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Irreducibility in algebraic groups and regular unipotent elements

Authors: Donna Testerman and Alexandre Zalesski
Journal: Proc. Amer. Math. Soc. 141 (2013), 13-28
MSC (2010): Primary 20G05, 20G07
Published electronically: August 16, 2012
MathSciNet review: 2988707
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Abstract: We study (connected) reductive subgroups $ G$ of a reductive algebraic group $ H$, where $ G$ contains a regular unipotent element of $ H$. The main result states that $ G$ cannot lie in a proper parabolic subgroup of $ H$. This result is new even in the classical case $ H = \mathrm {SL}(n,F)$, the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group $ H$ which contain a regular unipotent element.

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Donna Testerman
Affiliation: École Polytechnique Federale de Lausanne, FSB-MATHGEOM, Station 8, CH-1015 Lausanne, Switzerland

Alexandre Zalesski
Affiliation: Departimento di Matematica e Applicazioni, University of Milano-Bicocca, 53 via R. Cozzi, Milan, 20125, Italy

Received by editor(s): April 23, 2011
Published electronically: August 16, 2012
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.