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Complete reducibility and separability

Complete reducibility and separability
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 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.
0002-9947
4283-4311
Bate, M.
f6f06162-8a04-4ed0-80d8-3029995a86fc
Martin, B.
6a58db20-96d5-4c27-8e8a-8d2d8beaf65c
Roehrle, G.
062aed02-1c89-440f-a486-70b4f7828a0a
Tange, R.
cd58db15-f1a5-4b6a-aa52-5c37c7317f10
Bate, M.
f6f06162-8a04-4ed0-80d8-3029995a86fc
Martin, B.
6a58db20-96d5-4c27-8e8a-8d2d8beaf65c
Roehrle, G.
062aed02-1c89-440f-a486-70b4f7828a0a
Tange, R.
cd58db15-f1a5-4b6a-aa52-5c37c7317f10

Bate, M., Martin, B., Roehrle, G. and Tange, R. (2010) Complete reducibility and separability. Transactions of the American Mathematical Society, 362, 4283-4311. (doi:10.1090/S0002-9947-10-04901-9).

Record type: Article

Abstract

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.

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Submitted date: September 2007
Published date: 2010

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Local EPrints ID: 48511
URI: http://eprints.soton.ac.uk/id/eprint/48511
ISSN: 0002-9947
PURE UUID: 185e9fd7-971c-4192-adf6-d7113b10a9e5

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Date deposited: 08 Oct 2007
Last modified: 11 Nov 2019 18:43

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