1998 Fiscal Year Final Research Report Summary
Study of dualities and of "infinite sum=infinite product" identities from the trace formula viewpoint
| Project/Area Number |
09440022
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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, Professor, 大学院・数理学研究科, 教授 (40201149)
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| Co-Investigator(Kenkyū-buntansha) |
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)
KON-NO Takuya Kyushu Univ., Graduate School of Mathematics, Research Assistant, 大学院・数理学研究科, 助手 (00274431)
WAKIMOTO Minoru Kyushu Univ., Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (00028218)
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| Project Period (FY) |
1997 – 1998
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| Keywords | trace formulas / Paffian / Schur polynomial / Capelli identity / dual pair / oscillator representation / automorphic form / Lie algebra |
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| 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 so-called 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 negatively-curved 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 Kon-no 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 non-commutative 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
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