| Project/Area Number |
11440011
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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 | Tokyo Institute of Technology (2001)
Kyushu University (1999-2000) |
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Principal Investigator |
OCHIAI Hiroyuki Graduate school of Science and Engineering, Tokyo Institute of Technology, AP, 大学院・理工学研究科, 助教授 (90214163)
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| Co-Investigator(Kenkyū-buntansha) |
KANEKO Masanobu Kyushu University, Mathematics, P, 大学院・数理学研究院, 教授 (70202017)
OSHIMA Toshio The University Tokyo, Mathematical Science, P, 大学院・数理科学研究科, 教授 (50011721)
WAKIMOTO Minoru Kyushu University, Mathematics, P, 大学院・数理学研究院, 教授 (00028218)
KONNO Takuya Kyushu University, Mathematics, AP, 大学院・数理学研究院, 助教授 (00274431)
SHIMENO Nobukazu Okayama Science University, Science, AP, 理学部, 助教授 (60254140)
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| Project Period (FY) |
1999 – 2001
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| Keywords | semisimple / highest weight / degree / theta correspondence / nilpotent orbit / associated variety / prehomogeneous / Kazhdan-Lusztig |
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| Research Abstract |
The Bernstein degree, associated cycles and isotropy representation for a unitary highest weight representation of reductive Lie groups are important family of geometric invariants. For the symplectic groups, Ochiai with K. Nishiyama determines the Bernstein degree. For orthogonal and unitary groups, Nishiyama, Ochiai and K. Taniguchi determines the Bernstein degree and associated cycles. The general case including exceptional groups are done by Ochiai and Shohei Kato. This results are generalized to describe the degree of spherical orbits of a family of multiplicity-free representations by Ochiai and Kato. This is a new application of the Selberg-type integral.
A representation is considered as a quantization of a function space. Oshima introduces the notion of homogenized enveloping algebra, in order to deal with universal enveloping algebras and coordinate rings simultaneously. Using the Capelli operator, which is a quantization of minors, Oshima describes the annihilator ideals of generalized Verma modules of scalar type. Related to this work, Ochiai describes the difference of the centers of the enveloping algebras and the rings of invariant differential operators. Based on the work above, Oshima with Shimeno characterize the image of the Poisson transform associated to non-minimal parabolic.
Wakimoto discuss the modular invariance of characters of various affine algebras. Kaneko investigates the modular function *j from the geometry of singular moduli and supersingular elliptic curves over finite fields. Ochiai also discuss the quasi-modularity of the generating functions of an elliptic curve with a specified type of ramification indeces. Konno considers the unipotent representations of reductive groups over local fields, especially gives the description of CAP representations, which is a special subclass of unipotent representation, for low-rank groups.
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