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

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 $A_W=\bC[W]\ltimes S(V)$.