Distributions and Representations: The Kirillov Conjecture, Special Values Of L-functions and Fourier-Jacobi Models

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The investigator and his collaborators study several problems in the representation theory of reductive groups and its applications to number theory. The first problem is to prove the Kirillov conjecture which states that every irreducible unitary representation of Gl(n,R) or GL(n,C) remains irreducible when restricted to a certain subgroup. The second problem is to find a formula relating fourier coefficients of half integral weight modular forms with special values of L-functions of integral weight modular forms. This formula generalizes a classical formula of Kohnen and Zagier and is different from a formula given by Waldspurger. Other problems which are considered are the study of Bessel distributions and Bessel functions for quasi-split groups and the study of Fourier Jacobi models for representations of symplectic and unitary groups. The line of attack on these problems is to study certain invariant distributions and the spherical functions associated to them via regularity theorems. Unitary representations of semisimple Lie groups were studied by the famous physicists Wigner and Dirac among others in an attempt to understand and develop the theory of quantum mechanics. Later, mathematicians such as Gelfand and Harish Chandra developed and rigorized this beautiful theory. Many applications were found to physics, geometry, number theory and other fields. The purpose of the current proposal is to advance the understanding of representation theory and to continue to explore the connections between representation theory and number theory.