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

TL;DR

This paper investigates the structure of spherical nilpotent K-orbit closures in the isotropy representation of symmetric pairs for exceptional groups, revealing their normality and module structure in most cases.

Contribution

It provides a comprehensive analysis of the normality and regular function rings of spherical nilpotent orbit closures in exceptional symmetric pairs, including explicit module decompositions.

Findings

All but one orbit closure are normal.

Explicit K-module structures of regular functions are computed.

Identifies the unique non-normal case in type G2.

Abstract

Given an exceptional simple complex algebraic group G and a symmetric pair (G, K), we study the spherical nilpotent K-orbit closures in the isotropy representation of K. We show that they are all normal except in one case in type G2, and compute the K-module structure of the ring of regular functions on their normalizations.