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

本研究的目的是从迹公式的观点对对偶和“无限冲程=无限乘积”恒等式进行详细的研究，在此期间我们得到了以下良好的结果：1)首席研究员Wakayama和Ishikawa得到了许多关于Schur多项式的母函数的所谓Littlewood公式的推广。他们还建立了与经典群的计数组合学和表示有关的各种公式。2)首席研究员和合作者内田(京都)发展了对偶理论，特别是量子群对称下的球谐理论。3)首席研究员和合作者Sarnak(普林斯顿)证明了…的共轭类分布的一些稠密性定理负曲黎曼局部对称空间上的更多完整群。为了将这一结果推广到更精确的方向，研究人员Midorikawa研究并得到了半单李群的某些表示的特征标之间的显式关系。此外，Kon-no还构造了一些重要的酉型自同构表示。与迹公式相关，首席研究员和合作者Kurokawa(TIT)证明了Selberg Zeta函数的积分表示。4)首席研究员和合作者Parmegiani(Bologna)引入了由非对易谐振子所称的新的微分方程组，并通过连分式和振子表示给出了频谱的完整描述。5)首席研究员Wakimoto通过模变换对仿射李代数、超李代数和顶点算子的表示进行了非常精确的研究。最后，首席研究员代表研究人员对这一支持表示了特别的感谢。较少

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