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Cohomology of line bundles over compactifications of reductive groups. (Cohomologie des fibrés en droites sur les compactifications des groupes réductifs.) (French) Zbl 1061.14047

Let \(G\) be a connected reductive algebraic group considered as a homogeneous space for \(G\times G\). The author studies the cohomology groups of line bundles over \(G\times G\)-equivariant completions of \(G\). For a suitable finite covering \(\tilde G\) of \(G\), these cohomology groups are finite-dimensional \(\tilde G\times\tilde G\)-modules. The main result of the article is the determination of multiplicities of all simple \(\tilde G\times\tilde G\)-modules in these cohomology groups.
The formula obtained by the author applies, in particular, to wonderful compactifications of adjoint groups and complete toric varieties. The case of wonderful compactifications was considered independently by S. Kato [J. Algebra 259, 572–580 (2003; Zbl 1125.14302)]. The proof exploits explicit combinatorial descriptions of equivariant regular completions and invertible sheaves, and the Grothendieck-Cousin complex of \(\mathfrak g\times \mathfrak g\)-modules introduced by Kempf.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations

Citations:

Zbl 1125.14302
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References:

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