Variation on a theme of Richardson
Variation on a theme of Richardson
We consider the structure of parabolic subgroups P in general linear groups. The group P acts on its unipotent radical Pu and on all members of the descending central series Pu(l) via conjugation. By a fundamental theorem due to Richardson P acts on Pu with an open dense orbit. In fact, this density theorem holds for any reductive algebraic group. In this note we investigate the question of the existence of a dense P-orbit on Pu(l) for l1 using only most elementary methods. Despite the fact that for special P it is the case that P operates on Pu(l) with such a dense orbit for all l0, in general, however, this fails; we present a counterexample in GL15(k). Besides the general linear groups, we also study this question for other reductive algebraic groups.
linear algebraic groups, parabolic group actions, Richardson’s dense orbit theorem
239-246
Hille, Lutz
3ff2d97b-b3eb-487e-9066-d7449dc60d3b
Roehrle, Gerhard
85f9d4eb-d522-4a95-bde9-300e8e3e7886
May 2003
Hille, Lutz
3ff2d97b-b3eb-487e-9066-d7449dc60d3b
Roehrle, Gerhard
85f9d4eb-d522-4a95-bde9-300e8e3e7886
Abstract
We consider the structure of parabolic subgroups P in general linear groups. The group P acts on its unipotent radical Pu and on all members of the descending central series Pu(l) via conjugation. By a fundamental theorem due to Richardson P acts on Pu with an open dense orbit. In fact, this density theorem holds for any reductive algebraic group. In this note we investigate the question of the existence of a dense P-orbit on Pu(l) for l1 using only most elementary methods. Despite the fact that for special P it is the case that P operates on Pu(l) with such a dense orbit for all l0, in general, however, this fails; we present a counterexample in GL15(k). Besides the general linear groups, we also study this question for other reductive algebraic groups.
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Submitted date: April 2001
Published date: May 2003
Additional Information:
Special Issue on Linear Algebra Methods in Representation Theory
Keywords:
linear algebraic groups, parabolic group actions, Richardson’s dense orbit theorem
Identifiers
Local EPrints ID: 38489
URI: http://eprints.soton.ac.uk/id/eprint/38489
ISSN: 0024-3795
PURE UUID: 994ac3ed-74ea-4b50-88d1-03f4743e8964
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Date deposited: 12 Jun 2006
Last modified: 15 Mar 2024 08:08
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Author:
Lutz Hille
Author:
Gerhard Roehrle
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