2000 Fiscal Year Final Research Report Summary
Representation theory of algebraic groups through algebraic analysis
| Project/Area Number |
11440009
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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 | Hiroshima University |
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Principal Investigator |
TANISAKI Toshiyuki Hiroshima Univ., Graduate school of Science, Professor, 大学院・理学研究科, 教授 (70142916)
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| Co-Investigator(Kenkyū-buntansha) |
SHOJI Toshiaki Tokyo Science Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (40120191)
KASHIWARA Masaki Kyoto Univ., Research Institute of Natheuatical Science, Professor, 数理解析研究所, 教授 (60027381)
SAITO Yashihisa Hiroshima Univ., Graduate school of Science, Assistant, 大学院・理学研究科, 助手 (20294522)
KAWANAKA Norkki Osaka Univ., Graduate School of Science, Professor, 理学研究科, 教授 (10028219)
KANEDA Masahara Osaka City Univ., Faculty of Science, Professor, 理学部, 教授 (60204575)
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| Project Period (FY) |
1999 – 2000
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| Keywords | algebraic group / representation / algebraic analysis |
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| Research Abstract |
1. Highest weight modules over Kac-Moody Lie algebras
Kashiwara and Tanisaki tried to determine characters of irreducible modules over affine Lie algebras with critical highest weights. In particular, they considered relations of representations of affine Lie algebras with D-modules on the semi-infinite flag manifold. Through the investigation they noticed that the behavior of the equivariant line bundles is quite different from those on the ordinary flag manifold.
2. Flag manifold for quantum groups
Tanisaki constructed generalized flag manifold corresponding to general parabolic subgroups. The treatment of the unipotent radical is more delicate than the ordinary case. Besides Morita and Tanisaki tried to find a good formulation for the sheaf cohomologies and D-modules on the quantized flag manifold from the view point of non-commutative schemes.
3. Representations of toroidal Lie algebras
Saito constructed certain representations of toroidal Lie algebras usig Boson. He also investigated the automorphisms of the toroidal Lie algebras and found a connection with the moudular groups.
4. Representations of quantum groups over Laurent polynimial rings
Kaneda investigated representations of quantum groups over the Laurent polynimial rings, and have proved a version of a theorem of Kempf.
5. Solvable games
Kawanaka found a generalization of Sato's game, and investigated on it from the view point of representation theory.
6..Representations of comples reflection groups and their Hecke algebras
Shoji tried to extend the Frobenius formua for the Hecke algebra of type A to the Hecke algebras for complex reflection groups, and succeed in it in the case of Ariki-Koike algebra.
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