## Set-up for classical Springer Correspondence:

$G$ = reductive group (usually assumed to be semisimple of adjoint type) over $\mathbb{C}$, with Borel subgroup and maximal torus $B \supset T$, Weyl group $W=N_G(T)/T$.

Fix a unipotent $u \in G$ with component group $A(u) = C_G(u)/C_G(u)^\circ$.

$\mathcal{B} = G/B$ (flag variety), containing Springer fiber $\mathcal{B}_u$ = fixed points under $u$, $d=\dim \mathcal{B}_u$ (= half codimension of class of $u$ in unipotent variety)

Then $W \times A(u)$ acts on cohomology ($\ell$-adic or classical) $H^*(\mathcal{B}_u)$ with top cohomology in degree $2d$.

(Springer) Each irreducible representation of $W$ occurs, for some pair $(u,\phi)$ with $u$ unipotent and$\phi$ an irreducible character of $A(u)$, as an isotypic component of the $W \times A(u)$ representation on the top cohomology. Here all pairs $(u,1)$ occur.

Assume $A(u) \neq 1$ (possible except in type A).

(1) Must some pair $(u,\phi)$ with $\phi \neq 1$ occur?

(2) Is the representation of $A(u)$ on $H^*(\mathcal{B}_u)$ always a permutation permutation?

The answers to both questions seem to be yes, but I don't know any uniform approach using Springer theory. For example, (1) can be checked using case-by-case study of simple types, but is there a general reason for it? For (2) there is a sophisticated indirect argument using work of Bezrukavnikov, Mirkovic, Rumynin. All of this ties in naturally with some unsolved problems about representations of related Lie algebras in characteristic $p$.

notall pairs occur, but would have to hunt around a bit before justifying this comment! – Geordie Williamson Mar 14 '10 at 18:52allnontrivial characters of the component group failed to occur. – Jim Humphreys Mar 14 '10 at 19:22