1999 Fiscal Year Final Research Report Summary
Modular Representation Theory of Algebraic Groups
| Project/Area Number |
10640036
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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 Cith University |
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Principal Investigator |
KANEDA Masaharu Osaka City University, Faculty of Science, Professor, 理学部, 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, 理学部, 教授 (70142916)
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| Project Period (FY) |
1998 – 1999
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| Keywords | algebraic groups / representation theory / positive characteristic / D-modules / Weyl modules / infinitesimal method / quantum algebras / cohomology |
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| Research Abstract |
Unlike in characteristic 0, the D-module theory in positive characteristic remains in a dark. Inspired partly by recent introduction of holonomicity in positive characteristic by R.Bφgvad, I have tried to write down some basics of equivariant D-module theory with respect to an action of an algebraic group.
We say a module for a reductive group in positive characteristic has a good filtration if it admits a filtration whose subquotients are all Well modules. The questions of whether or not the tensor product of two modules with good filtrations remains to have a good filtration, and also of whether a module with good filtration admits a good filtration with respect to a Levi subgroup are two basic problems in the representation theory. After Wang J.-P. and S.Donkin solved the problems in most cases, O.Mathieu gave a brilliant, but a rather hard, proof in affirmative using the Frobenius splittings. It turned out that another entirely different proof was buried in Lusztig's theory of based modules. Expanding a little on Xi N.-H.'s observations, I have explained the alternative solution.
Some of fundamental cohomological results in the representation theory of algebraic groups hold over the ring of integers Z. I have quantized the Andersen-Haboush identity over the Laurent polynomial ring Z[v,vィイD1[-1ィエD1], that makes possible the quantization of classical cohomological results such as Kempf's vanishing theorem over Z[v,vィイD1-1ィエD1].
With the assistance in travel expenses I invited H. H. Andersen to work together, that has proved very fruitful. We can now describe the Jantzen filtration on infinitesimal Well modules by the Andersen filtration.
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