Highest Weights for Categorical Representations

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 $\tilde{\mathcal{D}}(B\backslash G/B)$. In other words, de Rham $G$-categories, i.e., 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 $\tilde{\mathcal{D}}(G/B)$

11 pages