The famous localisation theorem of Beilinson and Bernstein shows that the global sections functor gives an equivalence between the category of D-modules on the flag variety and certain modules for the enveloping algebra. In particular, D-modules on the flag variety have no higher cohomologies. Recently there has been much interest in generalizations of this story in the context of deformation quantizations of symplectic varieties. I will describe joint work with Tom Nevins in which we prove a general cohomology vanishing theorem for quantum Hamiltonian reductions. This, in conjunction with known derived equivalences implies localisation-type theorems for rings of invariant differential operators of interest in representation theory, and also provides a crucial tools in the categorical decomposition of D-modules which was described in Tom's talk. (Neither of the talks will assume knowledge of the other however!)