2002 Fiscal Year Final Research Report Summary
Modular Representation Theory of Algebraic Groups
| Project/Area Number |
12640034
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Osaka City University |
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Principal Investigator |
KANEDA Masaharu Osaka City University, Faculty of Science Professor, 大学院・理学研究科, 教授 (60204575)
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| Co-Investigator(Kenkyū-buntansha) |
TEZUKA Michishige Ryukyu University, Faculty of Science Professor, 理学部, 教授 (20197784)
YAGITA Nobuaki Ibaraki University, Faculty of Education Professor, 教育学部, 教授 (20130768)
TANISAKI Toshiyuki Hiroshima University, Faculty of Science Professor, 大学院・理学研究科, 教授 (70142916)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Lusztig's conjecture / Kempf's vanishing theorem / twining character / D-modules / Demazure modules / infinitesimal groups / quantum algebras |
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| Research Abstract |
Kempf's cohomology vanishng theorem, a fundamental result in the repreasentation theory, in the set up of quantum algebras over the laurent polynomial ring Z[v,v^<-1>] follows from the paper 4 cited below
The third paper studies a filtration on the hom-space of a standard module into a projective/injective in the category of G_1T-modules in relation to the Lusztig conjecture on the irreducible characters of a reductive algebraic group G over a field of positive characteristic. We formulate a conjecture from which Lusztig's conjecture will follow, and finds a relation to the Jantzen fitration on the standard modules
The second paper was meant to be a tiny step forward toward a proof of the D-affinity of the flag variety in positive characteristic. Last summer, however, Kashiwara and Lauritzen found a counter example to the D-affinity. Also last summer Bezrukavnikov, Mirkovic and Rumynin introduced a ring of crystalline differential operator U-modules of finite type with Harish-Chandra central character 0 and coherent D-modules, which is a generalization of the Beilinson-Bernstein correspondence in characteristic 0. They further show that the latter category with generalized Frobenius central characteristic 0. They further show that the latter category with generalized Frobenius central character X is equivalent to the category of coherent modules on the correponding Springer fiber, and that the latter is computed by transferring to characteristic 0. This opens up a new geometric way of attacking the Lusztig conjecture. We had luxury of Rumynin's visit by the grant and had very informative discussion
The first paper obtains a twining character formula of Demazure modules
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