Filtrations of cohomology modules for Chevalley groups. (English) Zbl 0545.20030
Let G be a Chevalley group, B a Borel subgroup of G, k a field of characteristic \(p>0\) and L a line bundle on G(k)/B(k) induced by a character of L. The object of this paper is to construct and study certain filtrations of certain subquotients of \(H^*(G(k)/B(k),L)\). The formal characters of the filtrations are shown to satisfy a ”sum formula”. In some cases, these filtrations and their ”sum formula” reduce to the corresponding things of J. C. Jantzen [J. Reine Angew. Math. 290, 117-141 (1977; Zbl 0342.20022)].
Let R be the local ring of the integers at p. Translation functors are defined in the category of G(R)-modules. These functors are combined with the filtrations to reduce a conjecture of Lusztig on the characters of irreducible G(k)-modules to a conjecture that certain homomorphisms of Weyl modules respect the filtrations up to a shift by 1.
This highly technical paper uses a very careful working out of the necessary cohomology theory over the integers, including a universal coefficient theorem, which is nicely presented in the first section and may be of independent interest.
Let R be the local ring of the integers at p. Translation functors are defined in the category of G(R)-modules. These functors are combined with the filtrations to reduce a conjecture of Lusztig on the characters of irreducible G(k)-modules to a conjecture that certain homomorphisms of Weyl modules respect the filtrations up to a shift by 1.
This highly technical paper uses a very careful working out of the necessary cohomology theory over the integers, including a universal coefficient theorem, which is nicely presented in the first section and may be of independent interest.
Reviewer: A.Ash
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 20G05 | Representation theory for linear algebraic groups |
| 14L35 | Classical groups (algebro-geometric aspects) |
Keywords:
Chevalley group; Borel subgroup; line bundle; filtrations; formal characters; sum formula; Translation functors; characters of irreducible G(k)-modules; Weyl modulesCitations:
Zbl 0342.20022References:
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