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Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]^{\ast}//W$). Hence, to each dominant root lattice element $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the highest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.

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In the sentence before your question, "dominant root" isn't what you intend. – Jim Humphreys May 2 '13 at 23:52
Yes, of course. Thanks for pointing that out! Cheers, George – George Melvin May 3 '13 at 0:48

This is mainly a suggestion about references that I recall offhand. The study of Weyl group actions on zero weight spaces in type $A_n$ goes back a long way, to work of Kostant, Gutkin, and others. One clarification: there is a nonzero weight space in a finite dimensional simple module with (dominant) highest weight $\lambda$ iff $\lambda$ lies in the root lattice. In one direction this is easy, but in the othr direction it's best understood in the wider context of Bourbaki's notion of "saturated" set of weights.

You might find work by Mark Reeder somewhat relevant to your question. In particular, he extended the result you mention on type $A_n$ to other simply-laced cases in his paper Zero weight spaces and the Springer correspondence, Indag. Math. 9 (1998). Obviously it's tricky to move beyond the partition notation for highest weights. (It's also nontrivial to track down all the work done that has implications for zero weight spaces and Weyl groups.)

Though it diverges from your line of questioning, I should also mention a very recent arXiv preprint by S. Kumar and D. Prasad here.

ADDED: Concerning your main question, I'm not sure how far the analogy with the adjoint module can be pressed (nor do I have enough insight into Borel's connection between the coinvariant algebra of the Weyl group and the cohomology of the flag variety). However, you need to be cautious in general about the nature of $W$-invariants in the algebra $S$. In the classical situation, you start with a realization of the reflection representation of $W$ and can then appeal to Chevalley's structure theorem for the algebra of polynomial invariants, etc. In general, how much control do you have over the action of $W$ on zero weight spaces relative to invariant theory? (Of course it's far more delicate to study arbitrary Coxeter groups and their possible reflection representations, as in the work of Soergel, Elias-Williamson.)

In the characteristic 0 setting, there has been a lot of work relating reductive algebraic groups, algebraic geometry, and invariant theory. This might (or might not) shed light on your question. The Russian school (Vinberg, Popov, Panyeshev, et al.) have been especially active, though I'm not close enough to their papers to identify what might be relevant here. Another source to consider would be the papers and lectures of Michel Brion.

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Thanks for the clarification and references, Prof. Humphreys. I had discovered Reeder's work (and the more recent work of Achar-Henderson-Riche) and it seems quite surprising that progress on this problem is recent; after all, it seems like a reasonably straightforward question to consider. – George Melvin May 3 '13 at 0:51

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