Symmetric Subgroup Actions on Isotropic Grassmannians

TL;DR

This paper explicitly classifies the orbits of symmetric subgroups acting on isotropic Grassmannians, detailing their invariants, orderings, and decompositions, thus advancing understanding of symmetry actions in geometric representation theory.

Contribution

It provides a complete parameterization of H-orbits in isotropic Grassmannians and describes their order structures and decompositions explicitly.

Findings

Explicit parameterization of H-orbits using invariants

Description of Bruhat order via majorization of invariants

Analysis of orbit decompositions and stabilizers

Abstract

Let G be the group preserving a nondegenerate sesquilinear form on a vector space V, and H a symmetric subgroup of G of the type G1 x G2. We explicitly parameterize the H-orbits in the Grassmannian of r-dimensional isotropic subspaces of V by a complete set of H-invariants. We describe the Bruhat order in terms of the majorization relationship over a diagram of these H-invariants. The inclusion order, the stabilizer, the orbit dimension, the open H-orbits, the decompositions of an H orbit into H\cap G_0 and H_0 orbits are also explicitly described.