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.

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

2010

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, .
(doi:10.1090/S0002-9947-10-04901-9).

## 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.

Text

** separability.pdf
- Author's Original**
## More information

Submitted date: September 2007

Published date: 2010

## Identifiers

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: 08 Jan 2022 04:15

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## Contributors

Author:
M. Bate

Author:
B. Martin

Author:
G. Roehrle

Author:
R. Tange

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