Distinguished unipotent elements and multiplicity-free subgroups of simple algebraic groups

Martin W. Liebeck, Gary M. Seitz and Donna M. Testerman

Vol. 279 (2015), No. 1-2, 357–382

DOI: 10.2140/pjm.2015.279.357

Abstract

For $G$ a simple algebraic group over an algebraically closed field of characteristic $0$, we determine the irreducible representations $\rho :G\to I\left(V\right)$, where $I\left(V\right)$ denotes one of the classical groups $SL\left(V\right)$, $Sp\left(V\right)$, $SO\left(V\right)$, such that $\rho$ sends some distinguished unipotent element of $G$ to a distinguished element of $I\left(V\right)$. We also settle a base case of the general problem of determining when the restriction of $\rho$ to a simple subgroup of $G$ is multiplicity-free.

Keywords

algebraic groups, unipotent elements, multiplicity-free representations

Mathematical Subject Classification 2010

Primary: 20G05

Secondary: 20G07, 22E46

Milestones

Received: 7 November 2014

Revised: 30 August 2015

Accepted: 31 August 2015

Published: 21 December 2015