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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.

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    $\begingroup$ 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 $\endgroup$
    – wky
    Commented Oct 22, 2015 at 6:50

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This is an extended comment (in community-wiki mode) with a link to the 1989 paper by Monty McGovern which you start with. William Graham, a somewhat later student of Vogan, wrote a paper which is archived in the same location.

If you can access MathSciNet, you will find there some (but probably not all) of the later papers citing McGovern's paper. This illustrates the range of work done in this direction pioneered by Vogan. Your questions are rather broad and may not yet have satisfying answers, but for example you might get some direct advice from McGovern (U. Washington) or Graham (U. Georgia) as well as from Achar (L.S.U.) and Sommers (UMass Amherst). Most of them were actually here at UMass last weekend for a workshop (as was Vogan) though this part of the subject didn't come up in the formal program.

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