For a Weyl group , we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the -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 and the Dirac operator for graded affine Hecke algebras play key roles.
"Characters of Springer representations on elliptic conjugacy classes." Duke Math. J. 162 (2) 201 - 223, 1 February 2013. https://doi.org/10.1215/00127094-1961735