1996 Fiscal Year Final Research Report Summary
A study on idenities of the form "infinite sum=infinite product" by an evaluation of determinants.
| Project/Area Number |
08454010
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
WAKAYAMA Masato Kyushu Univ.Graduate School of Mathematics associate professor, 大学院・数理学研究科, 助教授 (40201149)
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| Co-Investigator(Kenkyū-buntansha) |
KONNO Takuya Kyushu Univ.Graduate School of Mathematics research associat, 大学院・数理学研究科, 助手 (00274431)
YOSHIDA Eiji Kyushu Univ.Graduate School of Mathematics assistant professor, 大学院・数理学研究科, 講師 (20220626)
INOUE Junko Kyushu Univ.Graduate School of Mathematics assistant professor, 大学院・数理学研究科, 講師 (40243886)
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| Project Period (FY) |
1996
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| Keywords | trace formula / Pfaffian / dual pair / automorphic form / holonomy group / quantum group |
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| 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 (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
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