Quantization of Special Symplectic Nilpotent Orbits and Normality of their Closures

TL;DR

This paper investigates the structure and normality of closures of symplectic nilpotent orbits with even column sizes, using quantization models and representation theory to derive criteria for normality.

Contribution

It introduces a new criterion for the normality of orbit closures based on quantization and representation multiplicities, extending understanding of symplectic nilpotent orbits.

Findings

Derived a formula for multiplicities of fundamental representations in the function ring

Provided a criterion for the normality of orbit closures

Connected quantization models with geometric properties of orbits

Abstract

We study the regular function ring $R(\mathcal{O})$ for all symplectic nilpotent orbits $\mathcal{O}$ with even column sizes. We begin by recalling the quantization model for all such orbits by Barbasch using unipotent representations. With this model, we express the multiplicities of fundamental representations appearing in $R(\mathcal{O})$ by a parabolically induced module. Finally, we will use this formula to give a criterion on the normality of the Zariski closure $\bar{\mathcal{O}}$ of $\mathcal{O}$.