On the cohomology supported by line bundles over complete symmetric varieties (Q601760)

Let \(G\) be a semisimple linear algebraic group over an algebraically closed field of characteristic zero. If \(X\) is a smooth projective \(G\)-variety and \(L\) is a \(G\)-linearized line bundle over \(X\), then all cohomology groups \(H^d(X,L)\) are finite-dimensional \(G\)-modules. In this article, the author studies the case where \(X\) is a complete symmetric variety. Given a Bialynicki-Birula cell \(C\) of codimension \(c\), the Lie algebra \({\mathbf g}\) of \(G\) acts naturally on the cohomology group \(H_C^c(L)\). In this article, the author gives a necessary condition on the cell for the existence of a finite dimensional simple subquotient of this \({\mathbf G}\)-module. As an application, one can calculate the Euler-Poincaré characteristic of \(L\) over \(X\) and give estimates of the higher cohomology groups \(H^d(X,L)\).