Pure Maths Colloquium: Dirac operators for graded Hecke algebras and projective representations of finite reflection groups
30 January 2017 16:00 in CM221
The graded affine Hecke algebras were introduced by Lusztig in the study of smooth representations of reductive p-adic groups and Iwahori-Hecke algebra modules and, independently, by Drinfeld in connection with the representation theory of certain classes of quantum groups. Motivated by the classical Dirac operator theory in the representation theory of real reductive groups, we defined, in joint work with Barbasch and Trapa, Dirac operators for graded Hecke algebra modules. In this setting, the kernels (and indices) of these operators are modules for a certain Pin cover of finite reflection groups. The irreducible characters of this cover had been classified by Schur for symmetric groups and A. Morris and others for the other simple Coxeter groups, but in this new picture, they are naturally related to the geometry of the nilpotent cone of semisimple complex Lie algebras (in the case of Weyl groups) via Springer theory. I will explain the construction of the Dirac operator and the relation between the character theory of Hecke algebras and of these double covers of the reflection groups.
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