Characters of finite reductive groups and modular tensor categories

Lusztig's description of the character table of finite reductive groups of Lie type is a
remarkable achievement. His initial observation is that the characters for the general
linear group are given by a "geometric principle", best described via intersection cohomology
methods. For other groups (even in cases like SL_2 or Sp_4) the situation is more complicated
and certain "non-commutative Fourier transform" matrices appear. Over the last ten years,
work of Bezrukanikov-Finkelberg-Ostrik and Lusztig has explained these matrices in terms
of certain remarkable modular tensor categories. I will try to outline this theory and recent
work with Elias which takes the first steps towards understanding these categories for arbitrary
Coxeter groups. The end result should be a rich supply of exotic modular tensor categories
(though actually understanding what one gets is still a long way off.)