[Geom., Alg. & Phys.] 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. (https://sites.google.com/view/gapkias) (Zoom Meeting ID: 896 1004 4057, Passcode: 163412, link https://kias-re-kr.zoom.us/j/89610044057?pwd=QFeBYHcRG6bJBTVupKux4a2cEaZzVh.1)