Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras

Abstract

Abstract. 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 Gcomplete 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 also consider some rationality questions.

Cocharacter-closure and spherical buildings

“…The group G acts on g n via the simultaneous adjoint action for any n ∈ N. The next result follows from [5,Lem. 3.8] We now give the applications of our earlier results to G-complete reducibility over k for Lie algebras.…”

“…(ii). In analogy with the first assertion of (i) (although it is slightly more subtle), the key observation is that if λ, µ ∈ Y k (H) are R u (P λ (G))(k)-conjugate, then they are in fact R u (P λ (H))(k)-conjugate (see [5,Lem Henceforth, we write ∆ k (K) ⊆ ∆ ks (K) ⊆ ∆(K) and ∆ k (H, K) ⊆ ∆ k (K) without any further comment. One note of caution: the inclusion ∆ k (H, K) ⊆ ∆ k (K) does not in general respect the simplicial structures on these objects.…”

“…This theory has a counterpart for Lie subalgebras of g := Lie(G). The basic definitions and results were covered for algebraically closed fields in [13] and [5,Sec. 3.3], but the extension to arbitrary fields is straightforward (cf.…”

Complete reducibility and separable field extensions

“…In [5,Thm. 4.13], we prove a generalization of the reverse implication of Theorem 1.1 in the setting of "relative complete reducibility".…”

Closed orbits and uniform $S$-instability in geometric invariant theory

“…As well as being of interest in their own right, our general results on G-orbits and rationality have applications to the theory of G-complete reducibility, introduced by Serre [33] and developed in [1], [2], [4], [5], [17], [18], [19], [31], [32], [34], [35]. In particular, we are able to use them to answer a question of Serre about how G-complete reducibility behaves under extensions of fields (Theorem 5.11).…”