Event Details

Speaker: Dan Ciubotaru (Oxford) Title: The Dirac operator and representations of Coxeter groups

Abstract: The classical Dirac operator acting on sections of spinor bundles over a Riemannian symmetric space has led to spectacular applications to the representation theory of semisimple Lie groups in the works of Parthasarathy, Atiyah and Schmid, Kostant, Wallach in the 1970s and 1980s. Owing to new ideas of D. Vogan at the end of 1990s, a new algebraic approach to the Dirac operator (`Dirac cohomology') for real reductive groups was developed by Huang and Pandzic, Barbasch, Renard, and others. By analogy with this theory, about 12 years ago, in joint work with D. Barbasch and P. Trapa, we defined an algebraic Dirac operator for a class of convolution (Hecke) algebras and studied its connections with Springer representations of Weyl groups. I will give an overview of our construction and the old results, and then present some new observations relating the Dirac index in this setting, the spin representations of the Weyl group, and the partition of the set of irreducible Weyl group representations defined by Lusztig in the context of finite reductive groups.