Module categories over affine supergroup schemes

Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${rm sCoh}_{rm f}(mathcal{G})$ of (coherent sheaves of) finite dimensional $mathcal{O}(mathcal{G})$-supermodules in terms of $(mathcal{H},Psi)$-equivariant coherent sheaves on $mathcal{G}$. We deduce from it the classification of indecomposable {em geometrical} module categories over $sRep(mathcal{G})$. When $mathcal{G}$ is finite, this yields the classification of {em all} indecomposable exact module categories over the finite tensor category $sRep(mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra $kmathcal{G}$ of a finite supergroup scheme $mathcal{G}$, and then combine it with cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over $k$.