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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 $n\geq 0$ 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).

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I don't have a complete answer, but let me just note that your question 2) would be true if there is a Frobenius splitting of the cotangent bundle $T^*_{X^n}$ of $X^n$ which compatibly splits the diagonal copy $\Delta$ of $T^*_X$, the cotangent bundle of $X$. Since $T^*_{X^n}$ is the cotangent bundle of the flag variety of the semisimple algebraic group $G^n$, there is a candidate for this splitting, namely the splitting of Kumar, Lauritzen, and Thomsen that you mention. I don't know, though, if their splitting compatibly splits $\Delta$; that is an interesting question.

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Thanks; I understand this argument, but I don't know if the above-mentioned splitting is indeed compatible with the diagonal. – Alexander Braverman Nov 30 '11 at 21:53
Kumar, Lauritzen, and Thomsen give a more explicit version of their splitting in the type A case and mention that it's the same as a splitting due to Mehta and van der Kallen. In the paper of Mehta and van der Kallen they show that various subvarieties of the cotangent bundle are compatibly split in type A. They don't consider the diagonal case, but it may follow from their construction, which has the benefit of being more hands-on. You might want to contact one of them and ask. (Also, van der Kallen is active on MathOverflow). – Chuck Hague Dec 4 '11 at 17:08

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