Dirac cohomology for symplectic reflection algebras

Abstract

We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld. We generalize in this way the Dirac cohomology theory for Lusztig's graded affine Hecke algebras. We apply these constructions to the case of symplectic reflection algebras defined by Etingof-Ginzburg, particularly to rational Cherednik algebras for real or complex reflection groups with parameters t,c. As applications, we give criteria for unitarity of modules in category 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 semisimple Lie groups.