Barbasch-Sahi algebras and Dirac cohomology

Abstract

Abstract. We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld in [Dr]. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over C with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic 0 with an orthogonal module.Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call BarbaschSahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology, generalizing a result called "Vogan's conjecture" for connected semisimple Lie groups which was proved in [HP1], analogous results for graded affine Hecke algebras in [BCT] and for symplectic reflection algebras and Drinfeld Hecke algebras in [Ci].