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

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## Overview

In the earlier monograph *Pseudo-reductive Groups*, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, *Classification of Pseudo-reductive Groups*, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions.

The results and methods developed in *Classification of Pseudo-reductive Groups *will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

## Product Details

ISBN-13: | 9780691167930 |
---|---|

Publisher: | Princeton University Press |

Publication date: | 11/10/2015 |

Series: | Annals of Mathematics Studies Series |

Pages: | 256 |

Product dimensions: | 9.10(w) x 6.10(h) x 0.60(d) |

## About the Author

**Brian Conrad **is professor of mathematics at Stanford University. **Gopal Prasad **is the Raoul Bott Professor of Mathematics at the University of Michigan.

## Read an Excerpt

#### Classification of Pseudo-Reductive Groups

**By Brian Conrad, Gopal Prasad**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 2016 Princeton University Press**

All rights reserved.

ISBN: 978-1-4008-7402-6

All rights reserved.

ISBN: 978-1-4008-7402-6

CHAPTER 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(G[bar.k]) [subset] G[bar.k] 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) [subset] 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 *k*-group *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 *pseudo- semisimple.* 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 = Rk'/k(G') for finite extensions *k'/k* and connected reductive *k'*-groups *G'* are pseudo-reductive [CGP, Prop. 1.1.10]. If *G'* is nontrivial and *k'/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 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 GL*n* of a subgroup of GL*n*(*k*), and the maximal smooth *k*-subgroup of a schematic stabilizer (as in local-global problems). For questions not amenable to study over [bar.*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 [subset] 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 = [bar.k]* for any imperfect *k* and non-reductive pseudo-reductive G). Taking *K = ks* shows that *G* is pseudo-reductive if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 *k*-groups 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 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 {*Gi*} 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 pseudo-split pseudo-reductive groups and of the Existence Theorem for pseudo-split pseudo-simple groups?

(ii) The *standard construction* (see §2.1) is exhaustive when *p* := 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* = 2;3 provided that [*k : k*2 = 2 if *p* = 2; see [CGP, Thm. 10.2.1, Prop. 10.1.4]. More examples exist if *p* = 2 and [*k : 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 defining a notion of "pseudo-inner" *ks/k*-form via Autsm*G/k*)0)?

(iv) What can be said about existence and uniqueness of pseudo-split *ks/k*-forms, 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 scheme-theoretic center ZG can be a proper *k*-subgroup of Autsm*G/k*)0.)

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 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' := T* [intersection] *D*(*G*) in *D*(*G*) [CGP, Lemma 1.2.5]. Suppose *T* is *split*, so the set Φ:= Φ(*G,T*) of nontrivial *T*-weights on Lie(*G*) injects into X(*T'*) via restriction.

The pair (Φ,X(*T'*)Q) is always a root system (coinciding with Φ(*D*(*G*),*T'*) 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, (Φ,X(*T'*)Q) can be non-reduced when *k* is imperfect of characteristic 2 (the non-multipliable roots are the roots of the maximal geometric reductive quotient *G*red[bar.k]). A pseudo-split 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 Aut*G/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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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**

If *p* = 2 and [*k : k*2] > 2 then there exist families of *non-standard* absolutely pseudo-simple *k*-groups of types B*n*, C*n*, and BC*n* (for every *n* ≥ 1) with no analogue when [*k : k*2] = 2. Their existence is explained by a construction with certain degenerate quadratic spaces over *k* that exist only if [*k : k*2] > 2:

**Example 1.3.1.** Let (*V,q*) be a quadratic space over a field *k* with char(*k*) = 2, *d* := dim *V* ≥ 3, and *q* ≠ 0. Let *Bq : (v,w) [??] q(v + w)_- q(v)_- q(w)* be the associated symmetric bilinear form and *V*[perpendicular to] the defect space consisting of *v [member of] V* such that the linear form *Bq(v*, •) on *V* vanishes. The restriction *q|V]*perpendicular to] is 2-linear (i.e., additive and *q(cv) = c2q(v)* for *v [member of] V, c [member of] k*) and dim (*V/V*[perpendicular to]) = 2*n* for some *n* ≥ 0 since *Bq* induces a non-degenerate symplectic form on *V/V*[perpendicular to].

*(Continues...)*

Excerpted fromClassification of Pseudo-Reductive GroupsbyBrian Conrad, Gopal Prasad. Copyright © 2016 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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## Table of Contents

- Frontmatter, pg. i
- Contents, pg. v
- 1. Introduction, pg. 1
- 2. Preliminary notions, pg. 15
- 3. Field-theoretic and linear-algebraic invariants, pg. 28
- 4. Central extensions and groups locally of minimal type, pg. 57
- 5. Universal smooth k-tame central extension, pg. 66
- 6. Automorphisms, isomorphisms, and Tits classification, pg. 79
- 7. Constructions with regular degenerate quadratic forms, pg. 108
- 8. Constructions when Φ has a double bond, pg. 138
- 9. Generalization of the standard construction, pg. 171
- A. Pseudo-isogenies, pg. 181
- B. Clifford constructions, pg. 187
- C. Pseudo-split and quasi-split forms, pg. 206
- D. Basic exotic groups of type F4 of relative rank 2, pg. 230
- Bibliography, pg. 239
- Index, pg. 241