Serre functors and local duality for affine quotients

Abstract

The purpose of this short note is to study Serre functors of categories of quasicoherent sheaves on stacks of the form $\mathcal{Y} = \mathrm{Spec} A/G$ where $G$ is a reductive group acting on $\mathrm{Spec} A$ with a unique closed orbit. We show that the Serre functor is given by tensoring with the local cohomology of $ω_\mathcal{Y}$ at the unique closed orbit. Using this description, we develop analogues of the Matlis and local duality theorems for local rings.