Multiplicity-free representations of the principal $A_1$-subgroup in a simple algebraic group

Aluna Rizzoli and Donna Testerman

Vol. 336 (2025), No. 1-2, 433–470

DOI: 10.2140/pjm.2025.336.433

Abstract

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. For $p\ge h$, the Coxeter number of $G$, any regular unipotent element of $G$ lies in an $A_1$-subgroup of $G$; there is a unique $G$-conjugacy class of such subgroups and any member of this class is a so-called “principal $A_1$-subgroup of $G$”. Here we classify all irreducible $kG$-modules whose restriction to a principal $A_1$-subgroup of $G$ has no repeated composition factors, extending the work of Liebeck, Seitz and Testerman which treated the same question when $k$ is replaced by an algebraically closed field of characteristic zero.

We dedicate this paper to the memory of the esteemed mathematician, Gary Seitz,whose work and mentorship have a continuing impact on the field and on our lives

Keywords

algebraic group, principal $A_1$, representation theory

Mathematical Subject Classification

Primary: 20G05, 20G07

Milestones

Received: 4 April 2024

Revised: 2 November 2024

Accepted: 20 November 2024

Published: 26 May 2025