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

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 $V$.