Speaker: Ben Brubaker, University of Minnesota

Title: Combinatorial Solutions to Automorphic Problems

Introduction: These two talks will explain some interesting combinatorial problems and solutions that arise when studying automorphic forms and L-functions. The talks will present two very different combinatorial approaches, so are meant to stand alone, and will not assume any prior knowledge of automorphic forms. Instead they are meant as an invitation to a very rich dictionary between the two subjects, with implications in both directions.

Lecture 1: Generating functions for Schur polynomials and automorphic forms

Abstract: Character values of highest weight representations of Lie groups may be represented using the Weyl character formula or as a generating function over a combinatorial basis for the representation -- e.g. Young tableaux or Gelfand-Tsetlin patterns. We'll explain a simultaneous deformation of these two expressions for the character, and how it relates to certain matrix coefficients used in the construction of automorphic L-functions. We'll also briefly discuss several methods for proving the generating function identity, including crystal bases, square ice, and via Hamiltonians on fermions in a Dirac sea.

Lecture 2: Hecke algebra modules and automorphic forms

Abstract: The Iwahori-Hecke algebra arises naturally as a space of functions on an algebraic group over a local field, so it is not so surprising that it should play a starring role in automorphic forms. We'll explain how, given one single critical fact, the Hecke algebra allows one to easily compute spherical functions arising in automorphic forms, a class of special functions closely related to symmetric and non-symmetric Macdonald polynomials. At the end, we propose a recipe for determining the one magic fact.