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Classification of Pseudo-Reductive Groups
By Brian Conrad, Gopal Prasad PRINCETON UNIVERSITY PRESS
Copyright © 2016 Princeton University Press
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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 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 [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 : k2 = 2 if p = 2; see [CGP, Thm. 10.2.1, Prop. 10.1.4]. More examples exist if p = 2 and [k : k2 > 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 AutsmG/k (thereby defining a notion of "pseudo-inner" ks/k-form via AutsmG/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 AutsmG/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 Gred[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 AutG/k for pseudo-semisimple G, and properties of the identity component and component group of its maximal smooth closed k-subgroup AutsmG/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 : k2] > 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 [k : k2] = 2. Their existence is explained by a construction with certain degenerate quadratic spaces over k that exist only if [k : k2] > 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]) = 2n for some n ≥ 0 since Bq induces a non-degenerate symplectic form on V/V[perpendicular to].
(Continues...)
Excerpted from Classification of Pseudo-Reductive Groups by Brian Conrad, Gopal Prasad. Copyright © 2016 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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