Dirac cohomology for graded affine Hecke algebras

Dan Barbasch Department of Mathematics, Cornell University Dan Ciubotaru Department of Mathematics, University of Utah Peter E. Trapa Department of Mathematics, University of Utah

K-Theory and Homology Representation Theory mathscidoc: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 $$ \mathbb{H} $$ .
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@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.
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