Regular orbits of symmetric subgroups on partial flag varieties

TL;DR

This paper introduces a new way to parameterize symmetric subgroup orbits on partial flag varieties using Spaltenstein varieties and nilpotent orbits, aiding in representation theory and orbit closure understanding.

Contribution

It provides a novel parameterization of symmetric subgroup orbits on partial flag varieties connecting geometric objects and representation theory.

Findings

New parameterization of orbits using Spaltenstein varieties and nilpotent orbits

Enhanced understanding of orbit closure relations

Applications to unipotent representation enumeration

Abstract

The main result of this paper is a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of the motivations is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits.