2006 Fiscal Year Final Research Report Summary
Modular Representations of Algebraic Groups
| Project/Area Number |
15540040
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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, Graduate School of Science, Professor, 大学院理学研究科, 教授 (60204575)
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| Co-Investigator(Kenkyū-buntansha) |
TANISAKI Toshiyuki Osaka City University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (70142916)
YAGITA Nobuakl Ibaraki University, Department of Education, Professor, 教育学部, 教授 (20130768)
HASHIMOTO Yoshitake Osaka City University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (20271182)
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| Project Period (FY) |
2004 – 2006
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| Keywords | localization / triangulated localization / tilting |
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| Research Abstract |
After the spectacular success of Bezrukavnikov, Mirkovic and Rumynin, BMR for short in what follows, in extending the localization theorem of D-modules on the flag variety to positive characteristic, we started to investigate the localization of ＼bar D-modules.
On a smooth variety X in positive characteristic BMR's ＼mathcal{D}, which they call the sheaf of crystalline differential operators, is Berthelot's sheaf of arithmetic differential operators of level 0. In "Localization of ＼bar D-modules" written with Hashimoto and Rumynin we first gave a simple presentation of ＼mathcal{D}-{(m)} the sheaf of arithmetic differential operators of level m. After BMR we proved that each＼mathcal{D}"{(m)} is Azumaya. ＼bar ＼mathcal{D"{(m)}} is the endomorphism ring of the structure sheaf of X over its (m+1)-st Probenius twist, and is a central reduction of ＼mathcal{D"{(m)}}. We observed that the triangulated localization theorem for ＼bar＼mathcal{D}-{(m)} holds almost iff the direct image F"{m+1}_* ＼mathcal{O}_X of the structure sheaf under the (m+1)-st Frobenius morphism is tilting, and verified that on the projective space F"{m+1}_*＼mathcal{O}_X is tilting if the characteristic is large enough. On the flag variety, as the direct image of the ＼bar＼mathcal{D*{(m)}} is the whole of the sheaf of classical differential operators ＼mathcal {Diff} and as the cohomology vanishing of ＼mathcal {Diff} fails in the case of SL_5 by Kashiwara-Lauritzen, one cannot expect the triangulated localization theorem holds for all ＼bar＼mathcal{D-{(m)}1. Nevertheless, in view of BMR there may something special happening when m-0, and indeed, we found that F_*＼mathcal{O}_{G/B} is tilting in the case of SL 3 in sufficiently large characteristic.
In a joint paper with Ye "Equivariant localization of ＼bar D modules on the flag variety of the symplectic group of degree 4" we also verified that F_*＼mathcal{O}_{G/B} is tilting insufficiently large characteristic in case Sp_4 also.
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