Modules for algebraic groups with finitely many orbits on totally singular 2-spaces

TL;DR

This paper classifies the pairs of subgroups in classical algebraic groups that have finitely many double cosets, focusing on the case where the subgroup stabilizes totally singular 2-spaces, extending previous work on the case k=1.

Contribution

It determines all faithful irreducible self-dual modules where the simple subgroup has finitely many orbits on totally singular 2-spaces, specifically addressing the case k=2.

Findings

Classified all modules with finitely many orbits on totally singular 2-spaces.

Extended the double coset classification to the case k=2.

Provided explicit conditions for modules with finitely many orbits.

Abstract

This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in $G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and $J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case $k=1$ was solved in a previous paper and here we deal with $k=2$. We solve this case by determining all faithful irreducible self-dual $H$-modules $V$, such that $H$ has finitely may orbits on totally singular $2$-spaces of $V$.