Study of dualities and of "infinite sum=infinite product" identities from the trace formula viewpoint

本研究计划的目的是从迹公式的观点出发，对对偶和“无穷surn =无穷乘积”恒等式进行详细的研究。在此期间，我们得到了如下的好结果：1）研究组长和歌山和研究者石川得到了关于Schur多项式生成函数的所谓Littlewood公式的许多推广。他们还建立了各种公式有关的枚举组合和代表性的经典群。2）首席研究员和合作者梅田（京都）开发的理论的对偶对，特别是理论的球谐函数下的量子群对称性。此外，他与他的学生Kinoshita一起回答了关于斜Capelli元的显式构造的公开问题。3）首席研究员和合作者Sarnak（普林斯顿大学）证明了关于斜Capelli元的共轭类分布的一些密度定理。 ...更多信息 负曲局部对称空间上的完整群。为了将这一结果发展到更精确的方式，研究者Midorikawa研究并获得了半单李群的某些表示的特征之间的显式关系。此外，调查Kon-no构建了某些重要的自守表示的单幂型。与微量配方有关，主要研究者和合作者Kurokawa（TIT）证明了Selberg zeta函数的积分表示。4）主要研究者和合作者Parmeggiani（Bologna）引入了称为非对易谐振子的新微分方程系统，并通过连分式和振子表示给出了谱的完整描述。5）调查员Wakimoto提出了一个非常precisattion的representation的仿射李代数，超李代数和顶点算子的模块化的转换。最后，代表调查员的首席调查员要表示特别感谢这种支持。少

The aim of this research project was to make a detailed study of dualities and of "infinite surn = infinite product" identities from the trace formula viewpoint, During the period we obtained good results as follows :1) The head investigator Wakayama and the investigator Ishikawa obtained quite a number of generalizations of the so-called Littlewood formulas concerning the generating functions of Schur ploynomials. They also established various formulas relating the enumerative combinatorics and represenattions of the classical groups.2) The head investigator and the collaborator Umeda (Kyoto) developed the theory of the Dual Pair, especially the theory of spherical harmonics under the quantum group symmetries. Moreover, he gave an answer of the open problem concerning the explicit construction of skew Capelli elements with his student Kinoshita.3) The head investigator and the collaborator Sarnak (Princeton) proved some density theorem for the distribution of the conjugacy class of th … More e holonomy group on negatively-curved Riemannian locally symmmetric spaces. In order to develop this result to more precised way, the investigator Midorikawa studied and obtained an explicit relation among characters of certain representations of semi simple Lie groups. Also the investigator Kon-no constructed certain important automorphic representations of the unipotent type. Related to the trace formula, the head investigator and the collaborator Kurokawa (TIT) proved the integral represenattion of the Selberg zeta function.4) The head investigator and the collaborator Parmeggiani (Bologna) introduced the new system of differntial equation called by the non-commutative harmonic oscillator and gave a complete description of the spectrum by means of continued fractions and the oscillator represenations.5) The investigator Wakimoto made a very precised study of the represenattion of the affine Lie algebra, the super Lie algebra and vertex operators by means of modular transforms.Finally, on behalf of the investigators the head investigator would like to express their special thanks to this support. Less