Rings of regular functions on nilpotent orbits and their covers

Abstract

LetG be a complex semisimple algebraic group with Lie algebra $$\mathfrak{g}$$ . Let $$\mathcal{O} \subset \mathfrak{g}$$ be a nilpotentG-orbit, $$R(\mathcal{O})$$ its ring of regular functions. We derive a formula for $$R(\mathcal{O})$$ as aG-module and prove some partial results on $$R(\tilde {\mathcal{O}}),\tilde {\mathcal{O}}$$ a cover of $$\mathcal{O}$$ . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.