Let G be a connected linear algebraic group acting on a smooth complex variety X with finitely many orbits. In this case, the category of G-equivariant D-modules is equivalent to the category of finite dimensional representations of a quiver with relations, and one may take advantage of this structure to study the local cohomology modules with support in the G-stable subvarieties of X.
This thesis is dedicated to investigating categories of G-equivariant D-modules and local cohomology on Vinberg representations, i.e. when X is an irreducible representation of G, thought of as an affine space, and G is reductive. Such representations have been classified, and correspond to a choice of Dynkin diagram and vertex.
When X is the space of 2x2x2 hypermatrices endowed with the natural action of G= GL_2 x GL_2 x GL_2, or when X is the space of alternating senary 3-tensors endowed with the natural action of G=GL_6, our analysis entails: classifying and explicitly realizing the simple equivariant D-modules, and determining the corresponding quiver with relations. The latter case is joint work with András C. Lőrincz. As an application, we calculate local cohomology with support in the orbit closures, and obtain the Lyubeznik numbers.
In another direction, we determine the D-module structure of local cohomology with support in Pfaffian varieties, in which case the simple composition factors were known by past work of Raicu-Weyman. This information, combined with careful use of graded local duality allows us to calculate the Lyubeznik numbers for Pfaffian varieties. A major step in this work is our computations of Ext^j_S(S/I,S), where S is the coordinate ring of the space of skew-symmetric matrices and I is a G-invariant ideal. As another application of these Ext results, we determine the Castelnuovo-Mumford regularity of powers and symbolic powers of ideals of Pfaffians.