Complete reducibility: Variations on a theme of Serre

Complete reducibility, Kulshammer's question, conjugacy classes: a D_4 example

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present a new example of subgroup $H$ of $G$ of type $D_4$ in characteristic $2$ such that $H$ is $G$-completely reducible but not $G$-completely reducible over $k$ (or vice versa). This is new: all known such examples are for $G$ of exceptional type. We also find a new counterexample for Külshammer's question on representations of finite groups for $G$ of type $D_4$. A…

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 3

Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly non-connected) reductive group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In our previous work, we constructed examples of subgroups $H$ of $G$ that are $G$-completely reducible but not $G$-completely reducible over $k$ (and vice versa). In this paper, we give a theoretical underpinning of those constructions. To illustrate our result, we present a new such example in a non-connected reductive group of type $D_4$ in characteristic $2$. Then using Geometri…

Complete Reducibility and Conjugacy classes of tuples in Algebraic Groups and Lie algebras

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G^n, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G^n, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre's notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We …

Relative complete reducibility and normalised subgroups

We study a relative variant of Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^{\circ}$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ}$ is completely reducible. This allows us to generalise a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, …

On relative complete reducibility

Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre's notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility which directly generalizes equi…

A Geometric Approach to Complete Reducibility

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert--Mumford--Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove …

1998 Moursund Lectures at the University of Oregon

These informal notes are closely based on a series of eight lectures given by J.-P. Serre at the University of Oregon in October 1998. Professor Serre gave two talks per week for four weeks. The first talk each week was concerned with constructing embeddings of finite groups, especially $\PSL_2(p)$ and $\PGL_2(p)$, into Lie groups. The second talk each week was about generalizations of the notion of complete reducibility in group theory, especially in positive characteristic. The notes are divided into two parts, one for each of the topics of the lecture series. At the end of the notes, there …

$G$-complete reducibility and semisimple modules

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field %$k$ of characteristic $p > 0$. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre's notion of $G$-complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index $(H:H^\circ)$ is prime to $p$ and that $p > 2\dim V-2$ for some faithful $G$-module $V$. Then the following hold: (i) $V$ is a semisimple $H$-module if and only if $H$ is $G$-completely reducible; (ii) $…

Complete Reducibility and Separability

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very go…

$G$-完全可约性与饱和性

$G$-complete reducibility and saturation

本文研究连通约化线性代数群中完全可约性与饱和性的关系，在特征限制下证明 $H$-完全可约子群在 $G$ 中亦完全可约，并推广饱和性结果至一般群及李型有限子群。

Completely reducible Lie subalgebras

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H of G. In this note, we give a notion of G-complete reducibility -- G-cr for short -- for Lie subalgebras of Lie(G), and we show that if the closed subgroup H < G is G-cr, then Lie(H) is G-cr as well.

The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras

In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example i…