Serre functors and local duality for affine quotients

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