Regular Functions of Symplectic Spherical Nilpotent Orbits and their Quantizations

TL;DR

This paper investigates the algebraic structure of regular functions on classical spherical nilpotent orbits in symplectic groups, providing new verification of conjectures related to their quantizations and extending results to broader classes of orbits.

Contribution

It introduces a quantization model for certain nilpotent orbits in symplectic groups and verifies key conjectures by McGovern and Achar-Sommers for these orbits.

Findings

Verification of McGovern's conjecture for specific orbits

Verification of Achar-Sommers conjecture for these orbits

Extension of results to larger classes of nilpotent orbits assuming prior work

Abstract

We study the ring of regular functions of classical spherical orbits $R(\mathcal{O})$ for $G = Sp(2n,\mathbb{C})$. In particular, treating $G$ as a real Lie group with maximal compact subgroup $K$, we focus on a quantization model of $\mathcal{O}$ when $\mathcal{O}$ is the nilpotent orbit $(2^{2p}1^{2q})$. With this model, we verify a conjecture by McGovern and another conjecture by Achar and Sommers for such orbits. Assuming the results in [Barbasch 2008], we will also verify the Achar-Sommers conjecture for a larger class of nilpotent orbits.