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2012 Dirac cohomology for graded affine Hecke algebras
Dan Barbasch, Dan Ciubotaru, Peter E. Trapa
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Acta Math. 209(2): 197-227 (2012). DOI: 10.1007/s11511-012-0085-3


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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Dan Barbasch. Dan Ciubotaru. Peter E. Trapa. "Dirac cohomology for graded affine Hecke algebras." Acta Math. 209 (2) 197 - 227, 2012.


Received: 3 December 2010; Published: 2012
First available in Project Euclid: 31 January 2017

zbMATH: 1276.20004
MathSciNet: MR3001605
Digital Object Identifier: 10.1007/s11511-012-0085-3

Rights: 2012 © Institut Mittag-Leffler

Vol.209 • No. 2 • 2012
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