Abstract
We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld (Anal i Prilozhen 20(1):69–70, 1986). We generalize in this way, the Dirac cohomology theory for Lusztig’s graded affine Hecke algebras defined in Barbasch et al. (Acta Math 209(2):197–227, 2012) and further developed in Barbasch et al. (Acta Math 209(2):197–227, 2012), Ciubotaru et al. (J Inst Math Jussieu 13(3):447–486, 2014), Ciubotaru (J Reine Angew Math 671:199–222, 2012), Ciubotaru and He (Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index, Adv. Math. 283:1–50, 2015), Chan (On a twisted Euler-Poincaré pairing for graded affine Hecke algebras, preprint 2014, arXiv:1407.0956). We apply these constructions to the case of the symplectic reflection algebras defined by Etingof and Ginzburg (Invent Math 147:243–348, 2002), particularly to rational Cherednik algebras for real or complex reflection groups. As applications, we give criteria for unitarity of modules in category \({\mathcal {O}}\) and we show that the 0-fiber of the Calogero–Moser space admits a description in terms of a certain “Dirac morphism” originally defined by Vogan for representations of real reductive groups.
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It is a pleasure to thank B. Krötz and E. Opdam for the invitation to give a series of lectures on the theory of the Dirac operator for Hecke algebras at the Spring School “Representation theory and geometry of reductive groups”, Heiligkreuztal 2014, where some of these ideas crystallized. I also thank J.S. Huang, K.D. Wong, and the referee for corrections, helpful comments, and references.
