Some Microlocal Aspects of Perverse Coherent Sheaves and Equivariant D-Modules

We discuss microlocal aspects of two types of sheaves which are of interest to geometric representation theory: perverse coherent sheaves and equivariant D-modules.;The category of (constructible) perverse sheaves on a complex variety is characterized by exactness of the microlocal stalks (or vanishing cycles) functor. We prove an analogue of this characterization for the category of perverse coherent sheaves on a scheme with a group action. The main idea is to understand microlocal stalks via local cohomology along half-dimensional ("Lagrangian") subvarieties. We define "measuring subvarieties" as an analogue of these subvarieties in the coherent setting and show how they can be used to characterize perverse coherent sheaves.;The second part of this thesis is dedicated to understanding the (categorical) support theory of equivariant D-modules. We discuss how to compute the Hochschild cohomology of the category of D-modules on a quotient stack via a relative compactification of the diagonal morphism. We then apply this construction to the case of torus-equivariant D-modules and describe the Hochschild cohomology as the cohomology of a D-module on the loop space of the quotient stack.