Project/Area Number  09440022 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
Algebra

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
WAKAYAMA Masato Kyushu Univ., Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (40201149)

CoInvestigator(Kenkyūbuntansha) 
ISHIKAWA Masao Tottori University, Faculty of Education, Associate Professor, 教育学部, 助教授 (40243373)
HASHIZUME Michihiko University of Okayama, Faculty of Science, Science, Professor, 理学部, 教授 (50033890)
MIDORIKAWA Hisaichi Tsuda College, Faculty of Arts and Sciences, Professor, 学芸学部, 教授 (80055318)
KONNO Takuya Kyushu Univ., Graduate School of Mathematics, Research Assistant, 大学院・数理学研究科, 助手 (00274431)
吉田 英治 九州大学, 大学院・数理学研究科, 講師 (20220626)
WAKIMOTO Minoru Kyushu Univ., Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (00028218)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥7,700,000 (Direct Cost : ¥7,700,000)
Fiscal Year 1998 : ¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1997 : ¥5,000,000 (Direct Cost : ¥5,000,000)

Keywords  trace formulas / Paffian / Schur polynomial / Capelli identity / dual pair / oscillator representation / automorphic form / Lie algebra / 跡公式 / Schur 多項式 / Capelli 恒等式 / oscillator 表現 / affin Lie 環 / 不変微分作用素 / Schur多項式 / Capelli恒等式 / Dual pair / oscillator表現 / affin Lie環 / Schur関数 / Dual Pair / Oscillator表現 
Research Abstract 
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 socalled 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 negativelycurved 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 Konno 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 noncommutative 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
