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

本课题从迹公式的角度对对偶性和“无限n =无限积”恒等式进行了详细的研究。在此期间，我们取得了以下良好的成果：1)首席研究员Wakayama和研究员Ishikawa对所谓的关于Schur多项式生成函数的Littlewood公式进行了大量的推广。他们还建立了与经典群的枚举组合和表示有关的各种公式。2)首席研究员和合作者Umeda（京都）发展了对偶理论，特别是量子群对称下的球谐理论。3)在负弯曲黎曼局部对称空间上，主要研究者和合作者Sarnak （Princeton）证明了负弯曲黎曼局部对称空间上完整群共轭类分布的一些密度定理。为了更精确地发展这一结果，研究者Midorikawa研究并得到了半简单李群的某些表示的特征之间的显式关系。研究者Kon-no还构造了一些重要的单能型自同构表征。与示踪公式相关，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Parmeggiani（博洛尼亚）引入了非交换谐振子的微分方程新系统，并利用连分式和振子表示法给出了谱的完整描述。5)研究者Wakimoto用模变换的方法对仿射李代数、超李代数和顶点算子的表示进行了非常精确的研究。最后，首席研究员代表调查人员对这种支持表示特别的感谢。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