Complete reducibility and separability

By M. Bate, B. Martin, Gerhard Röhrle, R. Tange. 2007

51 citations recorded by the source corpus.

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 good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and G is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d33cea7b6ff51fd08dcf45c795b383adaf1065b

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In Discrete Mathematics and Combinatorics

1. Sebastian Herpel
2. R. Tange
3. B. Martin
4. Gerhard Röhrle
5. M. Bate

In Geometry and Topology

1. Sebastian Herpel
2. A. Jaikin‐Zapirain
3. Jay Taylor

In Mathematical Physics

1. Sebastian Herpel
2. R. Tange
3. B. Martin
4. Gerhard Röhrle
5. M. Bate
6. A. Jaikin‐Zapirain
7. Jay Taylor

Citations in the corpus

• On the Smoothness of Centralizers in Reductive Groups (2010, background): “l fixed points of Hin Lie(G). Here, in contrast to the later sections of the paper, we do not consider scheme-theoretic centralizers. A similar notion exists for subalgebras of Lie(G). See [BMR05] and [BMRT] for these concepts and their importance in the context of Serre’s notion of G-complete reduci”
• Closed Orbits and uniform S-instability in Geometric Invariant Theory (2009, background): “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 [32] and developed in [1], [2], [4], [5], [17], [18], [19], [30], [31], [33], [34].”