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

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We provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of the pin cover 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 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 AW=[W] S(V).