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“无限积=无限和”形式的几个重要恒等式的含义：1)Pfaffian的次和公式，2)对偶对，3)(Selberg)迹公式的精确研究。(1)我们在量子群的背景下发展了对偶对理论(Noumi-Umeda-W.，Comp.Math，104,1996)，并将其推广到一般秩情形(Umeda-W)。；再看一下量子矩阵空间上的微分算子及其应用/96)。此外，我们用Pfaffinas用计数组合的方法给出了特征标的Littlewood公式的一个新的证明，并建立了涉及Littilewood公式的几个恒等式(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，特别地，给出了椭圆角的乘积表示的表示理论解释(Ishikawa-W.；…(2)我们显式地研究了负曲局部对称黎曼空间上的迹公式，并解决了作为它的一个应用的完整的等分布性质问题(Sarnak-W.；关于闭合测地线的完整的均匀分布/96)。作为一个双积，我得到了一个关于(受限的)完整群的无穷小特征(与离散序列的最低K型有关的无穷小特征的不等性/97)的显著估计。同时，研究人员吉田研究了Krousterman Zeta，以获得关于Riemann曲面上闭测地线分布的余项的良好估计(关于Kuznetov迹公式的评论，见appaer)。Kon-no还研究了一些2阶经典群(即将出现U(2，2)和Sp(2)的剩余谱)上自同构表示的朗兰兹函数性和精确刻画尖角自同构表示的迹公式(GSP(2)I/97的Arthur迹公式)。为了帮助这些研究，井上建立了齐次Siegel域上简单传递作用的某些可解仿射变换群的全纯诱导表示的实现和不可约分解。较少

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