Dear Bugs, thanks for your answer! Your argument sounds ok. As I understand, we replace the eigensheaves $g\cdot s = \chi(g)\cdot s$ by the subsheaves consisting of all sections $s$ such that $\mu^* (s) = s\otimes x^i$ (for $i=0, \ldots, p-1$), where $\mu : X\times \mu_p \to X$ is the action and $\mu_p = \mathrm{Spec}\, k[x]/(x-1)^p$. Is this correct?

Dear Laurent, sorry for the delay. (1) $O_X(D)$ is the reflexive sheaf, the subsheaf $D+(f)\geq 0$ of the sheaf of rational functions, (2) Yes, you are right, I should assume $D$ stable under the action.

@Chris: I know, my "this was answered" wasn't intended to mean "this was answered YES". @AByer: both statements would follow from the existence of a diagonal Frobenius splitting of the toric variety in question (at least in positive characteristic, and maybe in characteristic 0 using the "toric" Frobenius morphism). Are there counterexamples to this?

I wanted to mention the question of existence of full exceptional collections on toric Fano varieties, but this was answered in a very recent paper by Efimov. Probably there are still interesting questions left regarding derived categories of toric varieties...

Maybe this will be helpful: Brian Conrad A modern proof of Chevalley's theorem on algebraic groups .

But the point is that e.g. the Bott-Samelson varieties are not toric in general, and I would like to include them as well. Probably I should consider such towers with a $B$-action instead of $T$-action (the toric ones would then have this $B$-action factored through the projection $B\to T$).

Dear Sasha, that's a good point! But maybe, as in the framework of toric varieties, we can restrict ourselves to some "uniform" or "equivariant" rank two bundles, in which case there would be some good description (a class wide enough to include examples that appear in nature)? Note that every uniform rank two vector bundle on the quadric splits and that equivariant bundles on toric varieties have a good description.

Actually, Pasquier's paper is a partial motivation for this question. His degeneration however relies heavily of the BSDH varieties being a quotient by an action of a product of Borels (he ,,degenerates'' the identity $B\to B$ into the projection $B\to T$).

What about the Ext groups? Is the function $e^i(y, F, G) = \dim_{k(y)} Ext^i(F_y, G_y)$ upper semicontinuous under some assumptions on $F$ and $G$? Of course for $F$ locally free this is just a cohomology group, but what about more general cases?