Hecke algebras, Whittaker functions and quantum groups

Abstract: I will give a brief overview of the Satake isomorphism and the Casselman-Shalika formula, which are basic tools in the representation theory of p-adic groups. These two results essentially state that the spherical Hecke algebra and the spherical Whittaker functions on a p-adic group can be understood in terms of the representation theory of the dual group. When passing from p-adic groups to their metaplectic covers, it was conjectured by Gaitsgory and Lurie (recently proved in different settings by Campbell-Dhillon-Raskin and Buciumas-Patnaik) that the dual group gets replaced by a certain quantum group at a root of unity. I will try to explain the conjecture of Gaitsgory-Lurie and if time permits some of the ideas of the proof in the algebraic setting, as well as some interactions to combinatorics and number theory.

mathematical physicsalgebraic geometrygeometric topologyquantum algebra

Audience: researchers in the discipline

Geometry, Algebra and Physics at KIAS