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

Ciubotaru, D
He, X

1 October 2015

## Last Updated:

2020-09-09T11:30:02.39+01:00

283

## DOI:

10.1016/j.aim.2015.07.002

1-50

## abstract:

© 2015 Elsevier Inc. 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).

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