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

研究者和他的合作者研究了约化群表示理论中的几个问题及其在数论中的应用。第一个问题是证明基里洛夫猜想，该猜想表明Gl（n，R）或Gl（n， C）的每一个不可约酉表示在限定于某一子群时仍然不可约。第二个问题是找出半整权模形式的傅立叶系数与整权模形式的l函数的特殊值之间的关系式。这个公式推广了Kohnen和Zagier的经典公式，不同于Waldspurger给出的公式。其他考虑的问题是准分裂群的贝塞尔分布和贝塞尔函数的研究以及辛群和酉群表示的傅里叶雅可比模型的研究。解决这些问题的思路是通过正则性定理来研究某些不变分布和与之相关的球函数。著名物理学家维格纳和狄拉克等人研究了半单李群的酉表示，试图理解和发展量子力学理论。后来，像Gelfandand Harish Chandra这样的数学家发展并完善了这个美丽的理论。在物理、几何、数论和其他领域发现了许多应用。本研究的目的是促进对表征理论的理解，并继续探索表征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.