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

In this paper, we give a uniform construction of irreducible genuine characters of the Pin cover W˜ of a Weyl group W  , and put them into the context of theory of Springer representations. In the process, we provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of W˜, and an extended Dirac operator for graded Hecke algebras. 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 AW=C[W]⋉S(V)AW=C[W]⋉S(V).