Generically free representations, I: Large representations

Garibaldi, Skip; Guralnick, Robert

doi:10.2140/ant.2020.14.1577

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Generically free representations, I: Large representations

Skip Garibaldi and Robert Guralnick

Vol. 14 (2020), No. 6, 1577–1611

DOI: 10.2140/ant.2020.14.1577

Abstract

This paper concerns a faithful representation $V$ of a simple linear algebraic group $G$ . Under mild assumptions, we show that if $V$ is large enough, then the Lie algebra of $G$ acts generically freely on $V$ . That is, the stabilizer in $Lie (G)$ of a generic vector in $V$ is zero. The bound on $dim V$ grows like ${(rank G)}^{2}$ and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position.

Keywords

generically free, virtually free, Lie algebra, representation, generic stabilizer

Mathematical Subject Classification 2010

Primary: 20G05

Secondary: 17B10

Milestones

Received: 6 May 2019

Revised: 19 November 2019

Accepted: 6 February 2020

Published: 30 July 2020

Authors

Skip Garibaldi
Center for Communications Research San Diego, CA United States

Robert Guralnick
Department of Mathematics University of Southern California Los Angeles, CA United States

Vol. 14, No. 6, 2020