Modular representations of algebraic groups

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04789
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka City University
• Principal Investigator: KANEDA masaharu 大阪市立大学, 大学院理学研究科, 教授 (60204575)
• Research Collaborator: TANISAKI TOSHIYUKI ; YAGITA NOBUAKI ; TEZUKA MICHISHIGE ; FURUSAWA MASAAKI ; KIMURA YOSHIYUKI ; KAWATA SHIGETO
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: algebraic groups / Frobenius morphism / representation theory

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.

Free Research Field

代数学

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。