Admissible modules and normality of classical nilpotent orbits I

TL;DR

This paper constructs specific modules for complex symplectic and orthogonal groups that reflect the structure of functions on nilpotent orbit closures, supporting a version of the Orbit Method and providing a new proof of their normality classification.

Contribution

It introduces modules matching orbit closure functions for classical groups, advancing the Orbit Method and offering a novel proof of nilpotent orbit normality classification.

Findings

Modules with $K$-structure matching orbit closures

Supports the Orbit Method for classical groups

Provides a new proof of normality classification

Abstract

In the case of complex symplectic and orthogonal groups, we find $(\mathfrak{g}, K)-$modules with the property that their $K-$structure matches the structure of regular functions on the closures of nilpotent orbits. This establishes a version of the Orbit Method of Kirrilov-Kostant-Souriau as proposed by Vogan. In the process we give another proof of the classification of nilpotent orbits with normal closure in the Lie algebra of a classical group first established by Kraft-Procesi.