| Project/Area Number |
15K04789
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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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 大阪市立大学, 大学院理学研究科, 教授 (60204575)
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| Research Collaborator |
TANISAKI TOSHIYUKI
YAGITA NOBUAKI
TEZUKA MICHISHIGE
FURUSAWA MASAAKI
KIMURA YOSHIYUKI
KAWATA SHIGETO
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | algebraic groups / Frobenius morphism / representation theory / Frobenius contraction / Steinberg module / injective modules / good filtration / exceptional sequence / Frobenius direct image / G_1P-Verma module / graded induction / translation functor / Koszulity / geometrically split / Frobenius splitting / Frobenius cosplitting / geometric splitting |
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| Outline of Final Research Achievements |
Let G denote a reductive algebraic group split over a field of characteristic p>0. In joint work with Abe noriyuki we determined for p>>0 the Loewy structure of the G_1T-Verma modules of singular highest weights, G_1 the Frobenius kernel of G and T a maximal torus of G. On the Grassmannian Gr(2,5) and on G/P with G in type G_2 and P a maximal parabolic subgroup of G we verified for p>>0 our conjecture that the the Frobenius direct image of the structure sheaf of G/P contains a Karoubian complete strongly exceptional poset of coherent sheaves. In type G_2 we also found that the Frobenius direct image has a nonzero self-extension, implying that the sheaf of rings of small differential operator on G/P has a non-vanishing 1st cohomology. In joint work with Michel Gros we described a new characterization, thanks to Donkin, of the Frobenius contraction, showing in particular that the Frobenius contraction preserves injectivity and the existence of a good filtration.
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| Academic Significance and Societal Importance of the Research Achievements |
上記中，GがG_2型でPのLevi部分群がGのshort simple rootを持つときには，G/Pの構造層のFrobenius direct imageは，我々の予想通りKaroubian complete strongly exceptional posetを持つだけで無く，余分な直既約成分を持ち，そのself-extensionは消えないことを発見した。新たなFrobenius contractionの特徴付けにより，Frobenius contractionがinjectivityやgood filtrationの存在を保つことが分かる。
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