A categorical approach to real Lie groupsMPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
The representation theory of real reductive Lie groups, such as SL(2,R), has played a fundamental role in mathematics for decades, with beautiful connections to number theory (through automorphic forms and the Langlands program), mathematical physics (through conservation laws and quantum states) and harmonic analysis (through nonabelian Fourier transform). Because of its varied applications, it has been studied through many different lenses – there is a rich set of geometric, algebraic, and analytic tools to describe admissible and unitary representations. In this talk, I will discuss a categorical perspective on this classical story, which emphases the combinatorial role of the Weyl group and its subgroups. Specifically, I will introduce an algebraic family of categories constructed from only Weyl group data, which encode information about characters of admissible representations of a real reductive group. These categories arise as module categories over the monoidal category of Soergel bimodules.