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....