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 \( \mathbb{H} \).
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Barbasch, D., Ciubotaru, D. & Trapa, P.E. Dirac cohomology for graded affine Hecke algebras. Acta Math 209, 197–227 (2012). https://doi.org/10.1007/s11511-012-0085-3
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DOI: https://doi.org/10.1007/s11511-012-0085-3