Wednesday, July 25, 15:00, 8.122

Johannes Flake (Rutgers), Dirac cohomology for generalized Hecke algebras.

Abstract:
Dirac operators have been important tools in representation theory, especially for unitary representations. Following a conjecture of David Vogan, it was shown more recently that for many classes of possibly non-unitary representations, the cohomology of a Dirac operator determines the central character, both in the context of real reductive Lie groups and for various types of Hecke algebras. I will explain how these cases can be studied uniformly, using smash products of Hopf algebras, PBW deformations of Koszul algebras, and superalgebras, and how an analog of Vogan's conjecture can be obtained in this general setting. As a novel application, we will discuss infinitesimal Cherednik algebras and their representations. Finally, I will present some open problems, including classification problems for classes of generalized Hecke algebras.