Dirac cohomology for graded affine Hecke algebras

Abstract

We define an analogue of the Casimir element for a graded affine Hecke algebra $ \mathbb{H} $, and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H^D (X) of an $ \mathbb{H} $-module X, and show that H^D (X) carries a representation of a canonical double cover of the Weyl group $ \widetilde{W} $. Our main result shows that the $ \widetilde{W} $-structure on the Dirac cohomology of an irreducible $ \mathbb{H} $-module X determines the central character of X in a precise way. This can be interpreted as p-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of $ \mathbb{H} $.