Highest Weights for Categorical Representations

Source abstract

Abstract We present a criterion for establishing Morita equivalence of monoidal categories and apply it to the categorical representation theory of reductive groups $G$. We show that the “de Rham group algebra” $\mathcal D(G)$ (the monoidal category of $\mathcal D$-modules on $G$) is Morita equivalent to the universal Hecke category $\mathcal D({N}\backslash{G}/{N})$ and to its monodromic variant $\widetilde{\mathcal D}({B}\backslash{G}/{B})$. In other words, de Rham $G$-categories, that is, module categories for $\mathcal D(G)$, satisfy a “highest weight theorem”—they all appear in the decomposition of the universal principal series representation $\mathcal D(G/N)$ or in twisted $\mathcal D$-modules on the flag variety $\widetilde{\mathcal D}(G/B)$.