Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index

TL;DR

This paper establishes a new connection between Springer theory, Green polynomials, and the extended Dirac operator for graded Hecke algebras, providing a uniform construction of irreducible genuine characters of a double cover of Weyl groups.

Contribution

It introduces a novel uniform construction of irreducible genuine characters of the pin cover of Weyl groups and develops a $q$-elliptic pairing related to Springer theory and Kostka systems.

Findings

Constructed a $q$-elliptic pairing generalizing Reeder's elliptic pairing.

Established a connection between Green polynomials and the extended Dirac operator.

Provided a new construction of irreducible genuine $ ilde{W}$-characters.

Abstract

We provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of the pin cover $\wti W$, a certain double cover of the Weyl group $W$, and an extended Dirac operator for graded Hecke algebras. Our approach leads to a new and uniform construction of the irreducible genuine $\wti W$-characters. In the process, we give a construction of the action by an outer automorphism of the Dynkin diagram on the cohomology groups of Springer theory, and we also introduce a $q$-elliptic pairing for $W$ with respect to the reflection representation $V$. These constructions are of independent interest. The $q$-elliptic pairing is a generalization of the elliptic pairing of $W$ introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product $A_W=\bC[W]\ltimes S(V)$.