Projective generation for equivariant $D$-modules

Abstract: We investigate compact projective generators in the category of equivariant $D$-modules on a smooth affine variety. For a reductive group $G$ acting on a smooth affine variety $X$, there is a natural countable set of compact projective generators indexed by finite dimensional representations of $G$. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof in the general case goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest. We also provide an alternative (more elementary) proof in the case that $G$ is a torus.