Dirac cohomology for graded affine Hecke algebras

By D. Barbasch, Dan Ciubotaru, Peter E. Trapa. 2010

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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 $$ \mathbb{H} $$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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