Characters of Springer representations on elliptic conjugacy classes

Abstract

For a Weyl group $W$, we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of $W$ in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the $W$-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of $W$ and the Dirac operator for graded affine Hecke algebras play key roles.