Cohomologie des fibrés en droites sur les compactifications des groupes réductifs

Abstract

We consider equivariant compactifications of some reductive complex and connected group, G, considered as an homogeneous space for G× G. For a finite covering G ̃ of G, the cohomology groups of line bundles over those compactifications are naturally finite dimensional G ̃ × G ̃ -modules. We determine, in this article, all those G ̃ × G ̃ -modules by calculating their multiplicities along each simple G ̃ × G ̃ -module. The achieved formula is in particular valid for the wonderful compactification of adjoint groups and also generalizes the well known description of the cohomology of line bundles over complete toric varieties. Our method is based upon the Grothendieck–Cousin complex which, if g is the Lie algebra of G, is a g× g -module complex. We analyse those g× g -modules by giving some filtrations in the associated graduate of which, some “generalized Verma modules” occur.