Suppose $U$ is a complex algebraic unipotent group. Let $X$ be a projective variety with a $U$-action. For simplicity, we may assume that there are only finite many $U$ orbits on $X$. The primary example that I have in mind is when $X=G/B$, the flag variety associated to a reductive group $G$, with $Usubset B$ the unipotent radical of a Borel subgroup acting on the left. In this case, the $U$-orbits are indexed by the Weyl group elements.
My question is:
Is there a concrete description of $D^b(Coh^U(X))$, the derived category of $U$-equivariant coherent sheaves on $X$? By concrete, I mean is there a way to construct a collection of generators and explicitly calculate the morphisms between them? I’m also particularly interested in the case for $X=G/B$ as above.