## Acta Mathematica

### 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 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 $\mathbb{H}$.

#### Article information

Source
Acta Math., Volume 209, Number 2 (2012), 197-227.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892660

Digital Object Identifier
doi:10.1007/s11511-012-0085-3

Mathematical Reviews number (MathSciNet)
MR3001605

Zentralblatt MATH identifier
1276.20004

Rights

#### Citation

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. https://projecteuclid.org/euclid.acta/1485892660

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