rt.representation theory - Role of nontrivial component groups in Springer Correspondence?

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 $p$.

• Next riemannian geometry - Hermitian symmetric spaces vs Hermitian homogeneous spaces
• Previous ag.algebraic geometry - Proper definition of a moduli problem