2010
DOI: 10.1007/s00209-010-0763-9
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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 z… Show more

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Cited by 14 publications
(39 citation statements)
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“…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.…”
Section: Applications To G-complete Reducibilitymentioning
confidence: 90%
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“…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.…”
Section: Applications To G-complete Reducibilitymentioning
confidence: 90%
“…(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.…”
Section: Spherical Buildings and Tits' Centre Conjecturementioning
confidence: 98%
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“…In [5,Thm. 4.13], we prove a generalization of the reverse implication of Theorem 1.1 in the setting of "relative complete reducibility".…”
Section: Remark 31mentioning
confidence: 97%
“…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).…”
mentioning
confidence: 99%