Generic stabilizers for simple algebraic groups acting on orthogonal and symplectic Grassmannians

TL;DR

This paper investigates the generic stabilizers of simple algebraic groups acting on orthogonal and symplectic Grassmannians, providing conditions for the existence of dense orbits and classifying subgroup pairs with dense double cosets.

Contribution

It establishes the existence of generic stabilizers for most cases and completes the classification of subgroup pairs with dense double cosets in classical groups.

Findings

Generic stabilizers are conjugate to a fixed subgroup in most cases.

Conditions for the existence of dense orbits are determined.

Complete classification of subgroup pairs with dense double cosets.

Abstract

We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the dimension of the variety. We prove that in all but a finite list of cases, there is a dense open subset where the stabilizer of any point is conjugate to a fixed subgroup, called the generic stabilizer. We use these results to determine whether there exists a dense orbit. This in turn lets us complete the answer to the problem of determining all pairs of maximal connected subgroups of a classical group with a dense double coset.