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

这位研究者和他的合作者研究了约化群的表示理论及其在数论中的应用中的几个问题。第一个问题是证明Kirillov猜想，即当限制到某个子群时，GL(n，R)或GL(n，C)的每个不可约酉表示仍然是不可约的。第二个问题是求出半整重模形式的傅里叶系数与整重模形式的L函数的特定值之间的关系式。这个公式推广了Kohnen和Zagier的一个经典公式，不同于Waldspurger给出的公式。所考虑的其他问题包括拟分裂群的Bessel分布和Bessel函数的研究，以及表示辛群和酉群的Fourier Jacobi模型的研究。研究这些问题的主线是通过正则性定理研究某些不变分布和与之相关的球面函数。为了理解和发展量子力学理论，著名物理学家Wigner和Dirac等人研究了半单李群的么正表示。后来，盖尔凡和哈里什·钱德拉等数学家发展并使这一美丽的理论严格化。它在物理学、几何学、数论等领域都有广泛的应用。本提案的目的是增进对表象理论的理解，并继续探索表象理论与数论之间的联系。

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.