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

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