Structure of cohomology of line bundles on G/B for semisimple groups. (English) Zbl 0732.20028
This paper contains a number of results on the G-structure of \(H^ i(G/B,L(\lambda))\). Here G is a semi-simple algebraic group, B is a Borel subgroup of G and L(\(\lambda\)) is the line bundle on G/B associated with the B-character \(\lambda\). The key idea is to obtain a connection between these representations and certain (appropriately twisted) infinitesimally induced representations. Among the results obtained is a proof of a conjecture by J. E. Humphreys [see CMS Conf. Proc. 5, 341-349 (1986; Zbl 0582.17004)] saying that generically the socle (as well as radical) series of the two representations correspond. Assuming Lusztig’s conjecture on the modular irreducible characters, this implies that the modules are rigid (via a theorem of M. Kaneda and the reviewer [Proc. Lond. Math. Soc., III. Ser. 59, 74-98 (1989; Zbl 0681.20029)]).
Reviewer: H.H.Andersen (Aarhus)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 20G05 | Representation theory for linear algebraic groups |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 17B45 | Lie algebras of linear algebraic groups |
| 17B50 | Modular Lie (super)algebras |
| 14M17 | Homogeneous spaces and generalizations |
Keywords:
socle and radical of cohomology modules; Loewy series; Frobenius subgroups; semi-simple algebraic group; line bundle; infinitesimally induced representations; modular irreducible charactersReferences:
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