Speaker: Jared Weinstein, UCLA

The Jacquet-Langlands Correspondence via Fourier Analysis

Oct. 20, 4:00 - 4:50 in MS 5148

Let F be a local nonarchimedean field, and let B/F be its unique quaternion (division) algebra. There is a natural correspondence, due to Jacquet and Langlands, between representations of the unit group of B and those of GL(2,F). The correspondence is characterized by an agreement of quantities known as L- and epsilon-factors. We will provide a new construction of the Jacquet-Langlands correspondence which reduces this required property to a Fourier-analytic calculation on a finite ring. We will start by discussing a few representations of finite groups, and the behavior of their matrix coefficients under Fourier transforms. We will then "inflate" these representations to arrive at representations of the product group GL(2,F) x B* which must realize the Jacquet-Langlands correspondence in its decomposition.