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

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The investigator and his collaborators study several problems in therepresentation theory of reductive groups and its applications to numbertheory. The first problem is to prove the Kirillov conjecture which statesthat every irreducible unitary representation of Gl(n,R) or GL(n,C)remains irreducible when restricted to a certain subgroup. The secondproblem is to find a formula relating fourier coefficients of halfintegral weight modular forms with special values of L-functions ofintegral weight modular forms. This formula generalizes a classicalformula of Kohnen and Zagier and is different from a formula givenby Waldspurger. Other problems which are considered are the study ofBessel distributions and Bessel functions for quasi-split groups and thestudy of Fourier Jacobi models for representations of symplectic andunitary groups. The line of attack on these problems is to study certaininvariant distributions and the spherical functions associated to them viaregularity theorems. Unitary representations of semisimple Lie groups were studied by the famous physicists Wigner and Dirac among others in an attempt to understand anddevelop the theory of quantum mechanics. Later, mathematicians such as Gelfandand Harish Chandra developed and rigorized this beautiful theory. Manyapplications were found to physics, geometry, number theory and otherfields. The purpose of the current proposal is to advance the understanding of representation theory and to continue to explore the connectionsbetween representation theory and number theory.