Characters of Springer representations on elliptic conjugacy classes

TL;DR

This paper derives explicit formulas for Springer representation characters and discrete series characters of graded affine Hecke algebras on elliptic conjugacy classes, utilizing the Pin double cover and Dirac operator.

Contribution

It provides new closed-form formulas for characters on elliptic classes, connecting Springer theory and affine Hecke algebra representations.

Findings

Closed formula for Springer representation characters on elliptic classes

Explicit W-character formula for discrete series of graded affine Hecke algebras

Use of Pin double cover and Dirac operator in character formulas

Abstract

For a Weyl group W, we give a simple closed formula (valid on elliptic conjugacy classes) for the character of the representation of W in each A-isotypic component of the full 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.