  To any element of a connected, simply connected, semisimple complex algebraic
group G and a choice of an element of the corresponding Weyl group there is an
associated Lusztig variety. When the element of G is regular semisimple, the
corresponding variety carries an action of the Weyl group on its (equivariant)
intersection cohomology. From this action, we recover the induced characters of
an element of the Kazhdan-Lusztig basis of the corresponding Hecke algebra. In
type A, we prove a more precise statement: that the Frobenius character of this
action is precisely the symmetric function given by the characters of a
Kazhdan-Lusztig basis element. The main idea is to find celular decompositions
of desingularizations of these varieties and apply the Brosnan-Chow
palindromicity criterion for determining when the local invariant cycle map is
an isomorphism.
  This recovers some results of Lusztig about character sheaves and gives a
generalization of the Brosnan-Chow solution to the Sharesian-Wachs conjecture
to non-codominant permutations, where singularities are involved. We also
review the connections between Immanants, Hecke algebras, and Chromatic
quasisymmetric functions of indifference graphs.
