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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Barbasch-Sahi algebras and Dirac cohomology

Author: Johannes Flake
Journal: Trans. Amer. Math. Soc. 371 (2019), 6883-6902
MSC (2010): Primary 16T05; Secondary 20C08
Published electronically: February 22, 2019
MathSciNet review: 3939564
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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 1986. 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 $ \mathbb{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 Barbasch-Sahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology. This theorem is a common generalization of a result called ``Vogan's conjecture'' for connected semisimple Lie groups which was proved by Huang and Pandžić; analogous results for graded affine Hecke algebras were proved by Barbasch, Ciubotaru and Trapa, and for symplectic reflection algebras and Drinfeld Hecke algebras by Ciubotaru.

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Additional Information

Johannes Flake
Affiliation: Department of Mathematics, Rutgers University, Hill Center – Busch Campus, 110 Frelinghuysen Road, Piscataway, New Jersey 08854–8019
Address at time of publication: Lehrstuhl B für Mathematik (Algebra), RWTH Aachen University, Pontdriesch 10-16, 52062 Aachen, Germany

Keywords: Hopf-Hecke algebras, Barbasch-Sahi algebras, Dirac cohomology, PBW deformations, central characters, Vogan's conjecture, pin cover
Received by editor(s): November 1, 2016
Received by editor(s) in revised form: October 22, 2017
Published electronically: February 22, 2019
Article copyright: © Copyright 2019 American Mathematical Society