Highest Weights for Categorical Representations

TL;DR

This paper establishes Morita equivalences between categories of D-modules related to reductive groups, demonstrating that de Rham G-categories satisfy a highest weight theorem analogous to classical representation theory.

Contribution

It introduces a criterion for Morita equivalence of monoidal categories and applies it to show de Rham G-categories satisfy a highest weight theorem, connecting them to universal Hecke categories.

Findings

Morita equivalence between D(G) and Hecke categories

De Rham G-categories satisfy a highest weight theorem

Decomposition of principal series 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 $\widetilde{\mathcal D}(B \backslash G / B)$. 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 $\widetilde{\mathcal D}(G/B)$