Sur la cohomologie à support des fibrés en droites sur les variétés symétriques complètes

Let X be a complete symmetric variety, i.e., the wonderful compactification of a symmetric G-homogeneous space (where G is a simply connected semi-simple linear algebraic group). If L is a line bundle over X and if C is a Bialynicki-Birula cell of codimension c in X, then the Lie algebra g of G operates naturally on the cohomology group with support HC c (L). A necessary condition on C for the existence of a finite-dimensional simple subquotient of this g-module is given. As applications one calculates the Euler-Poincaré characteristic of L over X, estimates the higher cohomology group Hd(X, L), d ≥ 0, and obtains the exact formulas in some cases including that of the complete conic variety. © 2010 Springer Science+Business Media, LLC.