Rings of regular functions on nilpotent orbits and their covers (Q1105053)

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