Regular Functions of Symplectic Spherical Nilpotent Orbits and their Quantizations

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.