Resumen de Dirac cohomology for symplectic reflection algebras

Dan Ciubotaru

• 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 OO 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.