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

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 $\mathsf{GL}_n$ 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 Pandžić 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.

Keywords

Hopf–Hecke algebras, Barbasch–Sahi algebras, Dirac cohomology, PBW deformations, infinitesimal Cherednik algebras

2010 Mathematics Subject Classification

Primary 16T05. Secondary 20C08.

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In fond memory of Bert Kostant: friend, philosopher, and guide.

The research of the second author was partially supported by a Simons Foundation grant (509766) and a National Science Foundation grant (DMS-2001537).

Received 16 April 2020

Accepted 13 September 2020

Published 22 December 2021