ScalingStacks

Remark 1.10. One can find a precursor to the above in the work of Müger [M] and Ostrik [Os].

Given a semisimple abelian monoidal category $\mathcal{C}$ and a module category $M$, consider the monoidal category $\mathcal{C}_{M}^{*}$ consisting of $\mathcal{C}$-linear endofunctors of $M$. Then independently of $M$, there is a canonical identification of the Drinfeld centers of $\mathcal{C}_{M}^{*}$ and $\mathcal{C}.$

A motivating example is when $H\subset G$ are finite groups, and one takes $\mathcal{C}=\operatorname{Rep}(G)$ and $M=\operatorname{Rep}(H)$, so that $\mathcal{C}^{*}_{M}\simeq\operatorname{Vect}(H\backslash G/H)$. Then independently of $H$, the center of $\operatorname{Vect}(H\backslash G/H)$ is the category of adjoint equivariant vector bundles on $G$.

The above theorem extends this picture from finite groups to algebraic groups.