# ag.algebraic geometry – Equivariant coherent sheaf category for unipotent group actions

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.