Mukai Duality for abelian stacks

An abelian stack is a stacky generalization of an abelian variety that was introduced by Brochard. Just as an abelian variety has a dual, an abelian stack $\mathcal{A}$ has a dual $\mathfrak{D}(\mathcal{A})$ which generalizes the classical dual. In general, $\mathfrak{D}(\mathcal{A})$ is no longer an abelian stack but a commutative group scheme which is an extension of a finite, flat, and finitely presented commutative group scheme by an abelian scheme. We show that Fourier-Mukai duality holds for tame abelian stacks and their duals. Our approach is as follows. Let $QC_\infty(\mathcal{A})$ be the stable infinity category of quasi-coherent sheaves on $\mathcal{A}$. We define a Poincare bundle on $\mathcal{A}\times \mathfrak{D}(\mathcal{A})$ and use this to show that $QC_\infty(\mathcal{A})$ and $QC_\infty(\mathfrak{D}(\mathcal{A}))$ are dual as objects in the infinity category of stable infinity categories. By a result of Ben-Zvi,Francis and Nadler we have that $QC_\infty(\mathcal{A})$ is self dual, giving that $QC_\infty(\mathcal{A})\cong QC_\infty(\mathfrak{D}(\mathcal{A}))$ which gives the statement for the derived categories. In addition we give new examples of tame abelian stacks.