# MathSciDoc: An Archive for Mathematician ∫

#### K-Theory and HomologyRepresentation Theorymathscidoc:1701.20001

Acta Mathematica, 209, (2), 197-227, 2010.12
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} \$\$ .
No keywords uploaded!
[ Download ] [ 2017-01-08 20:33:59 uploaded by actaadmin ] [ 555 downloads ] [ 0 comments ] [ Cited by 11 ]
```@inproceedings{dan2010dirac,
title={Dirac cohomology for graded affine Hecke algebras},
author={Dan Barbasch, Dan Ciubotaru, and Peter E. Trapa},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203359595300761},
booktitle={Acta Mathematica},
volume={209},
number={2},
pages={197-227},
year={2010},
}
```
Dan Barbasch, Dan Ciubotaru, and Peter E. Trapa. Dirac cohomology for graded affine Hecke algebras. 2010. Vol. 209. In Acta Mathematica. pp.197-227. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203359595300761.
Please log in for comment!