Regular functions on spherical nilpotent orbits in complex symmetric pairs: classical Hermitian cases

TL;DR

This paper investigates the structure of spherical nilpotent orbit closures in isotropy representations of Hermitian symmetric pairs, proving their normality and detailing their regular functions.

Contribution

It establishes the normality of all such orbit closures and characterizes their ring of regular functions in classical Hermitian symmetric pairs.

Findings

All orbit closures are normal.

The K-module structure of their regular functions is explicitly described.

Provides a comprehensive understanding of spherical nilpotent orbits in this setting.

Abstract

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe the K-module structure of their ring of regular functions.