Spherical nilpotent orbits and abelian subalgebras in isotropy representations

TL;DR

This paper classifies Borel subgroup orbits in abelian subalgebras within isotropy representations of symmetric pairs, using affine Weyl group combinatorics and strongly orthogonal roots to analyze sphericity.

Contribution

It provides a classification of Borel orbits in abelian subalgebras of isotropy representations and characterizes when these orbits are spherical, employing new combinatorial methods.

Findings

Classification of Borel orbits in abelian subalgebras

Characterization of sphericity of $G_0$-orbits

Use of $\sigma$-minuscule elements and orthogonal roots

Abstract

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $\sigma$-minuscule elements in the affine Weyl group of $\mathfrak g$ and that of strongly orthogonal roots in Hermitian symmetric spaces.