A study on idenities of the form "infinite sum=infinite product" by an evaluation of determinants.

本研究项目的目的是通过对各种行列式的评价，阐明“无穷积=无穷和”形式的几个重要恒等式的意义：1)Pfaffians的次要和公式，2)对偶对，3)（Selberg）迹公式的精确研究。(1)在量子群（Noumi-Umeda-W）的背景下发展了对偶对理论。, Comp.Math。， 104,1996)，并将其扩展到一般级别案件(Umeda-W。;再看量子矩阵空间上的微分算子及其应用[96]。在此基础上，利用Pfaffinas，用列举组合法对Littlewood的字符公式进行了新的证明，并建立了几个涉及Littlewood公式的恒等式（Ishikawa-Okada-W）。, J.Alg。、183、1996)。我们还得到了包含各种对称函数的生成函数，特别是给出了椭圆的积表示的表示理论解释（Ishikawa-W）。;…次要求和公式的更多应用II，Pfaffians和Schur多项式/96，新Schur函数级数/97)。(2)用显式方法研究了负弯曲局部对称黎曼空间上的迹公式，并应用它解决了完整度的等分布性质问题（Sarnak-W）。;封闭测地线完整度的均一分布[96]。作为一个双积，我得到了关于（受限）完整群的无穷小特征的一个显著估计（与离散级数的最低k型相关的无穷小特征的不等式/97）。与此同时，研究者吉田研究了Kloosterman zeta，以获得关于黎曼曲面上封闭测地线分布的剩余项的良好估计（关于库兹涅托夫迹公式的注释，待见）。此外，Kon-no还研究了一些经典2阶群上自同构表示的Langlands泛函(U （2,2) & Sp(2)的残差谱，将出现）和对一个显式描述逆自同构表示的迹公式的精确研究（GSp(2)的Arthur迹公式I/97）。为了帮助这些研究，Inoue建立了在齐次Siegel域上简单传递的仿射变换的可解群的全纯诱导表示的实现和不可约分解。少

The purpose of the present research project is to clarify a meaninmg of several important identities of the form "infinite product=infinite sum" by an evaluation of various determinants via the ideas : 1) the minor summation formula of Pfaffians, 2) dual pairs, 3) a precise study of (Selberg) trace formulas.(1) We developed the theory of dual pairs in a context of quantum groups (Noumi-Umeda-W., Comp.Math., 104,1996) and extended it to general rank cases (Umeda-W. ; Another look at the differential operators on the quantum matrix spaces and its applications/96). Further, we gave a new proof of Littlewood's formulas for characters by enumerative combinatorics using Pfaffinas and established several identities which involve Littilewood's formulas (Ishikawa-Okada-W., J.Alg., 183,1996). We obtained also generating functions including various symmetric functions and, in particular, gave a representation theoretic interpretation of the product representation of elliptic theta (Ishikawa-W. ; … More Applications of minor summation formulas II,Pfaffians and Schur Polynomials/96, New Schur function series/97).(2) We studied the trace formula on negatively curved locally symmetric Riemannian spaces in an explicit way and solved a problem concerning an equidistribution property of holonomy as an application of it (Sarnak-W. ; Equidistribution of holonomy about closed geodesics/96). As a biproduct, I got a ceratin remarkable estimate concerning an infinitesimal character of the (restricted) holonomy group (An in-equality of infinitesimal characters related to the lowest K-types of discrete series/97). Parallely, the investigator Yoshida studied the Kloosterman zeta for obtaining a good estimate of a remainder term with respect to a distribution of closed geodesic on a Riemann surface (Remarks on the Kuznetov trace formula, to appaer). Also Kon-no studied a Langlands' functoriality for automorphic representations on some classical groups of rank 2 (The residual spectrum of U (2,2) & Sp (2), to appear) and the precise study of trace formula for an explicit description of cuspidal automorphic representations (The Arthur trace formula for GSp (2) I/97). To help these studies, Inoue established a realization and an irreducible decomposition of holomorhically induced representation of some solvable group of affine transformations which act simply transitively on a homogeneous Siegel domain. Less