Vol. 14, No. 6, 2020

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Generically free representations, I: Large representations

Skip Garibaldi and Robert Guralnick

Vol. 14 (2020), No. 6, 1577–1611
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