Compatibility of t-structures for quantum symplectic resolutions

Abstract

Let $W$ be a smooth complex variety with the action of a connected reductive group $G$. Adapting Teleman’s stratification approach to a microlocal context, we prove a vanishing theorem for the functor of $G$-invariant sections—that is, of quantum Hamiltonian reduction—for $G$-equivariant twisted $\mathcal{D}$-modules on $W$. As a consequence, when $W$ is affine we establish a sufficient combinatorial condition for exactness of the global sections functors of microlocalization theory. When combined with the derived equivalence results of our recent work, this gives precise criteria for “microlocalization of representation categories” in the spirit of Kashiwara–Rouquier and many other authors.