# Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This subspace admits a representation U of the Weyl group W of G. Is there any method for determining the character of the representation U from the character of V? I feel like this should be known at least for the classical groups, but can't find any sources on it.

This is a reasonable question, and there has been some relevant literature over many decades. But my impression is that no definitive answer has been given, even for the classical types. I've intended for a long time to write my own survey on this topic, but meanwhile here are a few references and links:

1) Probably the earliest published work was in a very short (and untranslated) 1973 Russian paper by E.A. Gutkin Representations of the Weyl group in the space of vectors of zero weight here. Using the tensor product framework of Schur-Weyl theory, Gutkin studied the group $G = \mathrm{SL}_n(\mathbb{C})$ and its Weyl group $W=S_n$. He concluded that all irreducible representations of $S_n$ occur as zero weight spaces of irreducible $G$-representations. Of course there are infinitely many of the latter, but it's enough to consider those whose highest weights are (in the usual way) partitions of $n$; then the dual partitions parametrize the irreducible $W$-representations.

2) Though Gutkin's paper is not well referenced in other papers, there is a similar result in $\S4.1$ of a 1976 paper by B. Kostant here. Kostant went somewhat further with other simple Lie groups here.

3) Another 1976 paper, in the compact group framework of special unitary groups, also dealt with characters of symmetric groups in the setting of Schur-Weyl duality: D.A. Gay, Characters of the Weyl group of $\mathrm{SU}(n)$ on zero weight spaces and centralizers of permutation representations, Rocky Mt. J. Math. 6 (1976), 449-455.

4) A later paper by M. Reeder Zero weight spaces and the Springer correspondence, Indag. Math. 9 (1998), 431-441, extended the earlier results on symmetric groups to Lie types $D_n, E_n$. This occurs in the framework of his study of "small" representations, as in his paper Small representations and minuscule Richardson orbits, IMRN 2002, No. 5. He also gives numerous further references.

5) Yet another approach is found in a paper by S. Kumar and D. Prasad Dimension of zero weight space: An algebro-geometric approach, J. Algebra 403 (2014), 324-344 (not yet freely accessible online but posted in preprint form on arXiv:1304.4210).

The picture derived from this and related literature is still incomplete, it seems, but intriguing.

ADDED: I've been motivated to organize some notes online here. This is still a preliminary version, so email comments are welcome.

• Very useful scholarship... bravo. Oct 30 '14 at 23:44
• There's also a beautiful geometric approach to this construction (and to Reeder's results) by Achar, Henderson and Riche in arXiv:1205.5089. Oct 31 '14 at 3:52
• Thanks for the references! I'll take a look. I was actually hoping to extend some results on small representations, actually inspired by Broer's and Reeder's work, but that required knowing more about what the case of not-small representations looked like. Nov 2 '14 at 23:09
• @DavidBen-Zvi's comment references Achar, Henderson, and Riche, "Geometric Satake, Springer correspondence, and small representations II": clickable arXiv, MSN, and journal (free). Apr 27 '17 at 20:57
• There was a paper last year on arXiv with an update two days ago claimed to give more information about the Weyl group action on the zero weight spaces. In particular, the "longest element signatures" are computed for some case. Find arxiv.org/abs/1806.00347 Aug 28 '19 at 22:12