Combinatorial constructions in Representation Theory and Number Theory

Speaker: Dr Anna Puskas
Affiliation: University of Queensland

Abstract

Classical matrix groups over local fields play a central role in number theory. Their structure can be understood in terms of discrete data, and therefore some important objects in their representation theory have combinatorial constructions. In this talk, we will talk about some of these objects: metaplectic Iwahori-Whittaker functions. Formulas for Whittaker functions have been given in terms of a modified group action of the symmetric group (by work of Chinta-Gunnells, Chinta-Offen, McNamara), as well as using combinatorial devices such as highest-weight crystals. These can be related and better understood using representations of the Iwahori-Hecke algebra. This theory carries over to the infinite-dimensional setting, and connects with work on double affine Hecke algebras, where several intriguing open questions remain.