# What is known about line bundles on the tangent bundle of a flag variety?

Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. (I'm most interested in the positive characteristic case). Let $B \subseteq G$ be a Borel subgroup. Let $\cal T$ denote the tangent bundle of the flag variety $G/B$ and let $\pi : {\cal T} \to G/B$ be the projection.

For each integral weight $\lambda$ we have a line bundle $\cal L(\lambda)$ on $G/B$. A lot is known about the pullbacks of these bundles to the cotangent bundle of $G/B$ (see work of Broer, Kumar, Lauritzen, Thomsen, etc). For example, it is known which of these pullbacks to the cotangent bundle is ample, and there have a been a series of papers studying the $G$-module structures of the global sections of these pullbacks.

On the other hand, I haven't seen analogous results regarding the pullbacks $\pi^* \cal L(\lambda)$ of these bundles to the tangent bundle $\cal T$. To be more precise, I am most interested in knowing which of these pullbacks is ample. It would also be interesting to know if anyone has studied the $G$-module structure of $H^0( \cal T, \pi^* \cal L(\lambda) )$ for various $\lambda$.

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Is it clear what happens in the simplest cases (rank 1 or 2)? It would help to make the map $\pi$ explicit in any case. –  Jim Humphreys Sep 20 '10 at 10:53
That's a good question; I'll think about it. In general, sections of these pullback line bundles on the tangent bundle are isomorphic to sections of the equivariant bundle on $G/B$ with fiber $S( \mathfrak n ) \otimes k_\lambda$, where $S(\mathfrak n)$ is the symmetric algebra on the Lie algebra of the unipotent radical $U$ of $B$. Given that description, it shouldn't be too hard to consider some low-rank cases. –  Chuck Hague Sep 20 '10 at 15:13