Let $X$ denote the flag variety of a semi-simple group $G$ (in characteristic 0) and let $T_X$ denote its tangent bundle. I would like to ask the following question(s):
1) Is it true that for any $n$ n\geq 0$we have$H^i(X,T_X^{\otimes n})=0$for$i>0$? 2) More generally, let$\lambda$be a dominant weight of$G$and let$\mathcal O(\lambda)$be the corresponding line bundle on$X$. Is it true that$H^i(X, T_X^{\otimes n}\otimes \mathcal O(\lambda))=0$for$i>0$? When tensor powers of$T_X\$ are replaced by symmetric powers, this is known to be true (for example it is proved in a paper of Kumar, Lauritzen and Thomsen).