Cohomology of line bundles on Schubert varieties. I (Q1889877)

The authors investigate the cohomology modules \(H^i(X(w),L_\lambda)\), for \(\lambda\) non-dominant on a Schubert variety \(X (w)\) in the generalized flag variety, especially, the vanishing or non-vanishing of \(H^i(X(\omega), L_\lambda)\). For a generic \(\lambda\), the authors give a criterion for vanishing (resp. non-vanishing) of \(H^{l(w)}(X (w), L_\lambda)\) (resp. \(H^0(X(w), L_\lambda)\)). For a general \(j\), the authors give partial results on the vanishing of \(H^j(X(w), L_\lambda)\).