  W GL1 ' T be the isomorphism such that 2 is induced by composing the quotient map D  D=DŒ2 D T with the isomorphism GL1 ' D defined by t 7! diag.t; 1=t/. Let a W T ' GL1 be the inverse isomorphism. Define U˙ D URu;k .G/ .˙/ D Uh˙ai .Ru;k .G//, so Lie.U˙ / is the span of the T -weight spaces in Lie.Ru;k .G// for the T -weights having a fixed sign (relative to a). The K-subgroups N and SL2 in H  RK 0 =K .SL2 / have intersection equal to ker FSL2 =K , so RK=k .N / meets the evident Levi k-subgroup L WD SL2  RK=k .H / in ker FL=k (as we check by computing with points valued in kred algebras). Hence, the isomorphism RK=k .H /K 0 ' LK 0 carries RK=k .N /K 0 .2/

red onto .ker FL=k /K 0 , so GK 0 ' LK 0 ' SL2 . This provides a canonical k-homomorphism red f W G ! RK 0 =k .GK 0 / D RK 0 =k .SL2 /

carrying UG .˙/ into the root groups relative to the diagonal k-torus, and U˙  UG .˙/\ker f since Ru;k .G/  ker f . We will prove that UG .˙/\ker f ' RK=k .˛2 ˝K .V =K//. This has no nontrivial ks -points, so it forces U˙ D 1 and hence Ru;k .G/ is trivial. The quotient map q W RK=k .H / ! G carries the D-root groups RK=k .V / onto UG .˙/, and via the mapping property of RK 0 =k .
/ and computations with

200

Clifford constructions

ks -points of root groups we see that feWD f ı q is equal to the composite map RK=k .H / D HV;K 0 =k

/ RK 0 =k .SL2 /

RK 0 =k .FSL2 =K 0 /

/ RK 0 =k .SL2 /

carrying the D-root groups RK=k .V / into the root groups RK 0 =k .Ga / relative to the diagonal k-torus. These maps fe˙ W RK=k .V / ⇒ RK 0 =k .Ga / are given on points in a k-algebra A by the common map V ˝k A ! K 0 ˝k A defined by the inclusion V ,! K 0 and squaring on K 0 ˝k A. This squaring is valued in K ˝k A, p p and relative to the K-basis f1; u1 ; v2 g of V and the k-basis f1; u1 ; u2 ; u1 u2 g of K the resulting map V ˝k A ! K ˝k A is given by p p x0 C x1 u1 C x2 u2 7! x02 C u1 x12 C u2 x22 for x0 ; x1 ; x2 2 K ˝k A since xj2 2 A for all j . Clearly x02 C u1 x12 C u2 x22 D 0 if and only if each xj2 vanishes; i.e., x0 ; x1 ; x2 2 RK=k .˛2 / inside RK=k .Ga /. Hence, ker fe˙ D RK=k .˛2 ˝K V /. Since the map q˙ W RK=k .V / ! UG .˙/ is faithfully flat with kernel RK=k .˛2 /  RK=k .˛2 ˝K V / D ker fe˙ corresponding to j W K ,! V , yet this kernel coincides with the intersection of ker fe˙ and ker q D RK=k .N /, UG .˙/ \ ker f ' RK=k .˛2 ˝K .V =K// since j has a Klinear section. Step 3. Now we compute the root system ˆ.G; T / for the pseudo-split pseudo-semisimple k-group G. The positive and negative T -weight spaces in Lie.G/ are swapped under the adjoint action of the image w 2 G.k/ of the standard Weyl element in the Levi k-subgroup SL2  HV;K 0 =k D RK=k .H /, so these subspaces of Lie.G/ each have dimension .1=2/.25 7/ D 9. The D-root spaces in Lie.H / D Lie.RK=k .H // are V with weights ˙2, so on their images V =K in Lie.G/ the action of D=DŒ2 D T ' GL1 has weights ˙1. Note that dimk .V =K/ D 2ŒK W k D 8. Since RK=k .N / \ L D ker FL=k , we naturally identify L= ker FL=k D L.p/ ' SL2 with a k-subgroup of G containing T as its diagonal k-torus, so the Lie algebra of this k-subgroup provides 1dimensional T -weight spaces with weights ˙2 in Lie.G/. Thus, ˆ.G; T / D BC1 . Step 4. Finally, we prove that G is not of minimal type. Any absolutely pseudo-simple k-group of minimal type has trivial center if its root system over ks is non-reduced [CGP, Prop. 9.4.9], so it suffices to prove ZG is nontrivial. By considering the images V =K in Lie.G/ of the D-root spaces in the Lie

201

B.4 Type BC

algebra Lie.RK=k .H //, we see that the center ZG  ZG .T / D RK 0 =k .GL1 /=RK=k .N0 / is contained in the image in G of the visibly central k-subgroup RK 0 =k .2 /  RK=k .H /; the reverse containment is obvious. Since N0  RK 0 =K .GL1 /Œ2 D RK 0 =K .2 /, we conclude that ZG D RK 0 =k .2 /=RK=k .N0 /: But dim RK=k .N0 / D 9 and the dimension of t2

RK 0 =k .2 / D ker.RK 0 =k .GL1 /  RK=k .GL1 // is ŒK 0 W k ŒK W k D 12, so ZG is nontrivial. (Computing as in Step 1, ZG ' RK=k .˛2 / as k-groups.) Remark B.4.2. For G as in Example B.4.1, the absolutely pseudo-simple minimal type central quotient G=CG with a non-reduced root system must have trivial center [CGP, Prop. 9.4.9], so CG D ZG . In terms of the classification of minimal type pseudo-split absolutely pseudo-simple k-groups with a non-reduced root system (see [CGP, Thm. 9.8.6, Prop. 9.8.9(iii)]), G=ZG is classified by the pair .V .2/ ; V 0 / D .K 2 u1 C K 2 u2 ; k/. Example B.4.3. The construction in Example B.4.1 admits a higher-rank analogue (with root system BCn , n > 2), as follows. Let .K=k; K 0 ; V / be as in that rank-1 construction, choose n > 2, and let G denote the K-group Sp2n . Choose a split maximal K-torus T in G , and let G 0 D GK 0 , T 0 D TK 0 . Let  be a basis of ˆ WD ˆ.G ; T / with associated positive system of roots ˆC and fix a pinning C over K; this identifies Gc with SL2 for each c 2 . Let ˆC < D ˆ \ ˆ< and C C ˆ> D ˆ \ ˆ> . Step 1. By [CGP, Thm. C.2.29], there exists a unique K-subgroup H  RK 0 =K .G 0 / containing RK 0 =K .T 0 / that is pseudo-semisimple and satisfies:  ˆ.H; T / D ˆ,  Ha D HV;K 0 =K inside RK 0 =K .Ga0 / D RK 0 =K .SL2 / for the unique long root a 2  (and VK0 =K D RK 0 =K .GL1 / since dimK V D 3),  Hb D RK 0 =K .Gb0 / for every short root b 2 , so fHc gc2 generates H . Clearly the Levi K-subgroup G  RK 0 =K .G 0 / lies inside H , so G is a Levi K-subgroup of H by Proposition 2.1.2(i). In particular,

202

Clifford constructions

the subgroup Ru .HK /  HK admits K 0 as a field of definition over K, with HKss0 D G 0 , and RK 0 =K .T 0 /  H (so this K-subgroup of H is ZH .T /). Inside Lie.H / define the K-subspace X X na / C . nb /; nD. a2ˆC >

b2ˆC
, and clearly nb is AdH .Hb /-stable for all b 2 ˆ< . For distinct C 0 0 0 0 _ 0 b; b 2 ˆ< such that hb ; b i ¤ 0 the K -groups Gb and Gb 0 generate a K 0 -group P 0 0 Gb;b n  n. Hence, Ad.Hb / carries nb 0 0 of type A2 , so Lie.Gb;b 0 /  c2ˆC c
pairwise commute (as n ˆ> D A1 for ˆ D Cn ), to verify that n is AdH -stable it remains to check that C _ for a 2 ˆC > and b 2 ˆ< such that ha; b i ¤ 0, the adjoint actions of Hb on na and of Ha on nb are valued in n. We may replace G with the derived group Sp4 of the centralizer of the codimension-2 torus ..ker a/ \ .ker b//0red to reduce to the case n D 2. By negating b if necessary, we arrive at the situation that fa; bg is a basis of ˆ. Step 2. The chosen pinning of .G ; T / over K provides root group parameterizations uc W Ga ' Uc for c 2 ˆ such that the Chevalley commutation relations [Hum, §33.4] are satisfied. (There are no sign ambiguities since char.K/ D 2.) Let Xc WD Lie.uc /.@x jxD0 / and Zc D Lie.c _ /.t@ t j t D1 /. Consider the Hb -action on na . Clearly Hb is generated by the K-subgroups RK 0 =K ..U˙b /K 0 /, so it suffices to check that AdG .u˙b .x 0 //.na /  n for all x 0 2 K 0 . Since U b commutes with Ua , and Ub commutes with U a , it suffices to show that for v 2 V  K 0 , AdG .u˙b .x 0 //.X˙a /; AdG .u˙b .x 0 //.vZa / 2 n: The commutation relations give u˙b .x 0 /u˙a .y/u˙b .x 0 /

1

D .u˙b .x 0 /; u˙a .y//u˙a .y/ 2

D u˙.aC2b/ .x 0 y/u˙.aCb/ .x 0 y/u˙a .y/;

203

B.4 Type BC

so differentiating at y D 0 gives 2

AdG .u˙b .x 0 //.X˙a / D x 0 X˙.aC2b/ C x 0 X˙.aCb/ C X˙a 2 n since the roots ˙.a C b/ are short and x 0 2 2 K. Likewise, since hb; a_ i D 1, the vector AdG .u˙b .x 0 //.vZa / D vAdG .u˙b .x 0 //.Za / is equal to x 0 vX˙b C vZa , and this lies in n since b is short. Step 3. Next, consider the Ha -action on nb . Since Ha D HV;K 0 =K is generated by its root groups relative to the diagonal K-torus, it suffices to show that AdG .u˙a .v//.nb /  n for all v 2 V . The K 0 -vector space nb is spanned by fXb ; X b ; Zb g, and U a (resp. Ua ) commutes with Ub (resp. U b ), so it suffices to show K 0
AdG .u˙a .v//.X˙b /; K 0
AdG .u˙a .v//.Zb /  n for all v 2 V  K 0 . For y 0 2 K 0 we have u˙a .v/u˙b .y 0 /u˙a .v/

1

2

D u˙.aCb/ .y 0 v/u˙.aC2b/ .y 0 v/u˙b .y 0 /;

and differentiating at y 0 D 0 gives AdG .u˙a .v//.X˙b / D vX˙.aCb/ C X˙b since char.K/ D 2; its K 0 -span lies in n because the roots ˙.a C b/ are short. As ha; b _ i D 2 and char.K/ D 2, similarly AdG .u˙a .v//.Zb / D Zb . This proves that n is AdH -stable in Lie.H /, so it is a Lie ideal in Lie.H / and thus is a Lie subalgebra. For p D 2, Frobenius-semilinearity of the p-operation on Lie.H / implies that n is a p-Lie subalgebra of Lie.H / since v 2 2 K for all v 2 V . Hence, as in Example B.4.1, n D Lie.N / for a unique closed K-subgroup N  ker FH=K that is moreover a normal K-subgroup scheme of H . Step 4. Consideration of Lie algebras shows that T \ N D T Œ2. The ksubgroup scheme RK=k .N /  RK=k .H / is normal, and the calculations in the rank-1 case in Example B.4.1 immediately imply (by consideration of open cells) that the pseudo-reductive k-group RK=k .H / is perfect and that the split maximal k-torus S in RK=k .T / is a maximal k-torus of RK=k .H /. In particular, S \ RK=k .N / D S \ RK=k .T Œ2/ D SŒ2. The perfect smooth connected affine k-group G WD RK=k .H /=RK=k .N /

204

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contains the split maximal k-torus T WD S=SŒ2. We claim that G is pseudosemisimple with ˆ.G; T / D BCn and that G is not locally of minimal type (hence also not of minimal type). For each c 2 ˆ D ˆ.H; T / D ˆ.RK=k .H /; S/, let Sc  S and Tc  T D S=S Œ2 be the codimension-1 subtori corresponding to .ker c/0red  S, and let Nc  ker FHc =K correspond to nc  Lie.Hc /. Lie algebra considerations with [CGP, Lemma 7.1.1(1)] (and its proof) imply that Hc \ N D Nc for all c 2 ˆ, so clearly RK=k .Hc / \ RK=k .N / D RK=k .Nc /. Thus, since ZRK=k .H / .Sc / ! ZG .Tc / is surjective and RK=k .Hc / is perfect for all c 2 ˆ (clear for short c, and established early in Example B.4.1 for long c), it follows that D.ZG .Tc // D RK=k .Hc /=RK=k .Nc / for all c 2 ˆ, with maximal k-torus given by the image of c _ 2 X .T / D X .S / D 2X .T /. Step 5. We shall now prove that D.ZG .Tc // is pseudo-simple for every c. (This will be used in our proof of pseudo-reductivity of G in Step 6.) By the rank1 case treated in Example B.4.1, if a 2 ˆ> then D.ZG .Ta // is pseudo-simple with set of T -weights in X .T / D 2X .T / given by f˙a; ˙2ag. Likewise, if b 2 ˆ< then D.ZG .Tb // is identified with Q WD RK 0 =k .SL2 /=RK=k .ker FRK 0 =K .SL2 /=K /; and Q is naturally a k-subgroup of RK=k .RK 0 =K .SL2 /.p/ / (with p D 2) due to left exactness of RK=k . We claim that Q ' RK=k .SL2 /. Letting ' W K ,! K be the p-power inclusion (with p D 2), RK 0 =K .SL2 /.p/ WD RK 0 =K .SL2 / ˝K;' K D R.K 0 ˝K;' K/=K .SL2 / D R.K 0 p ˝K p K/=K .SL2 /; so applying RK=k to this yields R.K 0 p ˝K p K/=k .SL2 /. The composite map RK 0 =k .SL2 /  Q ,! R.K 0 p ˝K p K/=k .SL2 / evaluated on the Zariski-dense set of k-points is the map SL2 .K 0 / ! SL2 .K 0 p ˝K p K/ given by x 7! x p ˝ 1 on matrix entries, so this lands inside SL2 .K 0 p ˝K p k/. Thus, Q  R.K 0 p ˝K p k/=k .SL2 /. But the quotient map K 0 2 ˝K 2 k  kK 0 2 D K is between k-algebras with the same k-dimension, so it is an isomorphism. Hence, we have a closed immersion of k-group schemes j W Q ,! RK=k .SL2 /. Since RK=k .SL2 / is smooth and connected of dimension 12, j is an isomor-

B.4 Type BC

205

phism if dim Q D 12. The Frobenius kernel of RK 0 =K .SL2 / is .˛2 /12 as a Kscheme (not as a K-group scheme), so dim Q D ŒK 0 W k dim SL2

dim RK=k ..˛2 /12 /

D 16
3

12
dim RK=k .˛2 /

D 16
3

12
3 D 12:

Step 6. We next establish that the perfect smooth connected k-group G is pseudo-reductive. Note that the k-group ZG .T / D RK 0 =k .T 0 /=RK=k .T Œ2/ is commutative, and ZG .T / is a direct product of copies of the k-group RK 0 =k .GL1 /=RK=k .2 / that is pseudo-reductive (as this quotient is an extension of Œ2RK 0 =k .GL1 / D RK=k .GL1 / by the k-group scheme RK 0 =k .2 /=RK=k .2 / that has no nontrivial ks -points by [CGP, Ex. 5.3.7]). Thus, it suffices to show that Ru;k .G/  ZG .T /, or equivalently that no nontrivial T -weight occurs on Lie.Ru;k .G//. By [CGP, Rem. 2.3.6] applied to the quotient map RK=k .H /  G, any nontrivial T -weight that occurs on Lie.G/ is Q-linearly dependent on an element of ˆ. Suppose such a nontrivial T -weight exists on Lie.Ru;k .G//, so it lies in Qc for some c 2 ˆ. By [CGP, Thm. 3.3.11] there exists a T -stable nontrivial smooth connected k-subgroup U  Ru;k .G/ such that the T -weights occurring on Lie.U / lie in Qc f0g. Clearly U  ZG .Tc /, so U  D.ZG .Tc // since T -conjugation on the commutative quotient ZG .Tc /=D.ZG .Tc // is trivial. But Ru .D.ZG .Tc //k / D D.ZG .Tc //k \ Ru .Gk / by [CGP, Prop. A.4.8] (applied over k), so Uk  Ru .D.ZG .Tc //k /. By [CGP, Lemma 1.2.1] the pseudo-reductivity of D.ZG .Tc // thereby forces U D 1, a contradiction. It follows that no such weight c exists, as desired. Step 7. We have shown that G is pseudo-semisimple. The root system ˆ.G; T / that is non-reduced must be irreducible because the connected semisimple G ss is clearly simple of rank n. Thus, ˆ.G; T / D BCn . For any long a 2 ˆ, k the rank-1 pseudo-semisimple k-subgroup D.ZG .Ta // is as in Example B.4.1 and hence is not of minimal type. This k-subgroup is visibly generated by the ˙a-root groups (via the identification of the divisible root a 2 2X .T / as an indivisible character of T ), so G is not locally of minimal type and thus is not of minimal type.

Appendix C Pseudo-split and quasi-split forms

Let G be a pseudo-reductive group over an arbitrary field k. Motivated by the case of connected reductive groups, it is natural to wonder if G admits a pseudosplit ks =k-form; i.e., does there exist a pseudo-reductive k-group H such that Gks ' Hks and H admits a split maximal k-torus? In Proposition C.1.1 we show that if such an H exists then it is unique up to k-isomorphism. The existence problem for pseudo-split forms seems to be hopeless for commutative G in general, so for the study of existence we will focus on pseudo-semisimple G. In contrast with the reductive case, in Example C.1.2 we give (in every positive characteristic) many pseudo-semisimple G without a pseudo-split form due to an elementary field-theoretic obstruction that only arises for G that are not absolutely pseudo-simple. We also provide more subtle examples without a pseudo-split form in the absolutely pseudo-simple case in characteristic 2 and show that no such examples exist away from characteristic 2. For any pseudo-reductive k-group G, there is a finite Galois extension k 0 =k such that Gk 0 is pseudo-split. Put another way, for a given finite Galois extension k 0 =k and a pseudo-split pseudo-reductive k 0 -group G 0 , we may consider kdescents G of G 0 (i.e., k-groups G equipped with a k 0 -isomorphism Gk 0 ' G 0 ). The specification of such a G in terms of G 0 is an instance of Galois descent: a Gal.k 0 =k/-indexed collection of k 0 -isomorphisms f W G 0 ' G 0 satisfying the cocycle condition f ı .fı / D f ı for all ; ı 2 Gal.k 0 =k/. (See [BLR, §6.1, §6.2 Ex. B] for a general discussion of this formalism.) If H is a pseudo-split ks =k-form of G then Hk 0 is a pseudo-split ks0 =k 0 -form of the pseudo-split G 0 and hence Hk 0 ' G 0 due to the uniqueness of pseudo-split forms (Proposition C.1.1). Thus, exhibiting examples without a pseudo-split form is the same as giving a finite Galois extension k 0 =k and a pseudo-split pseudo-reductive k 0 -group G 0 that admits a descent to a k-group yet has no pseudo-split k-descent.

C.1 General characteristic We begin our discussion of ks =k-forms by settling the uniqueness problem for pseudo-split forms over a general field k.

C.1 General characteristic

207

Proposition C.1.1. If G and H are pseudo-split pseudo-reductive k-groups such that Gks ' Hks then G ' H . In particular, a pseudo-reductive k-group has at most one pseudo-split ks =k-form. Proof. The case of pseudo-semisimple groups is an immediate consequence of Theorem 6.3.11, so D.G/ ' D.H /. Thus, the main work is to incorporate a (commutative) Cartan subgroup. Choose split maximal k-tori S  G and T  H , and a ks -isomorphism f W Gks ' Hks carrying Sks onto Tks (found by composing an initial choice with some H.ks /-conjugation). For C WD ZG .S/ and C WD ZH .T /, the isomorphism  W Cks D ZGks .Sks / ' ZHks .Tks / D Cks induced by f descends to a k-isomorphism C ' C since the Gal.ks =k/-equivariance of  may be checked between the maximal tori (due to [CGP, Prop. 1.2.2]). By [CGP, Lemma 1.2.5, Prop. 1.2.6], G D C
D.G/ and C 0 WD C \D.G/ D ZD.G/ .S 0 / for the split maximal k-torus S 0 D S \ D.G/ of D.G/. Note that G D .D.G/ o C /=C 0 , and likewise H D .D.H / o C /=C 0 for the Cartan ksubgroup C 0 D ZD.H / .T 0 / of D.H / with T 0 WD T \ D.H /. The isomorphism Cks ' Cks induced by f clearly carries Ck0 s onto Ck0 s , and so the k-descent C ' C carries C 0 onto C 0 . This latter isomorphism is uniquely determined by its restriction between the split maximal k-tori S 0 and T 0 . As we have noted at the beginning, there exists a k-isomorphism  W D.G/ ' D.H /. By k-rational conjugacy of maximal split k-tori [CGP, Thm. C.2.3],  can be chosen to carry S 0 onto T 0 and hence C 0 onto C 0 . It suffices to arrange that this isomorphism C 0 ' C 0 agrees with the restriction of the k-isomorphism C ' C built from f , as then this latter isomorphism can be glued with  to build the desired k-isomorphism G ' H . Consider the composition of ks with the inverse of the restriction f 0 of f between derived groups. This is an automorphism of D.G/ks that preserves Sk0 s , and by precomposing  with conjugation by a suitable element of ND.G/ .S 0 /.k/ we can ensure that this automorphism preserves a chosen positive system of roots ˆC in ˆ WD ˆ.D.G/; S 0 /. By Propositions 6.3.4(i) and 6.3.7, we can precompose  with a suitable k-automorphism of .D.G/; S 0 ; ˆC / so that f 0 1 ı ks induces the identity automorphism of the Dynkin diagram and thus is the identity on Sk0 s . Hence, f 0 1 ı ks restricts to the identity on the commutative pseudo-reductive Ck0 s , so f and  now yield the same k-isomorphism C 0 ' C 0 , as required.  In every positive characteristic there are pseudo-reductive groups that do not admit a pseudo-split form, due to elementary field-theoretic obstructions: Example C.1.2. Let F=k be an extension of degree p 2 with p D char.k/ such that ŒF W ks D p and there is no degree-p subextension of F=k that is insepa-

208

Pseudo-split and quasi-split forms

rable over k (i.e., the inseparable part of F=k cannot be put “in the bottom”). For example, such an F exists when k D Fp .x; y/ (but not when Œk W k p  D p, nor when k D ks ). Let G 0 be a connected semisimple F -group that is absolutely simple and simply connected. The pseudo-reductive k-group G D RF =k .G 0 / is perfect. We claim that G has no pseudo-split ks =k-form. By [CGP, Prop. A.5.14], the ks =k-forms of G are the k-groups RE=k .H 0 /, where E is a ks =k-form of F (i.e., E ˝k ks ' F ˝k ks as ks -algebras) and H 0 is an E-group whose fibers over the factor fields of E are absolutely simple, simply connected, and of the same type as G 0 . The maximal k-tori in RE=k .H 0 / are in natural bijection with the maximal E-tori T 0 in H 0 by assigning to each T 0 the maximal k-torus T in RE=k .T 0 / [CGP, Prop. A.5.15(2)]. Such T cannot be k-split if some factor field of E is not purely inseparable over k, so if RE=k .H / is pseudo-split then the factor fields of E are purely inseparable over k. Now assume RE=k .H 0 / is pseudo-split, so since E is a ks =k-form of F we see that each factor field of E has degree p over k (as the inseparable degree is unaffected by passing to factor fields after scalar extension by k ! ks ). Qp Hence, E D i D1 Ei for purely inseparable degree-p extensions Ei of k. A ks isomorphism F ˝k ks ' E ˝k ks provides a k-embedding of F into E ˝k ks , and hence into a separable closure .E1 /s of E1 over k. Under this k-embedding, F is linearly disjoint from E1 over k since we assume F=k contains no degreep subextension purely inseparable over k. Thus, the subfield F ˝k E1 D FE1  .E1 /s has inseparable degree p 2 over k, contradicting that .E1 /s has inseparable degree p over ks . Hence, G has no pseudo-split ks =k-form. In the absolutely pseudo-simple case there is an affirmative result: Proposition C.1.3. Let G be an absolutely pseudo-simple k-group, with K=k the minimal field of definition for Ru .Gk /  Gk . Then G admits a pseudo-split ks =k-form in each of the following cases: (i) char.k/ ¤ 2, (ii) char.k/ D 2 and G is standard except possibly when the following all ss hold: GK is of outer type D2n .n > 2/, ZG is nontrivial, and G is not its own universal smooth k-tame central extension. (iii) char.k/ D 2 and G is either non-standard with root system over ks equal to F4 or has universal smooth k-tame central extension that is exotic with root system over ks of type Bn or Cn for some n > 2. A refinement if char.k/ D 2 with Œk W k 2  6 4 is given in Corollary C.2.12. Proof. By Theorem 3.4.2 and Proposition 3.4.3, in case (i) either G is standard or char.k/ D 3 with G D RK> =k .G / for a purely inseparable finite extension

C.1 General characteristic

209

K> =k and a basic exotic K> -group G of type G2 . In the latter case, by applying Theorem 3.3.8(ii) to Gks we see that the minimal field of definition over K> for the geometric unipotent radical of G is K (so kK 3  K> ( K). Likewise, in case (iii) we have G ' RK> =k .G /=Z where K> is a proper subfield of K containing kK 2 , G is a basic exotic K> -group, Z is a closed k-subgroup scheme of RK> =k .ZG /, and the minimal field of definition over K> for the geometric unipotent radical of G is K (see Proposition 3.2.6). Let G0 be the pseudo-split K>;s =K> -form of G , so RK> =k .G0 / is a pseudo-split ks =k-form of RK> =k .G /. Consider types B and C in (iii). The K> -group G is built from G0 through Galois-twisting against a continuous 1-cocycle on Gal.K>;s =K> / valued in AutG0 =K> .K>;s /. By Proposition 6.3.4(i), the K> -group Autsm G0 =K> is connected since there are no nontrivial automorphisms of Dynkin diagrams of type B or C. Hence, Autsm G0 =K> is as described as in Proposition 6.2.4, so its action on ZG0 is trivial. Consequently, the center RK> =k .ZG / of RK> =k .G / is naturally identified with RK> =k .ZG0 /, so Z corresponds to a closed k-subgroup scheme Z0  RK> =k .ZG0 /. Clearly RK> =k .G0 /=Z0 is a pseudo-split ks =k-form of G. Now we may assume G is standard with char.k/ arbitrary. By standardness and Proposition 3.2.6, G D RK=k .G 0 /=Z for an absolutely simple and simply connected K-group G 0 and a closed k-subgroup scheme Z of RK=k .ZG 0 / such that the Cartan k-subgroup RK=k .T 0 /=Z is pseudo-reductive for some (equivalently, any) maximal K-torus T 0 of G 0 ; this latter condition encodes that G is pseudo-reductive (see [CGP, Lemma 9.4.1]). 0 Let G00 be the split Ks =K-form of G 0 , and let f W .G00 /Ks ' GK be a s Ks -isomorphism. It suffices to construct a k-subgroup Z0 of the central ksubgroup RK=k .ZG00 / of RK=k .G00 / such that .Z0 /ks is carried over to Zks under RKs =ks .f /, as then G0 WD RK=k .G00 /=Z0 is a pseudo-split ks =k-form of G. The case Z D 1 is trivial, as is the case Z D RK=k .ZG 0 / (corresponding to when ZG D 1 by Lemma 4.1.1 since RK=k .ZG 0 / is the center of RK=k .G 0 / by [CGP, Prop. A.5.15(1)]), so we may assume G does not coincide with its universal smooth k-tame central extension RK=k .G 0 / and that ZG is nontrivial. We aim to construct a pseudo-split ks =k-form provided that char.k/ ¤ 2 when G 0 is of outer type D2n with n > 2. The K-group G 0 is obtained from the split G00 through Gal.Ks =K/-twisting against automorphisms of .G00 /Ks . If the root system of G00 is not of type D2n with n > 2 then any automorphism of .G00 /Ks acts on .ZG00 /Ks through either the identity or inversion. Since inversion preserves all subgroup schemes of a group scheme, it follows that away from outer type D2n (n > 2) every k-subgroup scheme Z of RK=k .ZG 0 / has base

210

Pseudo-split and quasi-split forms

change over ks that descends to a k-subgroup scheme Z0 of RK=k .ZG00 /. This settles cases where G 0 is not of outer type D2n (n > 2), and so settles (ii). Suppose G 0 is of outer type D2n (n > 2), so we may assume char.k/ ¤ 2 since (ii) is settled. Since ZG00 D 2  2 , so ZG 0 is étale, we have Z D RK=k .Z 0 / for a closed K-subgroup Z 0 of ZG 0 ; see [CGP, Prop. A.5.13]. Hence, G D RK=k .G 0 =Z 0 /, so a pseudo-split ks =k-form of G is obtained by applying RK=k to the split Ks =K-form of G 0 =Z 0 .  To explain the sense in which Proposition C.1.3 is optimal (apart from the case Œk W k 2  6 4 in characteristic 2), we consider absolutely pseudo-simple groups G over imperfect fields k with characteristic 2 such that G is not covered ss by Proposition C.1.3(ii),(iii). If G is standard then GK must be of outer type D2n for some n > 2, and we will construct many such G without a pseudo-split ks =kform in Proposition C.1.4. Suppose G is not standard, so its root system over ks is of type B, C, or BC with rank n > 1. If Œk W k 2  6 4 then in all non-standard cases there is a pseudo-split form; see Corollary C.2.12. If Œk W k 2  > 8 then the generalized exotic cases over k provide many examples without a pseudo-split ks =k-form, as we discuss in §C.3–§C.4. (Such generalized exotic cases for types Bn or Cn with n > 2 must lie beyond the exotic case, by Proposition C.1.3(iii).) Now consider standard absolutely pseudo-simple groups G over imperfect fields k with char.k/ D 2, so G D RK=k .G 0 /=Z for .G 0 ; K=k; Z/ as in the proof of Proposition C.1.3. Assume G 0 is outer type D2n (n > 2), so Gal.Ks =K/ maps onto a nontrivial subgroup of the diagram automorphism group of D2n (n > 2). The diagram automorphism group for D2n is Z=2Z when n > 2 and is S3 when n D 2. Thus, a necessary condition for the existence of such G which does not admit a pseudo-split ks =k-form is that K admits a quadratic Galois extension when n > 2 and admits a quadratic or cubic Galois extension when n D 2. Since K=k is purely inseparable, it is equivalent to say that k must admit a quadratic Galois extension when n > 2 and admit a quadratic or cubic Galois extension when n D 2. Over any such k admitting a quadratic Galois extension, examples without a pseudo-split form exist for any n > 2 (and Remark C.1.5 addresses k with a cubic Galois extension when n D 2): Proposition C.1.4. Assume k is imperfect with char.k/ D 2 and that k admits a quadratic Galois extension k 0 . Let K D k.a1=4 / for a 2 k k 2 . For n > 2, let G be the unique .up to isomorphism/ absolutely simple and simply connected non-split quasi-split k-group of type D2n that splits over k 0 . There is a closed k-subgroup Z of the center of RK=k .GK / such that G WD RK=k .GK /=Z is absolutely pseudo-simple with maximal geometric semisimple quotient G ss of adjoint type and G does not admit a pseudo-split ks =k-form. k

C.1 General characteristic

211

Proof. Let G be the absolutely simple and simply connected k-split group of type D2n , so Gk 0 ' Gk 0 since Gk 0 is split. Identify these k 0 -groups via a fixed k 0 isomorphism f W Gk 0 ! Gk 0 . For the nontrivial  2 Gal.k 0 =k/, ' WD f 1 ı f is an outer automorphism of Gk 0 (equivalently, it induces a nontrivial automorphism of the Dynkin diagram) and the k-descent of Gk 0 via ' is the quasisplit k-group G . The effect of ' on the center ZGk0 is an involution, and we can choose an identification ZG ' 2  2 over k such that this involution on .ZG /k 0 D ZGk0 swaps the two 2 -factors. Let  denote ZG . Let K 0 D k 0 ˝k K and let  also denote the nontrivial K-automorphism of K 0 . The center RK 0 =k 0 .K 0 / D RK 0 =k 0 .ZGK 0 / D RK 0 =k 0 .2 /  RK 0 =k 0 .2 / of RK 0 =k 0 .GK 0 / D RK 0 =k 0 .GK 0 / is identified with RK=k ./k 0 , and a closed k 0 subgroup scheme Z 0 of RK 0 =k 0 .2  2 / descends to a closed k-subgroup of RK=k .K / if and only if the K 0 =K-twist Z 0 is obtained from Z 0 by swapping of the two RK 0 =k 0 .2 /-factors. Likewise, Z 0 descends to a closed k-subgroup of RK=k .ZGK / D RK=k .2  2 / if and only if Z 0 D Z 0 . We shall construct an intermediate closed k-subgroup scheme   Z  RK=k .K / D ZRK=k .GK / such that: (i) RK=k .K /=Z has no nontrivial smooth connected k-subgroup, (ii) the k 0 -subgroup Zk 0  RK=k .K /k 0 D RK=k .ZGK /k 0 D RK=k .ZGK /k 0 (with the latter identification defined by RK 0 =k 0 .fK 0 / 1 ) does not descend to a closed k-subgroup scheme of RK=k .ZGK /. Assume such a Z exists. By Corollary 4.1.4, the perfect central quotient G D RK=k .GK /=Z is pseudo-reductive in view of (i), so it is absolutely pseudo-simple. Since   Z, the connected semisimple G ss is of adjoint type. k As G is standard, any ks =k-form of G is standard by [CGP, Cor. 5.2.3]. Thus, by [CGP, Prop. A.5.14] and Proposition 5.3.1(i) a pseudo-split ks =kform of G is isomorphic to RK=k .GK /=Z for a closed k-subgroup scheme Z of the center RK=k .ZGK / of RK=k .GK /. Assuming such a ks =k-form exists, we shall deduce a contradiction via condition (ii). Since the pseudo-split k 0 groups Gk 0 D RK=k .GK /k 0 =Zk 0 and .RK=k .GK /=Z/k 0 D RK 0 =k 0 .GK 0 /=Zk 0 are isomorphic over ks , they must be k 0 -isomorphic (by Proposition C.1.1). Such a k 0 -isomorphism arises from a unique k 0 -isomorphism  W RK=k .GK /k 0 !

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Pseudo-split and quasi-split forms

RK=k .GK /k 0 that carries Zk 0 onto Zk 0 . Since G is an outer form of G that splits over k 0 ,  1 ı   is an outer automorphism of RK=k .GK /k 0 . Consider the k 0 -automorphism D .RK 0 =k 0 .fK 0 // 1 ı  of RK 0 =k 0 .GK 0 /. We claim that the restriction of to the center RK 0 =k 0 .ZGK 0 / descends to a k-automorphism  of RK=k .ZGK /. The k 0 -automorphism of RK 0 =k 0 .GK 0 / 0 arises as the Weil restriction of a unique K -automorphism ˛ of GK 0 by [CGP, Prop. A.5.14], so it suffices to show that the action of ˛ on ZGK 0 D .ZGK /K 0 is defined over K. The effect of ˛ on the center only depends on the associated K 0 -point of 0 .AutGK 0 =K 0 / D 0 .AutGK =K /K 0 . But 0 .AutGK =K / is a constant K-group, so its K 0 -points are K-points and thus the desired k-descent  of on the center is obtained. It is harmless to replace Z with the closed central k-subgroup scheme .Z/, so the isomorphism RK 0 =k 0 .fK 0 / W RK=k .GK /k 0 ! RK=k .GK /k 0 carries Zk 0 onto Zk 0 . But Zk 0 descends to a closed k-subgroup scheme Z of RK=k .ZGK /, contradicting (ii). Thus, there is no pseudo-split ks =k-form of G when Z exists as above. We will find Z as an intermediate closed k-subgroup scheme Rk.pa/=k .k.pa/ /  Z  RK=k .K /:

(C.1.4)

Letting U D RK=k .2 /=Rk.pa/=k .2 /, we can express our problem in terms of a k 0 -subgroup scheme .Z=Rk.pa/=k .k.pa/ //k 0  .RK 0 =k 0 .2 /=Rk 0 .pa/=k 0 .2 //2 D Uk 0  Uk 0 as follows. Letting  be the nontrivial automorphism of k 0 =k, we seek a closed k 0 -subgroup Z 0  Uk 0  Uk 0 satisfying three conditions: (0)   .Z 0 / is obtained from Z 0 via the k 0 -automorphism of Uk 0  Uk 0 that swaps the factors, (1) .Uk 0  Uk 0 /=Z 0 contains no nontrivial smooth connected k 0 -subgroup, (2)   .Z 0 / ¤ Z 0 inside Uk 0  Uk 0 . Indeed, (0) implies that Z 0 arises from a k-subgroup scheme Z  RK=k .K / as in (C.1.4), and (1) and (2) say that Z satisfies the above desired properties (i) and (ii) respectively. To construct Z 0 , observe that Rk.pa/=k .˛2 / ' U via x 7! 1 C a1=4 x. We can choose t 2 k 0  such that .t/ ¤ t , so c WD .t/=t 2 k 0  satisfies Nk 0 =k .c/ D 1 (i.e.,  .c/ D 1=c) and c 62 k. Let Z 0  Uk 0  Uk 0 be the graph of the cmultiplication automorphism of Uk 0 D Rk 0 .pa/=k 0 .˛2 /. The k 0 -subgroup Z 0 satisfies (0) above because .x; .c/x/ D .x; x=c/ D .cy; y/ for y D x=c, and

213

C.1 General characteristic

it satisfies (2) because c 62 k. Finally, since Z 0 is the graph of a k 0 -group automorphism of Uk 0 , the quotient .Uk 0  Uk 0 /=Z 0 is identified as a k 0 -group with Uk 0 ' Rk 0 .pa/=k 0 .˛2 /, so (1) holds.  Remark C.1.5. Suppose n D 2 and k is imperfect of characteristic 2, admitting a cubic Galois extension k 0 . Let  be an order-3 diagram automorphism of D4 , and choose a generator  2 Gal.k 0 =k/. Let G be the unique (up to isomorphism) non-split quasi-split simply connected k-group of type D4 that splits over k 0 and let  D ZG . We may choose an identification k 0 D 2  2 so that the k 0 -isomorphism .k 0 / ' k 0 encoding the k-structure  is obtained from the order-3 automorphism . 11 10 / 2 Aut.2  2 / D GL2 .F2 /. Let K=k be as in Proposition C.1.4. For the graph Z 0 of multiplication by c 2 k 0  on Rk 0 .pa/=k 0 .˛2 / and the corresponding k-subgroup scheme Z  RK=k .K / as in the proof of Proposition C.1.4, the k-group G D RK=k .GK /=Z has exactly the same properties as in Proposition C.1.4 (with n D 2) provided that c 62 k  and  .c/ C 1 D 1=c. We may use c D .b/=b for b 2 k 0  such that .b/ 62 kb and Trk 0 =k .b/ D 0. To find such b, note that the differences .t/ t for t 2 k 0 constitute a k-plane in k 0 , and many b in this plane avoid eigenlines of  over k. The field-theoretic obstruction in Example C.1.2 to relaxing the absolutely pseudo-simple hypothesis in Proposition C.1.3 cannot arise when Œk W k p  D p. The following alternative example applies without restriction on Œk W k p  > 1. Example C.1.6. Let k be imperfect of characteristic p > 0 with a quadratic 2 Galois extension k 0 =k. Let K D k.a1=p / for a 2 k k p , K 0 D k 0 ˝k K, G D RK 0 =K .SLp /, and  D Rk 0 =k .p /. For U WD RK=k .p /=Rk.a1=p /=k .p / we claim that U ' Rk.a1=p /=k .˛p /p 1 as k-groups. Once this is shown, the construction in the proof of Proposition C.1.4 adapts without difficulty to build a central quotient G D RK=k .G /=Z with no pseudo-split ks =k-form. (This G is pseudo-simple but not absolutely pseudo-simple.) The Rk.a1=p /=k .p /-equivariant k-scheme isomorphism Rk.a1=p /=k .p /  Rk.a1=p /=k .˛p /p

1

' RK=k .p /

defined by .z; x1 ; : : : ; xp

1/

2

7! z.1 C x1 a1=p C


C xp

1a

.p 1/=p 2

/

implies U ' Rk.a1=p /=k .˛p /p 1 as k-schemes. Thus, any k-homomorphism h W Rk.a1=p /=k .˛p /p 1 ! U with trivial kernel may be viewed as a monic en-

214

Pseudo-split and quasi-split forms

domorphism of a finite type k-scheme, so h is an isomorphism by [EGA, IV4 , 17.9.6]. It therefore suffices to construct such an h. Let F WD k.a1=p /, so RK=F .p /=p D RK=F .GL1 /=GL1 is an fppf form p 1 p 1 of Ga and hence has Frobenius kernel that is an fppf-form of ˛p . This p 1 kernel is therefore F -isomorphic to ˛p (as ˛pr has no nontrivial fppf forms over fields, since its automorphism functor is represented by GLr ). Fix an F p 1 group inclusion  W ˛p ,! RK=F .p /=p onto the Frobenius kernel. We p 1  W ˛p ! RK=F .p / (not an claim that this lifts to an F -scheme morphism e F -homomorphism!). More generally, any F -scheme map f W ˛pr ! RK=F .p /=p carrying 0 to 1 lifts to an F -scheme map ˛pr ! RK=F .p /. Indeed, when viewed as a map ˛pr ! RK=F .GL1 /=GL1 , f lifts to an F -scheme map fe W ˛pr ! RK=F .GL1 / since there is no nontrivial GL1 -torsor over ˛pr . Multiplying fe by 1=fe.0/ arranges that fe.0/ D 1, so fe factors through a unique F -scheme morphism ˛pr ! RK=F .p / by the functorial meaning of RK=F . The map RF =k .e  / W RF =k .˛p /p 1 ! RK=k .p / yields a composite map RF =k .˛p /p

1 h

! RK=k .p /=RF =k .p / DW U ,! RF =k .RK=F .p /=p /

that is equal to the k-homomorphism RF =k ./ whose kernel is trivial (since ker  D 1), so h is a k-homomorphism with trivial kernel.

C.2 Quasi-split forms Let G be a pseudo-reductive group over a field k. Its minimal pseudo-parabolic k-subgroups constitute a G.k/-conjugacy class [CGP, Thm. C.2.5]. Such a ksubgroup P is called a pseudo-Borel k-subgroup if Pks is minimal among the pseudo-parabolic ks -subgroups of Gks , and we say G is quasi-split if it admits a pseudo-Borel k-subgroup. Example C.2.1. If G D Rk 0 =k .G 0 / for a nonzero finite reduced k-algebra k 0 and smooth affine k 0 -group G 0 with connected reductive fibers then P 0 7! Rk 0 =k .P 0 / is an inclusion-preserving bijection between the sets of parabolic k 0 -subgroups of G 0 and pseudo-parabolic k-subgroups of G [CGP, Prop. 2.2.13], so by extending scalars to ks we see that Rk 0 =k .P 0 / is a pseudo-Borel k-subgroup of G if and only if P 0 is a Borel k 0 -subgroup of G 0 . By [CGP, Prop. 3.5.1(4)] a pseudo-parabolic k-subgroup P of G is a pseudoBorel k-subgroup if and only if the pseudo-reductive quotient P =Ru;k .P / is

C.2 Quasi-split forms

215

commutative, or equivalently if and only if P is solvable. Thus, in the notation of the Tits classification in Theorem 6.3.11, if G is pseudo-semisimple then it is quasi-split if and only if M is commutative, or equivalently the anisotropic kernel D.M / is trivial. The triviality of the pseudo-semisimple D.M / is equivalent to that of its maximal central quotient D.M=ZM /, as well as to its canonical diagram Dyn.D.M=ZM // being empty. The k-isomorphism class of a quasisplit pseudo-semisimple k-group G is therefore determined by the isomorphism class of the pair .Gks ; Dyn.G// consisting of the ks -group Gks and its canonical diagram Dyn.Gks / equipped with the -action of Gal.ks =k/ arising from G. If B is a pseudo-Borel k-subgroup of G then for a maximal k-torus T in B the set ˆ.Bks ; Tks / is a Gal.ks =k/-stable positive system of roots in ˆ.Gks ; Tks / [CGP, Prop. 3.5.1]. Hence, the natural Gal.ks =k/-action on X.Tks / preserves the basis  associated to Bks , so this action coincides with -action of Gal.ks =k/ on  through diagram automorphisms (see Definition 6.3.3). Lemma C.2.2. Let G be a pseudo-semisimple k-group, and assume that the Dynkin diagram of the root system of Gks does not admit nontrivial automorphisms. Any quasi-split ks =k-form of G is pseudo-split. Proof. We may and do assume G is quasi-split and aim to prove that it is pseudosplit. Using notation as above, the absence of nontrivial diagram automorphisms implies that the Galois action on X.Tks / leaves the roots in  invariant. But  spans X.Tks /Q since G is pseudo-semisimple, so T is k-split.  The reader is referred to 4.1.2 and Proposition 6.1.4 for the definition and basic properties of the k-group ZG;C associated to a pseudo-reductive k-group G and Cartan k-subgroup C . 0 Lemma C.2.3. For a pseudo-reductive k-group G, the .Autsm D.G/=k / -action on D.G/ uniquely extends to an action on G.

Proof. The maxmal k-torus T in a Cartan k-subgroup C of G is an almost direct product of the maximal k-torus T WD T \ D.G/ of D.G/ and the maximal central k-torus Z in G [CGP, Lemma 1.2.5(ii)]. Thus, C WD C \ D.G/ D ZD.G/ .T / is a Cartan subgroup of D.G/. Since G D .D.G/ o C /=C and 0 .Autsm D.G/=k / D .D.G/ o ZD.G/;C /=C (see Proposition 6.2.4), the action of sm .AutD.G/=k /0 on D.G/ extends to an action on G that is the identity on the maximal central k-torus Z in G. Any k-automorphism of G that restricts to the identity on Z is uniquely determined by its restriction to D.G/ due to [CGP, Prop. 1.2.2], so it remains to show that any action on G by a smooth connected k-group H must restrict

216

Pseudo-split and quasi-split forms

to the identity on Z. We may and do assume k D ks , so considering k-points shows via smoothness of H that the H -action on G restricts to an action on Z. But H is connected and Z is a torus, so such an action on Z is trivial.  Let G be a pseudo-reductive group over a field k. Lemma C.2.3 provides 0 a Gal.ks =k/-equivariant inclusion .Autsm D.G/=k / .ks / ,! Aut.Gks /. This subgroup meets Aut.G/ in the group 0 PsInn.G/ WD .Autsm D.G/=k / .k/

(C.2.3)

whose elements are called pseudo-inner automorphisms of G (over k). In general .G=ZG /.k/  PsInn.G/. This inclusion is an equality if G is reductive but 0 generally not otherwise, and .Autsm D.G/=k / can fail to be perfect but its derived group is D.G/=ZD.G/ by Proposition 6.2.4, with D.G/=ZD.G/ D D.G=ZG / as quotients of D.G/. (Note that ZD.G/ is central in G. To see this, fix a Cartan subgroup C of G. Then C \ D.G/ is a Cartan subgroup of D.G/ and so ZD.G/ is contained in C \ D.G/  C . Now as C is commutative and G D C
D.G/, it is obvious that ZD.G/ is central in G.) Definition C.2.4. A pseudo-inner form of a pseudo-reductive k-group G is a ks =k-form obtained by twisting against a class in the image of the natural map H1 .ks =k; PsInn.Gks // ! H1 .ks =k; Aut.Gks //. For reductive G this recovers the notion of inner form, but for non-reductive G it is more general than twisting against a class in the image of H1 .k; G=ZG /. Any reductive G admits a unique quasi-split inner form. In the pseudo-reductive case all ks =k-forms of G are pseudo-inner when G is pseudo-semisimple and Autsm G=k is connected (as occurs when the Dynkin diagram of Gks has no nontrivial automorphisms, by Proposition 6.3.4(i)), and we will prove uniqueness of quasi-split pseudo-inner forms in general in Proposition C.2.8. The existence of quasi-split pseudo-inner forms will be proved provided that if char.k/ D 2 then either G is standard or Œk W k 2  6 4 (see Theorem C.2.10). Remark C.2.5. The restriction Œk W k 2  6 4 for existence of a quasi-split pseudoinner form in the non-standard case when char.k/ D 2 is necessary since the examples with no pseudo-split ks =k-form in §C.3–§C.4 over some k satisfying Œk W k 2  > 8 do not admit a quasi-split ks =k-form, due to Lemma C.2.2. In every positive characteristic, the pseudo-semisimple groups in Example C.1.6 without a pseudo-split ks =k-form are quasi-split, and the ones in Example C.1.2 without a pseudo-split ks =k-form admit a quasi-split pseudo-inner form (replace G 0 there with a quasi-split inner Fs =F -form); Lemma C.2.2 is not

C.2 Quasi-split forms

217

applicable to those groups because the Dynkin diagrams of their root systems over ks admit a nontrivial automorphism. Proposition C.1.4 provides absolutely pseudo-simple groups without a pseudo-split ks =k-form over any imperfect field k of characteristic 2 admitting a quadratic Galois extension, and those groups are quasi-split since G there is quasi-split (and Lemma C.2.2 is not applicable since the Dynkin diagram of D2n has nontrivial automorphisms). Lemma C.2.6. Let G be a quasi-split pseudo-reductive k-group and C a Cartan k-subgroup of G contained in a pseudo-Borel k-subgroup B. The group of kautomorphisms of G that arise from PsInn.G/ and preserve .C; B/ is ZG;C .k/. Proof. By Galois descent, without loss of generality we may assume that k is separably closed. Hence, by [CGP, Prop. 3.5.1] an automorphism of G carrying the unique maximal k-torus T of C into itself carries B into itself if and only if it carries the associated subset ˆ.B; T /  ˆ.G; T / into itself. Every automorphism of G restricting to the identity on C induces the identity automorphism of X.T /  ˆ.G; T / and so preserves B. Moreover, ZG;C .k/  PsInn.G/ by Lemma 6.1.2(i) (since ZG;C is connected, by Proposition 6.1.4). It remains to show that if a pseudo-inner automorphism f 2 PsInn.G/ preserves .C; B/ then f jC is the identity on C . Since C is commutative and pseudo-reductive, by [CGP, Prop. 1.2.2] it is equivalent to show that the automorphism f jT of T is the identity. Since B is solvable, the image of Bk in G red is k parabolic and solvable, hence a Borel subgroup with maximal torus Tk . Lemma 0 C.2.3 provides an action of .Autsm D.G/=k / on G uniquely extending its natural

action on D.G/, so now we may assume k D k, G is reductive, and f arises from Aut0G=k .k/ D .G=ZG /.k/. Preservation of T implies f 2 NG=ZG .T =ZG /. Since f preserves the Borel subgroup B, its image in W .G=ZG ; T =ZG / is trivial (as this Weyl group acts simply transitively on the set of positive systems of roots). Thus, f 2 T =ZG , so f acts as the identity on T . 

Lemma C.2.7. Let G be a quasi-split pseudo-reductive k-group. As S varies through the maximal k-split tori of G, the k-subgroups ZG .S/ constitute a G.k/-conjugacy class of Cartan k-subgroups of G and are precisely the Cartan k-subgroups of the pseudo-Borel k-subgroups of G. If C is a member of this distinguished G.k/-conjugacy class of Cartan ksubgroups then for any separable extension field k 0 =k the Cartan k 0 -subgroup Ck 0 is a member of the analogous distinguished G.k 0 /-conjugacy class. Proof. Maximal k-split tori of G constitute a single G.k/-conjugacy class, and likewise for pseudo-Borel k-subgroups of G [CGP, Thm. C.2.3, Thm. C.2.5].

218

Pseudo-split and quasi-split forms

By [CGP, Prop. C.2.4] (and the discussion immediately after its proof), some pseudo-Borel k-subgroup B contains S and moreover B D ZG .S/ n Ru;k .B/. Since B=Ru;k .B/ is commutative, so ZG .S/ is commutative, there is a unique maximal k-torus T contained in ZG .S/ and it is certainly maximal in G. Clearly T contains S and ZG .T / D ZG .S/, so ZG .S/ is a Cartan k-subgroup of G that is visibly also a Cartan k-subgroup of B. Since any maximal k-torus of B maps isomorphically onto a maximal ktorus in B=Ru;k .B/ ' ZG .S/, every such k-torus contains a maximal split k-torus of B. The B.k/-conjugacy of the maximal split k-tori of B then implies that the Cartan k-subgroups of B are a single B.k/-conjugacy class. The desired compatibility with separable extension k 0 =k is now immediate from the observation that Bk 0 is a pseudo-Borel k 0 -subgroup of Gk 0 since pseudo-Borel subgroups are precisely solvable pseudo-parabolic subgroups.  Proposition C.2.8. Up to k-isomorphism, a pseudo-reductive k-group G has at most one quasi-split pseudo-inner form. Proof. We must show that if G and G are quasi-split with G a pseudo-inner form of G then G ' G. Let B  G and B  G be pseudo-Borel k-subgroups, and let C  B and C  B be Cartan k-subgroups. By definition of “pseudoinner form”, there exists a ks -isomorphism f W Gks ' Gks such that for all 2 Gal.ks =k/ the ks -automorphism c. / D f ı . f / 1 of Gks lies in PsInn.Gks /. It is harmless to post-compose f with conjugation against an element of G.ks /, so by the G.ks /-conjugacy of pseudo-Borel ks -subgroups of Gks we can arrange that f .Bks / D Bks and then further arrange via composing f with conjugation by an element of B.ks / that f .Cks / D Cks . Each c. / belongs to PsInn.Gks / and preserves the pair .Cks ; Bks /, so c. / 2 ZG;C .ks / by Lemma C.2.6. Thus, it suffices to show that H1 .k; ZG;C / D 1. The pseudo-parabolic k-subgroup B 0 WD B \ D.G/ of D.G/ inherits solvability from B and thus is a pseudo-Borel k-subgroup (so D.G/ is quasi-split). The Cartan k-subgroup C 0 D C \ D.G/ of D.G/ is contained in B 0 , so C 0 is as described in Lemma C.2.7. Since D.G/ has trivial anisotropic kernel (as it is quasisplit), so ZD.G/;C 0 is identified with the kernel considered in Proposition 6.3.12 (applied to the pseudo-semisimple D.G/ and Cartan k-subgroup C 0  B 0 ), we have ZD.G/;C 0 ' RF =k .GL1 / for a nonzero finite reduced k-algebra F . But ZG;C ' ZD.G/;C 0 by Lemma 6.1.2(i), so we are done.  C.2.9. Now we address existence of a quasi-split pseudo-inner form of a pseudoreductive k-group. Ultimately this will be reduced to a refined Galois descent

C.2 Quasi-split forms

219

problem for a quasi-split pseudo-reductive group over a finite Galois extension of k. Thus, we first describe the setup for this refined descent problem. Let G 0 be a quasi-split pseudo-reductive group over a finite Galois extension k 0 =k, and let C 0 be a Cartan k 0 -subgroup contained in a pseudo-Borel k 0 -subgroup B 0 . Let S 0  C 0 be the maximal split k 0 -torus, so C 0 D ZG 0 .S 0 / by Lemma C.2.7. (We do not assume this data is equipped with a k-structure!) Suppose we are given k 0 -isomorphisms f W G 0 ' G 0 for all  2 Gal.k 0 =k/, and moreover assume f  2 PsInn.G 0 /
.f ı f /

(C.2.9.1)

in Isomk 0 .G 0 ; G 0 / for all ;  2 Gal.k 0 =k/. (One source of such ff g is a k 0 =k-descent datum on G 0 , but this will not be the only case we need.) The set of pairs .B 0 ; S 0 / in G 0 is permuted transitively by D.G 0 /.k 0 /, so we can adjust the choice of each f via D.G 0 /.k 0 /-conjugation such that f .C 0 / D C 0 and f .B 0 / D B 0 for all . After this adjustment, which retains (C.2.9.1), there exists z.;  / 2 PsInn.G 0 / preserving .B 0 ; C 0 / such that f  D z.; / ı f ı f :

(C.2.9.2)

By Lemma C.2.6 we have z.; / 2 ZG 0 ;C 0 .k 0 /, and the possible choices for such k 0 -isomorphisms G 0 ' G 0 are f0 D z ı f for varying z 2 ZG 0 ;C 0 .k 0 /. The commutativity of ZG 0 ;C 0 implies that the associated k 0 -isomorphisms h W .ZG 0 ;C 0 / D ZG 0 ;C 0 ' ZG 0 ;C 0 defined by  7! f ı  ı f 1 are independent of the choice of such ff g. It follows from (C.2.9.2) and the commutativity of ZG 0 ;C 0 that fh g satisfies the cocycle condition h D h ı  h for all ;  2 Gal.k 0 =k/; i.e., it is a Galois descent datum on the affine k 0 -group ZG 0 ;C 0 relative to k 0 =k. Let Z be the associated k-descent; this is a commutative pseudo-reductive k-group. Since B 0 .k 0 / \ NG 0 .S 0 /.k 0 / D ZG 0 .S 0 /.k 0 / D C 0 .k 0 /, any pseudo-Borel k 0 subgroup of G 0 containing C 0 has the form nB 0 n 1 for n 2 NG 0 .S 0 /.k 0 / that is unique modulo C 0 .k 0 /, say with image w 2 W .G 0 ; S 0 /.k 0 /. The analogue of Z using nB 0 n 1 is obtained from descent data defined by composing f with n-conjugation on G 0 and .n/ 1 -conjugation on G 0 . Thus, via the natural W .G 0 ; S 0 /-action on ZG 0 ;C 0 , the w-action descends to a k-isomorphism between the associated Z ’s. In this sense, Z depends only on .G 0 ; C 0 / and not on B 0 . By Lemma 6.1.2, Z only depends on .D.G 0 /; D.G 0 / \ C 0 / and likewise Z is unaffected by replacing G 0 with a pseudo-reductive central quotient.

220

Pseudo-split and quasi-split forms

If we replace f with f0 WD z ı f for z 2 ZG 0 ;C 0 .k 0 / D Z .k 0 / then the associated z 0 .;  / is z
.z
.f ı z ı f 1 //

1


z.; / D .z  =.z
.z ///
z.; /;

where x 7!  .x/ is the canonical action of Gal.k 0 =k/ on Z .k 0 /. In other words, the function .;  / 7! z.;  / 2 ZG 0 ;C 0 .k 0 / D Z .k 0 / changes exactly by a 2coboundary on Gal.k 0 =k/ valued in Z .k 0 /. By computing f./ D f.  / in two different ways, it is straightforward to see that .;  / 7! z.;  / 2 Z .k 0 / is a 2-cocycle. Thus, this 2-cocycle represents a class c.G 0 ; C 0 / 2 H2 .k 0 =k; Z .k 0 // that is independent of the choice of ff g (required to satisfy f .B 0 / D B 0 and f .C 0 / D C 0 ) and independent of the choice of B 0 (in the same sense that Z is independent of the choice of B 0 ). The construction shows that c.G 0 ; C 0 / vanishes if and only if we can find ff0 g that is a k 0 =k-descent datum. For such ff0 g, the associated k 0 =k-descent of G 0 is quasi-split since f0 .B 0 / D B 0 . Hence, the vanishing of c.G 0 ; C 0 / 2 H2 .k 0 =k; Z .k 0 // is exactly the condition that G 0 admits a quasi-split k-descent whose k 0 =k-descent datum is related to the initial ff g through composition against pseudo-inner k 0 -automorphisms of G 0 (due to Proposition 6.2.4, Lemma C.2.6, and the equality ZG 0 ;C 0 D ZD.G 0 /;D.G 0 /\C 0 ). Theorem C.2.10. A pseudo-reductive k-group G admits a quasi-split pseudoinner form provided that if char.k/ D 2 then G is standard or Œk W k 2  6 4. 0 Proof. Pseudo-inner forms involve Galois-twisting against .Autsm D.G/=k / , so we first reduce to the case where G is perfect. Clearly if G is standard then so is D.G/. Suppose D.G/ has a quasi-split ks =k-form arising from Galois0 twisting against a class  2 H1 .k; .Autsm D.G/=k / /, so the twist of G against  is a pseudo-inner ks =k-form H of G such that D.H / is quasi-split. But P 7! P \D.H / is a bijection between the sets of pseudo-parabolic k-subgroups of H and D.H / and it is inclusion-preserving in both directions (see the discussion [CGP] immediately after [CGP, Rem. 11.4.2]), so H inherits the quasi-split property from D.H /. Now we may replace G with D.G/ so that G is pseudo-semisimple. In particular, the quotient G=ZG is pseudo-semisimple of minimal type with trivial center (see Proposition 4.1.3) and it is standard if G is standard. The formation of 0 .Autsm G=k / is insensitive to replacing G with a pseudo-reductive central quotient (Corollary 6.2.6), and a pseudo-inner form H of G is quasi-split if and only if the pseudo-inner form H=ZH of G=ZG is quasi-split, so we may replace G with G=ZG to arrange that G is of minimal type. Now passing to the universal

C.2 Quasi-split forms

221

e (of minimal type by Proposition 5.3.3, and smooth k-tame central extension G standard if G is by Lemma 3.2.8), we can assume G ss is simply connected. k Let C  G be a Cartan k-subgroup, and let T be its maximal k-torus. Choose a finite Galois extension k 0 =k splitting T , so G 0 WD Gk 0 has split maximal k 0 torus T 0 WD Tk 0 and admits a pseudo-Borel k 0 -subgroup B 0 containing C 0 WD Ck 0 . Let € D Gal.k 0 =k/ and define the k-descent Z of ZG 0 ;C 0 via the construction in C.2.9 applied to the k 0 =k-descent datum ff g 2€ on G 0 defined by G, thereby defining a class c.G 0 ; C 0 / 2 H2 .k 0 =k; Z .k 0 //. It suffices to show that c.G 0 ; C 0 / D 0, as then G admits a quasi-split pseudo-inner k 0 =k-form H . Let   ˆ.G 0 ; T 0 / be the basis associated to B 0 , equipped with the -action of € as in Definition 6.3.3. Note that if G is standard then so is Ga0 for every a 2 ˆ.G 0 ; T 0 /. For each a 2 , let Ca0 WD ZGa0 .a_ / D Ga0 \ C 0 , so the natural map Y  0 W ZG 0 ;C 0 ! ZGa0 ;Ca0 a2

is an isomorphism by Proposition 6.1.4. Let ka  k 0 be the subfield over k such that Gal.k 0 =ka / is the stabilizer of a in Gal.k 0 =k/ under the -action. For 2 €, consider the unique w 2 W .G 0 ; T 0 / such that w . .// D . Define a k 0 -isomorphism f 0 W G 0 ' G 0 carrying . C 0 ; B 0 / onto .C 0 ; B 0 / by composing the standard k 0 =k-descent isomorphism G 0 ' G 0 with conjugation against a choice of representative n 2 NG 0 .T 0 /.k 0 / of w ; note that f 0 is only well-defined up to conjugation against C 0 .k 0 / (upon varying the choice of n ). The relation (C.2.9.1) is satisfied by ff 0 g 2€ due to Proposition 6.3.4(i) since the uniqueness of the w ’s ensures that w  D w
.w /. Note that the intervention of the n ’s is a possible obstruction to ff 0 g 2€ being a k 0 =k-descent datum (so we needed to formulate (C.2.9.1) with the intervention of PsInn.G 0 /). The k 0 -group Ga0 is generally not equipped with a descent to a ka -group, but it fits into the framework of (C.2.9.1) relative to k 0 =ka . Indeed, if 2 € then 0 conjugation by n on G 0 carries G .a/ onto G 0 a (see Definition 6.3.3), so f 0 restricts to a composite isomorphism 0 0 f ;a W .Ga0 / ' G .a/ ' G 0 a :

For the pseudo-Borel k 0 -subgroup Ba0 WD PGa0 .a_ / D Ga0 \ B 0 of Ga0 containing 0 carries .C 0 / onto C 0

0 0 the Cartan k 0 -subgroup Ca0 , f ;a a

a and .Ba / onto B a . 0 W .G 0 / ' G 0 carrying the pair Hence, for 2 €a WD Gal.k 0 =ka / we get f ;a a a

0

0 0 0 . .Ba /; .Ca // over to .Ba ; Ca / and satisfying (C.2.9.1) for Ga0 relative to k 0 =ka . 0 ’s for 2 € are As in C.2.9, the restrictions h ;a W .Ca0 / ' Ca0 of the f ;a a 0 independent of the choice of n ’s and constitute a k =ka -descent datum; we let

222

Pseudo-split and quasi-split forms

Za denote the associated ka -descent. For ;  2 €a we have 0 0 0 f;a D za .; /
.f;a ı .f;a //;

where the automorphism za .; / 2 ZGa0 ;Ca0 .k 0 / is the restriction to Ga0 ,! G 0 of z.;  / 2 ZG 0 ;C 0 .k 0 / D Z .k 0 /. In other words,  0 .z.; // D .za .; //a2 . Let fai g be a set of representatives in  for the orbits of the -action, and define ki D kai  k 0 , Gi0 D Ga0 i , Ci0 D ZGi0 .ai_ / D Gi0 \ C 0 , Bi0 D PGi0 .ai_ / D Gi0 \ B 0 , and Zi D Zai . Clearly Bi0 is a pseudo-Borel k 0 -subgroup of Gi0 and the k 0 -isomorphism  0 descends to a k-isomorphism Y  WZ ' Rki =k .Zi /: (C.2.10) i

Note that the natural quotient map Rki =k .Zi /ki  Zi is induced by the natural quotient map ki ˝k ki ! ki defined by multiplication. Thus, the canonical quotient k 0 of ki ˝k k 0 arising from the inclusion defining ki inside k 0 over k identifies Zi .k 0 / as the quotient of Z .k 0 / D ZG 0 ;C 0 .k 0 / induced by the natural projection ZG 0 ;C 0 ! ZGi0 ;Ci0 arising from restriction of automorphisms due the definition of Zi (using Bi0 ). Under the isomorphism Y Y H2 .k 0 =k; Z .k 0 // D H2 .k 0 =k; Zi .ki ˝k k 0 // ' H2 .ki ; Zi .k 0 // provided by (C.2.10) and Shapiro’s Lemma, the i th projection is induced by restricting 2-cocycles along Gal.k 0 =ki / ,! Gal.k 0 =k/ and composing with the map Z .k 0 /  Zi .k 0 / as just considered, so c.G 0 ; C 0 / D .c.Gi0 ; Ci0 //i . Hence c.G 0 ; C 0 / D 0 if c.Gi0 ; Ci0 / D 0 for all i ; we shall prove the latter. For each i , the k 0 -group Gi0 is not provided with a k-structure but for every 0

2 €i WD Gal.k 0 =ki / the -twist of Gi0 admits the k 0 -isomorphism f ;a onto Gi0 . i Moreover, Gi0 is pseudo-split over k 0 and is absolutely pseudo-simple of minimal type with rank 1 and maximal geometric reductive quotient SL2 . Since A1 and BC1 have no nontrivial diagram automorphisms, by Proposition 6.3.4(i) (applied to Gi0 ) we have PsInn.Gi0 / D Aut.Gi0 /, so trivially the condition (C.2.9.1) is 0 g satisfied by ff ;a for each i . i 2€i Lemma C.2.11 below is a rank-1 refinement of our original problem for pseudo-inner forms of k-groups (it concerns pseudo-split k 0 -groups not equipped with a k-structure), and it applies to each .Gi0 ; k 0 =ki / to prove that Gi0 admits a pseudo-split k 0 =ki -descent for all i , so every c.Gi0 ; Ci0 / vanishes. 

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223

Lemma C.2.11. Let k 0 =k be a finite Galois extension, and G 0 a pseudo-split and absolutely pseudo-simple k 0 -group of minimal type such that G 0 kss0 ' SL2 . If G 0 is not standard and char.k/ D 2 then assume Œk W k 2  6 4. If G 0 is k 0 -isomorphic to each of its Gal.k 0 =k/-twists then G 0 descends to a pseudo-split pseudo-semisimple k-group. Proof. The possibilities for such G 0 are classified up to k 0 -isomorphism (functorially in k 0 =k) using inseparable field extensions and some linear algebra data. The isomorphism hypothesis with Galois twists is expressable in terms of an equivalence relation on these fields and linear algebra data. We have to show that any such equivalence class of data over k 0 admits a representative that descends to k. Let ˆ be the root system for G 0 (either A1 or BC1 , the latter possible only if char.k/ D 2). Note that G is standard if char.k/ ¤ 2, by Proposition 3.1.9. Case (1) (Standard case, any characteristic): The k 0 -group G 0 is classified by the minimal field of definition K 0 =k 0 for the geometric unipotent radical, which can be any purely inseparable finite extension. Explicitly, G 0 D RK 0 =k 0 .SL2 /. Choose a k 0 -isomorphism G 0 ' G 0 for each  2 Gal.k 0 =k/. This defines k 0 isomorphisms between K 0 and its Gal.k 0 =k/-twists; such isomorphisms are unique (in contrast with the k 0 -group isomorphisms) since K 0 =k 0 is purely inseparable, so we get a k 0 =k-descent datum on K 0 . Hence, there is an extension K of k such that k 0 ˝k K ' K 0 as k 0 -algebras, so K=k is purely inseparable of finite degree and RK=k .SL2 / is a pseudo-split k-descent of G 0 . Case (2) (char.k/ D 2, Œk W k 2  6 2): The case of perfect k is trivial, so assume Œk W k 2  D 2. The cases with ˆ D A1 are part of Case (1) (see Proposition 3.1.8), and by [CGP, Thm. 9.9.3(1)] the cases with ˆ D BC1 are classified up to isomorphism over k 0 by the minimal field of definition K 0 =k 0 of the geometric unipotent radical (which can be any nontrivial purely inseparable finite extension). Hence, the BC1 cases proceed as in Case (1). Case (3) (char.k/ D 2, Œk W k 2  D 4, ˆ D A1 ): In these cases G 0 is classified by pairs .K 0 =k 0 ; V 0 / where K 0 is a purely inseparable finite extension of k 0 and V 0 is a nonzero k 0 K 0 2 -subspace of K 0 such that k 0 hV 0 i D K 0 . Explicitly, by Proposition 3.1.8, G 0 D HV 0 ;K 0 =k 0 with K 0 the minimal field of definition over k 0 for the geometric unipotent radical of G 0 , and k 0 -isomorphism classes correspond to K 0  -scaling on V 0 . Hence, .V 0 / is a multiple of V 0 (by an element of K 0 ) for every 2 Gal.k 0 =k/. As in Case (1) we have K 0 D k 0 ˝k K for a purely inseparable finite extension K=k. We seek a multiple of V 0 that descends to a kK 2 -subspace V of K. The case V 0 D K 0 is trivial, so we may assume V 0 ¤ K 0 . In particular, ŒK 0 W k 0 K 0 2  > 2 (since k 0 hV 0 i D K 0 ), so clearly dimk 0 K 0 2 .V 0 / > 2.

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The degree ŒK 0 W K 0 2  is equal to 4 (as ŒE W E p  is invariant under finite extension of any field E of characteristic p > 0, as explained in Remark 8.3.1), so k 0  K 0 2 since otherwise ŒK 0 W k 0 K 0 2  6 2. This forces dimK 0 2 .V 0 / D 3, so V 0 is a K 0 2 -hyperplane in K 0 . Since HomK 0 2 .K 0 ; K 0 2 / is 1-dimensional over K 0 , the descent after suitable K 0  -scaling is obvious. Case (4) (char.k/ D 2, Œk W k 2  D 4, ˆ D BC1 ): It follows from [CGP, Thm. 9.8.6, Prop. 9.8.9] (with n D 1) that G 0 is classified up to k 0 -isomorphism by triples .K 0 =k 0 ; V 0 ; W 0 / consisting of a nontrivial purely inseparable finite extension K 0 =k 0 , a nonzero k 0 K 0 2 -subspace V 0  K 0 , and a K 0 2 -subspace W 0  K 0 containing V 0 such that W 0 =V 0 is nonzero with finite K 0 2 -dimension and k 0 hW 0 i D K 0 . Such triples yield k 0 -isomorphic groups if and only if the field data coincide and the linear algebra data are related via scaling by a common element of K 0  . Once again there is a k-descent K of K 0 and ŒK W K 2  D Œk W k 2  D 4. By hypothesis .V 0 ; W 0 / is a K 0  -multiple of each of its Galois twists, and we seek a pair .V; W / relative to K=k such that .Vk 0 ; Wk 0 / is a K 0  -multiple of .V 0 ; W 0 /. First suppose the root field for V 0 is K 0 2 , so k  K 2 and K 0 D K 0 2 hW 0 i. The nonzero proper K 0 2 -subspace V 0 of the 4-dimensional K 0 2 -vector space K 0 is either a line or a hyperplane over K 0 2 since otherwise it is a line over a quadratic extension of K 0 2 inside K 0 , contradicting that its root field is K 0 2 . Since the sets of K 0 2 -lines and K 0 2 -hyperplanes in K 0 each constitute a single K 0  -homothety class, some K 0  -multiple of .V 0 ; W 0 / descends if W 0 D K 0 . Thus, assume W 0 ¤ K 0 , so W 0 is a K 0 2 -hyperplane in K 0 (any smaller K 0 2 dimension for W 0 would contradict that ŒK 0 W K 0 2  D 4 and K 0 2 hW 0 i D K 0 ). We may apply K 0  -scaling so that W 0 descends to a K 2 -hyperplane W in K. Thus, the only scaling factors in K 0  carrying the pair .V 0 ; W 0 / to a Galois twist are elements of .K 0 2 / , so V 0 is Gal.k 0 =k/-stable and hence descends. This settles all cases with V 0 having root field K 0 2 . Finally, assume V 0 is a subspace over a subfield F 0  K 0 strictly containing 2 K 0 , so ŒK 0 W F 0  D 2 and dimF 0 .V 0 / D 1 (since ŒK 0 W K 0 2  D 4 and W 0 =V 0 ¤ 0). Necessarily F 0 D K 0 2 hV 0 i, so F 0 descends to a quadratic extension F of K 2 inside K since V 0 is a K 0  -multiple of its Gal.k 0 =k/-twists. The quotient K 0 =V 0 is an F 0 -line (hence a K 0 2 -plane), and we can arrange V 0 D F 0 via K 0  -scaling. The nonzero K 0 2 -subspace W 0 =V 0 in the F 0 -line K 0 =V 0 is either the entire space or can be moved to any K 0 2 -line via suitable F 0  -scaling, so some F 0  multiple of .V 0 ; W 0 / descends.  The following result provides a refinement to Proposition C.1.3, and examples in §C.3–§C.4 show that there is no room for further improvement concern-

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ing the general existence of pseudo-split ks =k-forms. Corollary C.2.12. If k is imperfect with characteristic 2 and Œk W k 2  6 4 then an absolutely pseudo-simple k-group G admits a pseudo-split ks =k-form except possibly if G is standard with root system over ks equal to D2n for some n > 2. Proof. By Proposition C.1.3, we may assume that G is not standard and does not have root system F4 , so its irreducible root system is of type B, C, or BC with rank n > 1. Since Œk W k 2  6 4, a quasi-split ks =k-form is provided by Theorem C.2.10. By Lemma C.2.2, this ks =k-form is pseudo-split.  Remark C.2.13. Inspired by Proposition C.1.3 and Corollary C.2.12 (and Remark 6.3.16), a problem that naturally arises in connection with the Tits-style classification theorem (Theorem 6.3.11) is to determine when an absolutely pseudo-semisimple ks -group G descends to a pseudo-split k-group G. This is not a matter of constructing ks =k-forms since we are not given any k-group. Let ˆ be the irreducible root system of G , and let  be a basis of ˆ. A necessary condition is that for every c 2  the minimal field of definition over ks for the geometric unipotent radical of Gc descends (necessarily uniquely up to unique isomorphism) to a purely inseparable finite extension Kc =k. Assuming such a Kc =k exists for every c, let K=k denote their compositum (inside the perfect closure of k). In the standard case we have G D RK=k .G 0 /ks =Z for the split simply connected K-group G 0 with root system ˆ and a closed ks -subgroup Z  RK=k .ZG 0 /ks , so by [CGP, Prop. A.5.14] existence amounts to Z arising from a k-subgroup of RK=k .ZG 0 /. When G is not of minimal type, so ZG 0 is not K-étale, then such descent for Z appears to be difficult to control. Now assume G is of minimal type (standard or not); this avoids the ks -fiber of the type-D2n construction in Proposition C.1.4. By Theorem 3.4.1 (when ˆ is reduced) and [CGP, Thm. 9.8.6, Prop. 9.8.9] (when ˆ is non-reduced), it is necessary and sufficient that Gc descends to a pseudo-split k-group for all c 2 . However, we seek a more convenient criterion in terms of fields and linear algebra data. By Proposition 3.1.8 and Theorem 3.4.1, under the above running assumptions the pseudo-split k-descent exists except possibly when k is imperfect of characteristic 2 and Gc is non-standard for some c 2 , in which case ˆ is of type Bn , Cn , or BCn for some n > 1. In these remaining cases a further necessary condition is that the Ks -homothety class of linear algebra data inside Ks classifying the rank-1 group Gc for non-divisible c 2 ˆ should be stable (as a homothety class!) under the action of Gal.Ks =K/ D Gal.ks =k/. This additional necessary condition is not sufficient when Œk W k 2  > 4 and k admits a quadratic

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Pseudo-split and quasi-split forms

Galois extension: counterexamples of every type B, C, or BC arise via the ks fiber of the k-groups (without a pseudo-split ks =k-form) built in §C.3–§C.4 for such k. However, if Œk W k 2  6 4 then sufficiency holds, as shown by the arguments in Cases (2)–(4) at the end of the proof of Lemma C.2.11.

C.3 Rank-1 cases Let k be an imperfect field with char.k/ D 2, and consider absolutely pseudosimple k-groups G of minimal type with root system A1 over ks such that G ss D SL2 . Let K=k be the minimal field of definition for Ru .Gk /  Gk , and k ss define G 0 D GK , so iG W G ,! RK=k .G 0 / has trivial kernel and G 0 is simply connected of rank 1. If Œk W k 2  > 8 then under some Galois-theoretic hypotheses on k we will now construct such a G having root field k and having no pseudosplit ks =k-form. (In §C.4 this example is used to build analogues with root system over ks equal to Bn , Cn , BCn for any n > 1.) Example C.3.1. Assume Œk W k 2  > 8 and that k admits a quadratic Galois extension k 0 . (Eventually we will also assume that Br.k/ ! Br.k 0 / has nontrivial kernel, as can be arranged in many cases, but we postpone the introduction of Brauer groups until later in the construction.) We may and do choose K  k 1=2 such that ŒK W k D 8. Let ft1 ; t2 ; t3 g be a 2-basis for K=k. Choose a 2 k 0 k (so k 0 D k.a/). Let  be the nontrivial k-automorphism of k 0 ;  will also denote the nontrivial K-automorphism of K 0 WD k 0 ˝k K. The 4-dimensional k 0 -subspace V 0 D k 0 C k 0
t1 C k 0
.t2 C at3 / C k 0
t1 .t2 C .a/t3 /  K 0 satisfies  .V 0 / D t1 V 0 (since t12 2 k  ), and k 0 ŒV 0  D K 0 . We claim that F 0 WD f0 2 K 0 j 0
V 0  V 0 g is equal to k 0 . Choose x 2 F 0 . Since 1 2 V 0 , so x D x
1 2 V 0 , we can assume x D c1 t1 C c2 .t2 C at3 / C c3 t1 .t2 C .a/t3 / for c1 ; c2 ; c3 2 k 0 and we want x D 0. The k 0 -subspace V 0 contains xt1 , but xt1 2 W 0 WD k 0 C k 0
.t2 C .a/t3 / C k 0
t1 .t2 C at3 /: Using the 2-basis property for ft1 ; t2 ; t3 g relative to K 0 over k 0 we see that the intersection V 0 \ W 0 equals k 0 , so x 2 k 0 t1 yet t1 62 F 0 , so x D 0 as desired.

C.3 Rank-1 cases

227

The k 0 -subgroup HV 0 ;K 0 =k 0  RK 0 =k 0 .SL2 / admits a k-descent G (necessarily of minimal type) via the isomorphism   .HV 0 ;K 0 =k 0 / ' H .V 0 /;K 0 =k 0 D H t1 V 0 ;K 0 =k 0 ' HV 0 ;K 0 =k 0 defined in terms of conjugation on RK 0 =k 0 .SL2 / D RK=k .SL2 /k 0 by the k-point 

 0 t1 2 RK=k .PGL2 /.k/ D PGL2 .K/: 1 0

By Galois descent, the minimal field of definition over k for Ru .Gk /  Gk ss is K=k. Moreover, the maximal reductive quotient GK is the K-form of SL2 1 0 0 arising from the class in H .K =K; PGL2 .K // represented by the 1-cocycle c W  7! . 01 t01 / 2 PGL2 .K/  PGL2 .K 0 /. Suppose G admits a pseudo-split ks =k-form. This form is k-isomorphic to HV;K=k for a nonzero k-subspace V  K satisfying khV i D K, and Proposition C.1.1 applied over k 0 gives HVk0 ;K 0 =k 0 ' HV 0 ;K 0 =k 0 with Vk 0 WD k 0 ˝k V . Necessarily Vk 0 D V 0 for some  2 K 0  , but .Vk 0 / D Vk 0 and  .V 0 / D ./.V 0 / D ./t1 V 0 ; so . ./=/t1 2 F 0 D k 0 . Hence, t1 2 .=.//k 0  . Applying NK 0 =K then gives that the element t12 2 k  lies in Nk 0 =k .k 0  /. In view of how K and ft1 ; t2 ; t3 g were chosen, we can arrange for a contradiction (and hence conclude that G does not admit a pseudo-split ks =k-form) provided that Nk 0 =k .k 0  / ¤ k  (for then we may choose K to contain t1 2 k 1=2 such that t12 62 Nk 0 =k .k 0  /, and t1 62 k since .k  /2  Nk 0 =k .k 0  /); this is equivalent to the condition that ker.Br.k/ ! Br.k 0 // is nontrivial. For example, if k WD F.z1 ; z2 ; z3 / is a rational function field in 3 variables over a finite field F of characteristic 2 and k 0 D L.z2 ; z3 / for a separable quadratic extension L=F.z1 / then the kernel of Br.F.z1 // ! Br.L/ is nontrivial by global class field theory and it injects into ker.Br.k/ ! Br.k 0 //. ss ss The description of GK in terms of H1 .K 0 =K; PGL2 .K 0 // shows that GK is the algebraic group of norm-1 units in the quaternion algebra over K associated to the class t1 2 K  =NK 0 =K .K 0  /  Br.K/Œ2. But t1 is not a norm from K 0 ss since (by design) the element t12 2 k  is not a norm from k 0 , so GK is Kanisotropic.

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C.4 Higher-rank and non-reduced cases Now we adapt the construction in Example C.3.1 to give additional examples of absolutely pseudo-simple G over k of minimal type which do not admit a pseudo-split ks =k-form and whose root system (over ks ) is of type B, C, or BC of any rank. No such G has a quasi-split ks =k-form (by Lemma C.2.2, since the Dynkin diagrams of these root systems have no nontrivial automorphism). Example C.4.1 (B and C). Let k be an imperfect field of characteristic 2 such that Œk W k 2  > 8 and k admits a quadratic Galois extension k 0 =k (with nontrivial automorphism denoted as ). Also assume Br.k/ ! Br.k 0 / has nontrivial kernel. Choose K=k as in Example C.3.1 with a 2-basis ft1 ; t2 ; t3 g such that the element t12 2 k  is not a norm from k 0 , so we obtain a 4-dimensional k 0 -subspace V 0  K 0 WD k 0 ˝k K such that HV 0 ;K 0 =k 0 has a k-descent with no pseudo-split ks =kform and root field k. The k-descent is defined by using a descent datum on the k 0 -subgroup HV 0 ;K 0 =k 0  RK 0 =k 0 .SL2 / via  D . 01 t01 / 2 PGL2 .K 0 /. That is, we use the Galois descent datum on the k 0 -group HV 0 ;K 0 =k 0 obtained by composing the canonical k 0 -isomorphism   .HV 0 ;K 0 =k 0 / ' H .V 0 /;K 0 =k 0 D H t1 V 0 ;K 0 =k 0 with the natural action of  on RK 0 =k 0 .SL2 / (which carries H t1 V 0 ;K 0 =k 0 onto HV 0 ;K 0 =k 0 , restricting to inversion on the common diagonal Cartan k 0 -subgroup). For n > 2, consider generalized basic exotic pseudo-split absolutely pseudosimple k 0 -groups .G 0 ; T 0 / with root system ˆ D ˆ.G 0 ; T 0 / equal to Bn or Cn . By Proposition 8.2.5, up to k 0 -isomorphism there is a unique G 0 such that Gc0 ' HV 0 ;K 0 =k 0 for c 2 ˆ< when ˆ D Bn with n > 2 and for c 2 ˆ> when ˆ D Cn with n > 3. Let L0  G 0 be a Levi k 0 -subgroup containing T 0 ,  a basis of ˆ, and c0 the unique short root in  when ˆ D Bn with n > 2 and the unique long root in  when ˆ D Cn with n > 3. Choose a pinning of .L0 ; T 0 ; / so that the Levi k 0 -subgroup SL2 ' L0c0 ,! Gc0 0 identifies Gc0 0 with HV 0 ;K 0 =k 0 via iGc0 . 0 By Theorem 6.1.1, there is a unique Galois descent datum on G 0 relative to k 0 =k whose effect on Gc0 0 D HV 0 ;K 0 =k 0 is as above and whose effect on Gc0  RK 0 =k 0 .SL2 / for c 2  fc0 g is the composition of the canonical descent datum with transpose-inverse (having the effect of inversion on the diagonal Cartan k 0 -subgroup). This descent datum preserves the split k 0 -torus T 0 and defines a descent of .G 0 ; T 0 / to a pair .G; T / over k where G is of minimal type and T splits over k 0 with the nontrivial element of Gal.k 0 =k/ acting on X .Tk 0 / via inversion. In particular, for every c 2 ˆ.Gk 0 ; Tk 0 / D ˆ the absolutely pseudosimple k 0 -subgroup Gc0  G 0 D Gk 0 of rank 1 generated by the ˙c-root groups is preserved by the k 0 =k-descent datum and so descends to an absolutely pseudo-

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229

simple k-subgroup Gc  G of rank 1. Suppose there is a pseudo-split ks =k-form H of G, say with a pseudosplit maximal k-torus S, so Hk 0 and Gk 0 are pseudo-split and ks0 =k 0 -forms of each other. By Proposition C.1.1, there is a k 0 -isomorphism Hk 0 ' Gk 0 D G 0 . This isomorphism can be chosen to carry Sk 0 over to Tk 0 D T 0 , so we may identify ˆ.H; S / with ˆ. This defines Hc for all c 2 ˆ, and .Hc0 /k 0 ' Gc0 0 D HV 0 ;K 0 =k 0 , so Hc0 is a pseudo-split k-descent of HV 0 ;K 0 =k 0 . But by construction of .V 0 ; K 0 =k 0 / in Example C.3.1, HV 0 ;K 0 =k 0 has no such k-descent. Thus, there is no such H ; i.e., G does not admit a pseudo-split ks =k-form. As Gk 0 D G 0 , the minimal field of definition over k for the geometric unipotent radical of G ss is K=k and GK is a simply connected semisimple K-group that splits over K 0 (with root system ˆ). Example C.4.2 (BC). Let .K=k; V 0 ; / be as in Example C.3.1, and let W 0 be a k 0 -linear complement of V 0 in K 0 , so dimk 0 W 0 D 4. Define 2

2

V .2/ D K 0 w 0 ˚ K 0 t1 .w 0 /  k 0 w 0 ˚ k 0 t1 .w 0 / for a nonzero w 0 2 W 0 . To justify the description of V .2/  K 0 inside a direct sum over k 0 , note that if t1 .w 0 / D 0 w 0 for some 0 2 k 0  then .w 0 /=w 0 D 0 =t1 , so applying NK 0 =K yields t12 D Nk 0 =k .0 /, an absurdity. Observe that  .V .2/ / D t1 V .2/ since t12 2 k  , and t1 .w 0 / 62 V 0 (since  .V 0 / D t1 V 0 D t1 1 V 0 and W 0 \V 0 D 0), so V .2/ \V 0 is a proper K 0 2 -subspace of the K 0 2 -plane V .2/ . Thus, if V .2/ \ V 0 ¤ 0 then V .2/ \ V 0 is spanned over K 0 2  k 0 by w 0 C ˛ 2 t1  .w 0 / for some ˛ 2 K 0  . We may and do choose W 0 to contain w 0 WD t1 t2 . Since ft1 ; t2 ; t3 g is a 2basis for K 0 over k 0 , it follows that V .2/ \ V 0 D 0. Note that V .2/ ˚ V 0 contains 1 and generates K 0 as a k 0 -algebra since the same holds for V 0 . Hence, for arbitrary n > 1 there exists a pseudo-split pseudo-semisimple k 0 -group G 0 D D.GK 0 =k 0 ;V 0 ;V .2/ ;n / of minimal type with root system BCn as in [CGP, Thm. 9.8.1(1)]. The k 0 -group G 0 is equipped with a standard split maximal k 0 -torus D 0 , and 0 if  is a basis of ˆ.G 0 ; D 0 / with unique multipliable element c0 then G2c D 0 0 0 0 HV ;K =k . Thus, by using [CGP, Prop. 9.8.9(i)] with n D 1 (applied to Gc0 0 0 and its  -twist) and Theorem 6.1.1, the Galois-twisting construction on G2c in 0 0 0 Example C.3.1 extends to a descent datum on .G ; D / that is the composition of transpose-inverse with standard descent datum on Gc0 D RK 0 =k 0 .SL2 / for all c 2  fc0 g. By the same reasoning as in Example C.4.1, the k-descent G of G 0 does not have a pseudo-split ks =k-form.

Appendix D Basic exotic groups of type F4 of relative rank 2

Let k be an imperfect field of characteristic 2, and consider a basic exotic kgroup H such that Hks has root system F4 . The minimal field of definition K=k for the geometric unipotent radical of H is a nontrivial purely inseparable finite extension of k satisfying K 2  k, and H.k 0 / D Autk 0 .Hk 0 / for all separable extensions k 0 =k by [CGP, Cor. 8.2.5]. Fix a nontrivial purely inseparable finite extension K=k inside k 1=2 , so there exists a pseudo-split basic exotic k-group G with root system F4 such that the minimal field of definition for Ru .Gk /  Gk is K=k; G is unique up to k-isomorphism by Proposition C.1.1. The possibilities up to k-isomorphism for G with the associated K=k are exactly the ks =k-forms of G , and the nonpseudo-split forms are those G with k-rank less than 4. It is classical over an arbitrary field (see [Spr, 17.5.1–17.5.2]) that a connected semisimple group of type F4 which is neither anisotropic nor split has relative rank 1. In contrast, we will see below that for suitable k there do exist G as above with k-rank 2. Such k-groups give rise to Moufang quadrangles “of type F4 ” as defined in (16.7) of [TW] (see also [TW, Ch. 14, (17.4)] as well as [MV]). In this appendix we use techniques developed in this monograph and [CGP] to show that k-rank 3 cannot occur and to classify the isomorphism classes of ks =k-forms G of k-rank 2 in terms of conformal isometry classes of certain anisotropic regular degenerate quadratic spaces over k.

D.1 General preparations D.1.1. Let S be a maximal k-split torus in G and let P be a minimal pseudoparabolic k-subgroup of G containing S . Since G is pseudo-split, every pseudoparabolic ks -subgroup of Gks is G .ks /-conjugate to one that descends to a pseudo-parabolic k-subgroup of G . Thus, we can choose a pseudo-parabolic ksubgroup P of G such that there is a ks -isomorphism .Gks ; Pks / ' .Gks ; Pks /. Since G .ks / D Autks .Gks / and P is its own normalizer in G [CGP, Thm. 3.5.7],

231

D.1 General preparations

it follows that the pair .G; P / is obtained from .G ; P/ via twisting by a continuous P.ks /-valued 1-cocycle on Gal.ks =k/. Let U D Ru;k .P/, so U is k-split, and let M be a smooth closed ksubgroup of P such that U o M ! P is an isomorphism. (For example, we can use M D ZG ./ for a 1-parameter k-subgroup  of G such that P D PG ./.) Let M 0 be the derived group of M . Every ks =k-form of U is k-split by [CGP, Thm. B.3.4], so the twist of U by any continuous M .ks /-valued 1-cocycle on Gal.ks =k/ is k-split. Hence, the natural injective map H1 .k; M / ! H1 .k; P/ is bijective. In particular, we can find a continuous 1-cocycle c W Gal.ks =k/ ! M .ks / that twists .G ; P/ into .G; P /, and using that to define the initial isomorphism .Gks ; Pks / ' .Gks ; Pks / above ensures that this isomorphism carries Mks over to Mks for a smooth closed k-subgroup M of P . For U WD Ru;k .P /, clearly P D U o M and so the natural map M ! P =U is an isomorphism. We now have an isomorphism of triples .Gks ; Pks ; Mks / ' .Gks ; Pks ; Mks /: (D.1.1) Remark D.1.2. The only relevant property of the Galois extension ks =k above is that Gks is pseudo-split. Later we will encounter situations in which GF is pseudo-split for a quadratic Galois extension F=k, so we can use F in place of ks when building (D.1.1). By [CGP, Prop. C.2.4], minimality of P in G implies P D U o ZG .S/ for any maximal split k-torus S of G lying in P . The isomorphism M ' P =U implies that such an S can be found inside M , so we make such a choice of S. Choose a maximal k-torus T in M D ZG .S/, so S  T and (D.1.1) carries Tks over to a maximal ks -torus in Mks . Since Tks is ks -split and M contains a split maximal k-torus T , the M .ks /-conjugacy of split maximal ks -tori in Mks allows us to choose (D.1.1) so that it also carries Tks over to Tks . Fix a basis  of ˆ.Gks ; Tks / contained in the parabolic subset ˆ.Pks ; Tks /. We enumerate the basis  as fa1 ; a2 ; a3 ; a4 g corresponding to the Dynkin diagram

a1



a2



3+ a3

a4



Let 0 be the set of elements of  that are trivial on the subtorus Sks  Tks , and call an element of  distinguished if it lies in  0 . As shown in the proof of [CGP, Thm. C.2.15], the restrictions to S of the distinguished roots in  constitute the basis k  for the positive system of roots ˆ.P; S/ of k ˆ D

Basic exotic groups of type F4 of relative rank 2

232

ˆ.G; S /, and the fibers of  0  k  are the orbits under the -action of Gal.ks =k/ on  0 . The F4 -diagram has no nontrivial automorphisms, so the -action is trivial and hence  0  k  is bijective. Since (D.1.1) carries Tks over to Tks , it identifies  with a basis of ˆ WD ˆ.G ; T /. The root system ‰ D ˆ.M ; T / consists of the roots in ˆ which are Z-linear combination of roots in 0 , so the Dynkin diagram of ‰ is the subdiagram of  with vertices given by 0 . Let f‰i g be the set of irreducible components of ‰. By [CGP, Thm. 3.4.6] there exists a Levi k-subgroup L  G containing the k-split maximal torus T , and since L is simply connected we see that its k-subgroup Li generated by the root groups corresponding to the roots in ‰i is simply connected and ˆ.Li
T ; T / D ‰i . The set _ is a basis of X .T /, so the k-subgroup of L generated by T and the Li ’s is a semi-direct product Q Q of c2 0 c _ .GL1 / against i Li . The analogue then holds over K for the pair .LK ; TK /, so the inclusion iG W G ,! RK=k .GKss / D RK=k .LK / shows via inspection of the open cell of the basic exotic G of type F4 (see [CGP, Rem. 7.2.8]) that Y Y M D M 0
ZG .T / D Mi0 o RKc =k .GL1 / (D.1.2) i

c2 0

where Kc D K for short c 2 , Kc D k for long c 2 , and Mi0  RK=k ..Li /K / is the k-subgroup of M generated by the root groups Ua for a 2 ‰i . We will now show that G cannot be of k-rank 3. More generally: Proposition D.1.3. If a2 or a3 is distinguished then G is pseudo-split. Proof. Assume G is not pseudo-split, so H1 .k; M / ¤ 1 and ‰ is not empty (cf. Lemma C.2.2). The omission of a2 or a3 from 0 implies that each ‰i is of type A1 or A2 . Thus, each Li is either SL2 or SL3 . Define Ki D K if ‰i consists of short roots in ˆ.G ; T / and Ki D k if ‰i consists of long roots in ˆ.G ; T /, so Mi0 ' RKi =k ..Li /Ki /. Hence, (D.1.2) gives MD

Y i

RKi =k ..Li /Ki / o

Y

RKc =k .GL1 /;

c2 0

so the non-abelian Shapiro’s Lemma yields an exact sequence of pointed sets Y Y H1 .Ki ; .Li /Ki / ! H1 .k; M / ! H1 .Kc ; GL1 /: i

c

D.2 Forms of k-rank 2

The outer terms vanish, contradicting that H1 .k; M / ¤ 1.

233



Remark D.1.4. By the same method we see that if k is imperfect of characteristic 3 and G is a basic exotic k-group of type G2 that is k-isotropic (i.e., some root is distinguished) then G is pseudo-split. This is analogous to the classical fact [Spr, 17.4.2] that isotropic connected semisimple groups of type G2 over a field are always split.

D.2 Forms of k-rank 2 D.2.1. Assume G is k-isotropic but not pseudo-split, so by Proposition D.1.3 the roots a2 and a3 are non-distinguished. The k-rank of G is therefore equal to 1 or 2. We now focus on the case of k-rank 2, so the distinguished vertices are precisely a1 and a4 . The minimal pseudo-parabolic k-subgroup P is equal to U o ZG .S / with k-split U D Ru;k .P /, and the derived group M 0 of M WD ZG .S / is k-anisotropic and of type B2 , corresponding to the subdiagram of  with vertex set 0 D fa2 ; a3 g. We denote the split maximal k-torus T \ M 0 of M 0 by S 0 . The structure of open cells of pseudo-split basic exotic groups of type F4 (such as G ) and the Isomorphism Theorem for pseudo-split pseudo-reductive groups (Theorem 6.1.1) imply that M 0 ' Spin.q/ for the orthogonal sum .V ; q/ D H1 ? H2 ? .K; x 2 /

(D.2.1)

over k where Hi is a hyperbolic plane and K is equipped with the quadratic form given by the squaring map into k (making K the defect space V ? ). By the M 0 .k/-conjugacy of split maximal k-tori in M 0 and the natural bijection (respecting k-rank) between the sets of maximal k-tori in Spin.q/ and its maximal central quotient SO.q/, the isomorphism M 0 ' Spin.q/ can be chosen so that 0 S 0 corresponds to the split maximal k-torus S in SO.q/ that keeps stable each of the two hyperbolic planes H1 and H2 under its action on V . We identify M 0 with Spin.q/ using such an isomorphism. The nontrivial weights of S 0 in the standard representation of Spin.q/ (via its image in SO.q/) on V are the short roots. Swapping H1 and H2 if necessary, we may assume that the weights on H1 are ˙a3 and on H2 are ˙.a2 C a3 /. Let e1 2 H1 be a weight vector for a3 , let e2 2 H2 be a weight vector for a2 C a3 , and let fi 2 Hi be the unique weight vector for the opposite weight to that of ei such that the value Bq .ei ; fi / 2 k  is equal to 1.

234

Basic exotic groups of type F4 of relative rank 2

Lemma D.2.2. There exists an involution  2 NM 0 .S 0 /.k/ inducing inversion on S 0 such that fi D .ei /. Proof. The F2 -span V0 of the ei ’s, fi ’s, and h D 1 2 K D V ? defines a nondegenerate quadratic space .V0 ; q0 / over F2 such that Spin.q0 / is naturally an F2 descent of the Levi k-subgroup L 0 D Spin.qjH1 ˚H2 ˚kh / of M 0 D Spin.q/ that contains S 0 as a split maximal k-torus. Under the resulting identification of the 0 adjoint central quotient SO.q0 / as an F2 -descent of the Levi k-subgroup L D 0 SO.qjH1 ˚H2 ˚kh / of SO.q/, it is clear that S descends to the split maximal F2 torus S 0  SO.q0 / consisting of elements preserving the lines spanned by each of e1 ; f1 ; e2 ; f2 ; h (with trivial action on h). The corresponding split maximal F2 -torus S0  Spin.q0 / is an F2 -descent of S 0  L 0 . It suffices to find an involution in NL 0 .S 0 /.k/ inducing inversion on S 0 and carrying ei to fi for i D 1; 2, so it is even sufficient to find such an involution in NSpin.q0 / .S0 /.F2 /. Since F 2 D 1, the natural map NSpin.q0 / .S0 /.F2 / ! NSO.q0 / .S 0 /.F2 / is identified with the map of Weyl groups W .Spin.q0 /; S0 / ! W .SO.q0 /; S 0 /; and this is an isomorphism since Spin.q0 / ! SO.q0 / is a central isogeny. Thus, we just need to find an involution  in NSO.q0 / .S 0 /.F2 / carrying ei to fi for i D 1; 2, and acting on the isogenous quotient S 0 of S0 via inversion. Define  2 GL.V0 / to be the automorphism that fixes h and swaps ei and fi for i D 1; 2. Clearly  is an involution that lies in O.q0 /.F2 /, and O.q0 / D SO.q0 /  2 since dim V0 is odd, so O.q0 / and SO.q0 / have the same points valued in any field. By design we see that -conjugation restricts to inversion on S 0 , so we are done.  We begin our analysis of the possibilities for G of k-rank 2 by showing that M 0 is the spin group of an anisotropic ks =k-form of .V ; q/; an explicit description of the possibilities for such quadratic spaces will be given later. Lemma D.2.3. There exists a regular degenerate anisotropic quadratic space .V; q/ over k such that M 0 ' Spin.q/, and .V; q/ is unique up to conformal isometry. Moreover, there exists a separable quadratic extension F=k such that MF is pseudo-split .so MF ' MF and the derived group MF0 is pseudo-split/. Since S is unique up to G.k/-conjugacy, the conformal isometry class of .V; q/ is uniquely determined by the k-isomorphism class of G. Proof. Recall that we have a continuous 1-cocycle c W Gal.ks =k/ ! M .ks / that twists .G ; P; M / into .G; P; M /. This compatibly twists the derived group

D.2 Forms of k-rank 2

M 0 D Spin.q/ into the derived group M 0 . As in (D.1.2), we have Y M D M0o RKc =k .GL1 /;

235

(D.2.3)

c2 0

so H1 .k; M 0 / ! H1 .k; M / is surjective and hence we can replace c with a cohomologous 1-cocycle to arrange that it is valued in M 0 .ks /. Via the maximal central quotient map M 0 D Spin.q/ ! SO.q/, c is carried onto a 1-cocycle c valued in SO.q/.ks /, so M 0 =ZM 0 ' SO.q/ where q is the c-twist of q. This latter k-isomorphism uniquely lifts to a k-isomorphism M 0 ' Spin.q/ between universal smooth k-tame central extensions, and q is anisotropic since M 0 is k-anisotropic (see Proposition 7.1.6). The uniqueness of .V; q/ up to conformal isometry follows from Proposition 7.2.2(i). Since char.k/ D 2 and q written in terms of linear coordinates must have a nonzero cross-term (as its defect space is a proper subspace), we can find a plane on which the restriction of the anisotropic q is not additive and hence is nondegenerate. Thus, there exists a separable quadratic extension F=k such that qF is isotropic. Clearly .M 0 =ZM 0 /F is F -isotropic, so MF0 is also F -isotropic. But M 0 commutes with S and the intersection M 0 \ S is finite, so GF has F -rank at least 3 and hence is pseudo-split by Proposition D.1.3. In particular, the F -group MF D ZG .S /F is pseudo-split.  D.2.4. We fix F=k as in Lemma D.2.3, so GF is pseudo-split. By Remark D.1.2, after possibly changing M inside P we obtain an isomorphism of triples .GF ; PF ; MF / ' .GF ; PF ; MF /

(D.2.4)

arising from a 1-cocycle c W Gal.F=k/ ! M .F /. (This retains the property that MF is pseudo-split, and every M  P has the form ZG .S/ for some S  P . The maximal k-torus T will no longer play any role, so it does not matter whether or not the above F -isomorphism carries TF to an F -torus of MF that descends to a k-torus of M .) Via (D.1.2), Hilbert’s Theorem 90 gives the surjectivity of H1 .F=k; M 0 .F // ! H1 .F=k; M .F //; so replacing c with a cohomologous 1-cocycle makes it valued in M 0 .F /. Next, we seek a maximal k-torus S 0 of M 0 that splits over F . For this purpose, fix a minimal pseudo-parabolic F -subgroup Q of MF0 , and let be the nontrivial k-automorphism of F . By [CGP, Prop. 3.5.12(2), Prop. C.2.7], the intersection Q \  .Q/ is a smooth connected F -subgroup that contains a split

236

Basic exotic groups of type F4 of relative rank 2

maximal F -torus T 0 of MF0 . The F -group Q \  .Q/ visibly descends to a k-subgroup H 0 of M 0 , so Q \  .Q/ D HF0 : Lemma D.2.5. The split maximal F -torus T 0 in HF0 is central. In particular, T 0 is unique and so descends to a maximal k-torus in H 0 that splits over F . Proof. Suppose to the contrary, so T 0 has a nontrivial weight a0 on Lie.HF0 /. Consider the associated root group Ua0 in HF0 as in [CGP, Def. 2.3.4]. This is a nontrivial F -split smooth connected unipotent F -subgroup of HF0 . Q/. As Ua0 is F -split and Q is a minimal pseudo-parabolic F -subgroup of the pseudosplit pseudo-semisimple F -group MF0 , it follows from [CGP, Cor. C.3.9] that Ua0  Rus;F .Q/. The subgroup U 0 of Hk0 s generated by the ks -split unipotent groups h.Ua0 /ks h 1 for varying h 2 H 0 .ks / is contained in Rus;ks .Qks /, so it is a smooth connected unipotent normal ks -subgroup of Hk0 s . By [CGP, Thm. B.3.4] a smooth connected unipotent ks -group is ks -split if it is generated by ks -split subgroups. Thus, U 0 is ks -split and hence U 0  Rus;ks .Hk0 s /. Since Rus;k .H 0 /ks D Rus;ks .Hk0 s /, we see that Rus;k .H 0 / ¤ 1. But H 0  M 0 , so M 0 contains the nontrivial split smooth connected unipotent k-subgroup Rus;k .H 0 /. This contradicts the k-anisotropicity of the pseudoreductive M 0 , by [CGP, Cor. C.3.9].  D.2.6. Let S 0 be a maximal k-torus of M 0 that splits over F . Any two F split maximal tori of MF0 are conjugate under M 0 .F /, so by applying this to SF0 and SF0  MF0 ' MF0 (using (D.2.4)) we can replace c with a cohomologous 1-cocycle so that it takes values in NM 0 .S 0 /.F / and the associated k-isomorphism M 0 ' c M 0 carries S 0 over to c S 0 . As S 0 is anisotropic over k and splits over the separable quadratic extension F , the effect of conjugation by c. / on SF0 is inversion. Likewise, the image c. / 2 SO.q/.F / D O.qF / of c. / 2 M 0 .F / D Spin.q/.F / fixes the defect space KF WD F ˝k K pointwise (due to Proposition 7.1.2) and must move each weight space for the action of the split rank-2 torus SF0 on VF WD F ˝k V to the weight space for the opposite weight (since c. / acts on SF0 through inversion). The involution  2 NM 0 .S 0 /.k/ from Lemma D.2.2 acts on S 0 via inversion, so c. / is a point of _ ZM 0 .S 0 /.F /
 D ..a2 /_ .GL1 /.F /  RK=k ..a3 /K .GL1 //.F //
:

D.2 Forms of k-rank 2

237

Thus, c. / D a2_ .s/a3_ .st /
 for unique s 2 F  and t 2 .F ˝k K/ . As e1 has weight a3 and e2 has weight a2 C a3 , the identities ha2 ; a3_ i D 2 and ha3 ; a2_ i D 1 imply a3 .a2_ .s/a3_ .st// D st 2 ; .a2 C a3 /.a2_ .s/a3_ .st// D s: Since 2 D 1, so .fi / D ei for each i (as .ei / D fi for each i ), we obtain c. /.e1 / D .st 2 /

1

f1 ; c. /.f1 / D st 2 e1 ; c. /.e2 / D s

1

f2 ; c. /.f2 / D se2 :

Hence, the cocycle condition c. /
.c. // D 1 is equivalent to the combined conditions s 2 k  and t 2 K  , which is to say c. / D a2_ .s/a3_ .st/
 2 NM 0 .S 0 /.k/:

(D.2.6.1)

Conversely, for s 2 k  , t 2 K  , the element a2_ .s/a3_ .st/
 2 NM 0 .S 0 /.k/ has order 2, so we can define a 1-cocycle c W Gal.F=k/ ! NM 0 .S 0 /.k/ such that c. / is this element. The action of c. / on ei and fi is as given above. The quadratic space .V; q/ obtained by twisting .V ; q/ by c is the restriction of qF to the k-subspace V of VF consisting of v such that c. /.
v/ D v. Let Hi D .F ˝k Hi / \ V , so V D H1 ˚ H2 ˚ K. The elements of H2 are the F -linear combinations ye2 C zf2 such that ye2 C zf2 D s .z/e2 C s

1

.y/f2 ;

which is to say that the elements of H2 are s .z/e2 C zf2 with z 2 F . But q.s .z/e2 C zf2 / D sNF =k .z/ since Bq .e2 ; f2 / D 1, and likewise qF jH1 D st 2 NF =k . Hence, .V; q/ is isomorphic to st 2 NF =k ? sNF =k ? .K; x 2 / (D.2.6.2) for a separable quadratic extension field F=k and scalars s 2 k  and t 2 K  . Assume .V; q/ is anisotropic, so the twist c M 0 ' Spin.q/ is k-anisotropic (by Proposition 7.1.6). Since P WD c P D c U o c M is a pseudo-parabolic ksubgroup of G WD c G and c M 0 is k-anisotropic, we see that P must be minimal as a pseudo-parabolic k-subgroup of G. The maximal tori of M 0 are 2dimensional, so G has k-rank equal to 4 2 D 2. We have shown: Proposition D.2.7. For every separable quadratic extension field F=k and elements s 2 k  and t 2 K  such that (D.2.6.2) is anisotropic, the homomorphism

238

Basic exotic groups of type F4 of relative rank 2

c W Gal.F=k/ ! NM 0 .S 0 /.k/ defined by 7! a2_ .s/a3_ .st/
 yields a twist G D c G that has k-rank equal to 2. Moreover, all basic exotic pseudo-reductive k-groups of type F4 with k-rank 2 and minimal field of definition K=k for the geometric unipotent radical arise in this manner. Example 7.1.7 provides k, K, and F such that for some s 2 k  and t 2 K  the associated quadratic form (D.2.6.2) is k-anisotropic. Hence, for such k and K there exist basic exotic k-groups G of type F4 whose k-rank is 2. By Lemma D.2.3, a necessary condition on such triples .F=k; s; t/ and 0 .F =k; s 0 ; t 0 / for the corresponding ks =k-forms of G to be k-isomorphic is that the respective associated quadratic spaces .V; q/ and .V 0 ; q 0 / as in (D.2.6.2) are conformal. Such conformality is also sufficient. Indeed, the “anisotropic kernel” M 0 D D.ZG .S // of the ks =k-form G arising from .F=k; s; t/ satisfies M 0 =ZM 0 D SO.q/, so by Proposition 7.2.2(i) specifying the conformal isometry class of .V; q/ is the same as specifying the k-isomorphism class of M 0 =ZM 0 . But by Theorem 6.3.11, the k-isomorphism class of G is determined by K=k and the k-isomorphism class of M 0 =ZM 0 (since the ks -isomorphism class of Gks is determined by the extension Ks D ks ˝k K of ks , and the Dynkin diagrams of Gks and Mk0 s do not admit nontrivial automorphisms).

Bibliography

[Ar]

M. Artin, “Brauer–Severi varieties” in Brauer groups in ring theory and algebraic geometry, LNM 917, Springer-Verlag, New York (1982), 194–210.

[Bo]

A. Borel, Linear algebraic groups (2nd ed.), Springer-Verlag, New York, 1991, 288pp.

[BLR]

S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, SpringerVerlag, New York, 1990, 325pp.

[Bou]

N. Bourbaki, Lie groups and Lie algebras (Ch. 4–6), Springer-Verlag, New York, 2002.

[C1]

B. Conrad, Finiteness theorems for algebraic groups over function fields, Compositio Math. 148 (2012), 555–639.

[C2]

B. Conrad, “Reductive group schemes” in Autour des schémas en groupes I, Panoramas et Synthèses 42–43, Société Mathématique de France, 2014.

[CGP]

B. Conrad, O. Gabber, G. Prasad, Pseudo-reductive groups (2nd ed.), Cambridge Univ. Press, 2015.

[SGA3] M. Demazure, A. Grothendieck, Schémas en groupes I, II, III, Lecture Notes in Math 151, 152, 153, Springer-Verlag, New York, 1970. [EKM] R. Elman, N. Karpenko, A. Merkurjev, The algebraic and geometric theory of quadratic forms, AMS Colloq. Publ. 56, 2008. [EGA]

A. Grothendieck, Eléments de Géométrie Algébrique, Publ. Math. IHES 24, 28, 32, 1966–7.

240

BIBLIOGRAPHY

[SGA2] A. Grothendieck, Cohomologie Locale des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et Globaux, North-Holland, Amsterdam, 1968. [Hum]

J. Humphreys, Linear algebraic groups (2nd ed.), Springer-Verlag, New York, 1987.

[KMRT] M-A. Knus, A. Merkurjev, M. Rost, J-P. Tignol, The book of involutions, AMS Colloq. Publ. 44, Providence, 1998. [Mat]

H. Matsumura, Commutative ring theory, Cambridge Univ. Press, 1990.

[MV]

B. Mühlherr, H. Van Maldeghem, Exceptional Moufang quadrangles of type F4 , Canad. J. Math. 51 (1999), 347–371.

[PY]

G. Prasad, J-K Yu, On quasi-reductive group schemes, Journal of algebraic geometry 15 (2006), 507–549.

[R]

B. Rémy, Groupes algébriques pseudo-réductifs et applications, Séminaire Bourbaki, Exp. 1021, Astérisque 339 (2011), 259–304.

[Spr]

T. A. Springer, Linear algebraic groups (2nd ed.), Birkhäuser, New York, 1998.

[Tits]

J. Tits, “Classification of algebraic semisimple groups” in Algebraic groups and discontinuous groups, Proc. Symp. Pure Math.,vol. 9, AMS, 1966.

[TW]

J. Tits, R. Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002.

[T]

B. Totaro, Pseudo-abelian varieties, Ann. Sci. Ecole Norm. Sup. 46 (5), 2013, 693–721.

Index AutG;C , AutG;T , 58 AutG=k , 4 Autsm G=k , 3 Aut.X;D/=F , Aut.X;D/=F , 130–131 AutX=F , AutX=F , 130 B q , 110, 141 Bq , 109 CG , 21 CIsom.q/, 122 CO.q/, 122 > ; < , 14 k , 102 D.G/, 1 Dyn.G/, 94 Dyn.T; ˆC /, 94 EG , 95, 99 Fc , 42 k ˆ, 102 ˆ> , ˆ< , 143 F> ; F< , 45  f ,  X , 14 Gc , 37 0 G.a 0 / , 182 G .E 0 /, 70 red ss GK , GK , 14 g.T /, 41 e 8, 74 G, GX;D , 131 H pred , H prmt , 25

H sm , 14, 70 HV;C =k , 61 HV;K=k , 28 iG , 6, 15 Kc , 37 K> ; K< , 47 khV i, 28 nw , 154 PHV;K=k , 28, 113 q , 142 u , 142 PsInn, 216 qK , 115, 118 q ? , 143 Rk .G/, 14 Rk 0 =k , 1 Ru;k .G/, 1 SO.q/, 110 Spin.q/, 121, 234 -action, 95 Uc , 37 U.a/ , 181 UV˙ , 28 U w , 154 V , 28 V ? , 109  VK=k , 29 W .ˆ/, 14 G , 6, 38

242

INDEX

ZG , 14 ZG;C , 58 ZG .H /, 14 absolutely pseudo-simple group, 2 associated fields, 45 basic exceptional, 159–161 generalized basic exotic, 138 minimal type, 44 primitive pair, 174 pseudo-split descent, 223, 225 pseudo-split form, 225 pseudo-split minimal type classification, 51 rank-1, 29–38, 42, 43, 49, 61, 76, 82, 105, 197–201, 223, 226 root datum, 99 root field, 172 SO.q/, 110 standardness criterion, 40 type B, 128, 131, 188–189 type C, 146–149, 189–190 universal smooth k-tame central extension, 77 adjoint generalized basic exotic, 138 anisotropic kernel, 101 automorphism functor, 90 non-representable, 91 automorphism scheme, 90–93, 215 central quotient, 92 component group, 97–100 basic exceptional, 7, 159–161 center, 166–168 pseudo-parabolic subgroup, 162 pseudo-split classification, 162 ZG;C , 170 basic exotic group, 19–21 type F4 , 230

Bruhat decomposition, 30, 34, 152 canonical diagram, 94, 101 Cartan subgroup, 63–64, 135 generalized basic exotic, 150–151 HV;K=k , 29 SO.q/, 110 central quotient anisotropic kernel, 101 automorphism scheme, 92 generalized standard, 175 locally of minimal type, 76 maximal torus, 75 minimal field of definition, 39 minimal type, 10 pseudo-reductivity, 58 schematic center, 57, 75 standardness, 41 surjectivity of G , 59 ZG;C , 81 conformal isometry, 121–128 defect space, 5, 109 degree (inside Severi–Brauer variety), 130 Dynkin diagram, 94 distinguished vertices, 101, 231– 233 SO.q/, 126 -action, 95 exotic pseudo-reductive group, 21, 54 generalized basic exotic group, 7 ambient basic exotic group, 139– 140 Cartan subgroup, 150–151 center, 166–168

243

INDEX

maximal tori, pseudo-parabolic subgroups, 141, 151 pseudo-split classification, 150– 151 root groups, 149–150 type B, 138 type C, 149 very special quotient, 142 ZG;C , 170 generalized exotic group, 162 central extension, 194 characterization, 164 isomorphisms, 163 separable extension, 164 generalized standard group, 9, 174 derived group, 179 locally of minimal type, 179 generalized standard presentation, 175 rigidity, 177 Isogeny Theorem, 185 Isomorphism Theorem, 79, 89, 97, 228, 233 relative version, 101–103 k-radical, 14 ks =k-form, 11 k-tame, 7–8, 66–68 center, 174 k-unipotent radical, 1 Levi subgroup, 14, 134, 139 k-tame central extension, 74 intermediate group, 15, 18 SO.q/, 110, 115, 124 locally of minimal type, 10, 63–65 central extension, 76 counterexamples, 63, 187–190, 197–205 generalized standard, 179

long root field, 45, 145 maximal pseudo-reductive quotient, 25 normal subgroup, 27 maximal smooth closed subgroup, 14 minimal field of definition, 14 unipotent radical, 15–16, 29, 37– 40, 46, 113 minimal type, 10, 21–27 central quotient, 10, 76 centralizer and normal subgroup, 25 counterexamples, 61, 85, 197– 205 generalized exotic group, 163 G=ZG , 58 non-reduced root system, 62, 85– 87 non-standard, 139 primitive pair, 174 root field, 43 separable extension, 23 standardness, 36 Weil restriction, 25 ZG;C , 83 non-degenerate quadratic space, 5, 109 primitive pair, 173 pseudo-Borel subgroup, 214 pseudo-inner form, 13, 216 pseudo-isogeny, 181 ZG;C , 88 Pseudo-Isogeny Theorem, 181 pseudo-parabolic subgroup basic exceptional group, 162 central quotient, 75

244

generalized basic exotic group, 141, 151 pseudo-reductive group, 1 basic exceptional, 7 basic exotic, 19–21 Bruhat decomposition, 152, 154 Cartan subgroup, 135 center, 57 central quotient, 58 connectedness of ZG;C , 83 derived group, 2 Dyn.G/, 94–95 exotic, 54 generalized basic exotic, 7 generalized standard, 9, 174 Isomorphism Theorem, 79 locally of minimal type, 10, 63, 190–194 minimal type, 10, 21 non-standard, 164 normal subgroup, 2 pseudo-parabolic subgroup, 75 pseudo-split, 3 quasi-split, 3 root datum, 4, 18 root system, 4 separable extension, 2 standard, 17–18, 54–56 pseudo-semisimple group, 1 automorphism scheme, 90, 97 Cartan subgroup, 63 central quotient, 59 locally of minimal type, 76 structure of AutG=k , 91–93 Tits-style classification, 101 pseudo-simple group, 2 pseudo-split, 3 pseudo-split classification, 49, 51 pseudo-split form, 11

INDEX

counterexamples, 13, 207, 208, 210–214, 226–229 existence, 225 SO.q/, 110 uniqueness, 207 quadratic space defect space, 5 non-degenerate, 5 regular, 5, 117, 234 root field, 121 quasi-split, 3, 214 quasi-split form, 13 cohomological obstruction, 219– 220 counterexamples, 13, 228 existence, 220–224 uniqueness, 218 rank-1 absolutely pseudo-simple group, 29–37, 42, 43, 49, 61, 76, 105, 197–201, 223, 226 regular quadratic space, 5, 109 root datum, 18 root field, 138, 145 conformal isometry, 122 long, short, 45, 46 minimal type, 43 non-reduced root system, 172 quadratic space, 121 reduced root system, 42 relation with minimal field for unipotent radical, 46 separable extension, 42 SO.q/, 121, 123 Weil restriction, 43 ZG;C , 82 root system, 18 prmt GF , 173

245

INDEX

non-reduced, 7, 172, 197–205, 229 pseudo-reductive group, 4 reduced, 7, 40, 44, 45, 49, 51, 190–196 Weyl group, 14 scheme-theoretic center, 14 scheme-theoretic centralizer, 14 separable extension G prmt , 25 generalized basic exotic, 138, 149 generalized exotic, 164 k-tameness, 67 minimal type, 23 pseudo-reductivity, 2 regular quadratic space, 109 root field, 42 universal smooth k-tame central extension, 69 Severi–Brauer variety, 130 short root field, 45, 145 SO.q/ center, Cartan subgroup, 110 characterization, 128, 135 conformal isometry, 123 isotropic, 117–121 Levi subgroup, 115 long, short root fields, 121 root groups, 113 spin group Spin.q/, 121 standard presentation, 17 standard pseudo-reductive group, 17– 18, 40 central quotient, 41 standardness counterexamples, 61–62 universal smooth k-tame central extension, 78

-action, 95, 102, 106, 232 Tits-style classification, 101–103 universal smooth k-tame central extension, 8, 74, 76–78 semisimple dictionary, 69 Spin.q/, 121 very special isogeny, 20, 141 very special quotient, 142, 155 Weil restriction, 1 G prmt , 26, 44 G prmt , 172 HV;K=k , 32 minimal type, 25 primitive pair, 174 pseudo-Borel subgroup, 214 root field, 43 universal smooth k-tame central extension, 77–78 ZG;C , 87–89 ZG;C central quotient, derived group, 81 connectedness, 83 product decomposition, 84 rank-1, 82 Weil restriction, 87–89