Hopf–Hecke algebras, infinitesimal Cherednik algebras, and Dirac cohomology

Hopf-Hecke algebras and Barbasch-Sahi were defined by the first named author (2016) in order to provide a general framework for study of Dirac cohomology. The aim this paper is explore new examples these definitions contribute their classification. are distinguished an orthogonality condition PBW property. property such as ones considered here has been great interest literature we extend discussion further results on classification deformations class hitherto unexplored examples. We infinitesimal Cherednik $GL_n$ Etingof, Gan, Ginzburg [Transform. Groups, 2005] with generalized show that they fact algebras, is, version Vogan's conjecture analogous Huang Pandžic [J. Amer. Math. Soc., 2002] available them. derive explicit formula square operator use it finite-dimensional irreducible modules. find cohomology modules non-zero it, fact, determines uniquely.