Acta Mathematica

Dirac cohomology for graded affine Hecke algebras

Dan Barbasch, Dan Ciubotaru, and Peter E. Trapa

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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 HD (X) of an $ \mathbb{H} $-module X, and show that HD (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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Acta Math., Volume 209, Number 2 (2012), 197-227.

Received: 3 December 2010
First available in Project Euclid: 31 January 2017

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2012 © Institut Mittag-Leffler


Barbasch, Dan; Ciubotaru, Dan; Trapa, Peter E. Dirac cohomology for graded affine Hecke algebras. Acta Math. 209 (2012), no. 2, 197--227. doi:10.1007/s11511-012-0085-3.

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