Studies in Representation Theory

Harvard University, Cambridge MA

Investigators

Abstract

Abstract Schmid The proposal has two related themes: a new method for establishing the analytic continuation and functional equation of L-functions attached to automorphic representations, and a study of irreducible unitary representations of reductive Lie groups. Broadly speaking, functional equations have been proved either by the method of integral formulas or the Langlands-Shahidi method. These have complementary advantages and drawbacks, and seem to be approaching the limits of their applicability. In joint work with Steve Miller, I have begun to derive analytic properties of L-functions from an analysis of the corresponding automorphic distribution. In this approach, the Gamma factors arise from computations on arithmetic quotients of nilpotent groups, which are far more tractable than the analogous computations with Whittaker functions. We expect to complete the analysis of the exterior-square and symmetric square L-functions for GL(n,Z), possibly also for higher level. We shall then consider L-functions for other groups. Although there are many isolated results about irreducible unitary representations of reductive Lie groups, they do not yet suggest a general picture. Kari Vilonen and I shall apply Morihiko Saito's theory of mixed Hodge modules to the study of unitary representations. Saito's theory implies the existence of certain canonical filtrations on Harish-Chandra modules, which do not seem visible from any other point of view. We already have a concrete conjecture about the unitary dual of a reductive Lie group. We shall work on the proof of the conjecture and investigate its implications. Riemann's Zeta function encodes deep properties of prime numbers. Conjecturally automorphic L-functions do the same for primes in number fields. The behavior of prime numbers has fascinated mathematicians for centuries, and will likely continue to do so in the future. However, prime numbers now have practical, highly significant applications in cryptography. For this reason, L-functions have become important not only to analytic and algebraic number theorists, but also to cryptographers. Irreducible unitary representations of reductive groups constitute the "atoms" of Fourier analysis on such groups and their quotient spaces. They play a significant role in number theory, specifically the Langlands program. The mathematical physics literature abounds with examples of irreducible unitary representations. This aspect of mathematical physics would be clarified and unified by a systematic understanding of such representations.