Regular functions of symplectic spherical nilpotent orbits and their quantizations

Abstract

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