Quantization of Special Symplectic Nilpotent Orbits and Normality of their Closures

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 $\mathcal{O}$.