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