# Regular functions on nilpotent orbits and their covers

Let $G$ be a complex semisimple algebraic group with Lie algebra $\mathfrak{g}$. In 1989 McGovern described the structure (as $G$-module) of the ring of regular functions on a finite cover of the nilpotent coadjoint orbit $G\cdot e$ , $e \in \mathfrak{g}$ nilpotent element. More precisely, there are finite dimensional representation $F_i$ of Levi subgroup $L$ (arising in Levi decomposition of parabolic subgroup corresponding to $e$) $$R(\widetilde{G\cdot e}) \cong \sum\limits_{i} (-1)^i \mathrm{Ind}_L^G(\wedge^i \mathfrak{g}_1 \otimes F_i).$$

What is modern state of play of the question? Is there some description of $F_i$? I shall be vastly obliged for any references.

• for another perspective, see link.this is not published though. I think Barbasch also mentioned the results in his talk in Yale this year, see slide 29/35 of link – wky Oct 22 '15 at 6:50