Budget Amount *help 
¥3,800,000 (Direct Cost : ¥3,800,000)
Fiscal Year 1996 : ¥3,800,000 (Direct Cost : ¥3,800,000)

Research Abstract 
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 (NoumiUmedaW., Comp.Math., 104,1996) and extended it to general rank cases (UmedaW. ; 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 (IshikawaOkadaW., 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 (IshikawaW. ;
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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 (SarnakW. ; 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 inequality of infinitesimal characters related to the lowest Ktypes 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 Konno 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
