Potent categorical representations

TL;DR

The paper develops potent categorical representations of complex reductive groups, proposing a Langlands duality with the dual group$\check G$ encoded via a 2-categorical Fourier transform and higher sheaf theory. It builds a foundational framework using periodization (2IndCoh^π) and equivariant expansion (2IndCoh^ε), defines the potent Hecke monad, and formulates a multiplicative Fourier transform that links de Rham potent categories for $G$ and $\check G$, predicting dual realizations of a single governing monad. The authors connect these ideas to extended 3d topological quantum field theories, Rozansky–Witten theory, and 3d gauge theories, showing how potent representations capture both geometric and physical data, including Coulomb branches and mass deformations via the $\Omega$-background. They outline conjectures and constructions relating traces, character sheaves, Hilbert schemes, and DAHA, aiming for a unified picture of local/global Langlands across Betti, de Rham, and graded settings, with potential applications to real, relative, and finite Langlands. Overall, the work provides a broad, highly structured categorical framework that ties geometric representation theory, TFT, and Langlands duality through potent representations and their symplectic parameterization.

Abstract

We introduce and motivate -- based on ongoing joint work with Germán Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent categorical representations of $G$ and its Langlands dual $\check G$. We emphasize the symplectic nature of potent categorical representations in their simultaneous dependence on parameters in maximal tori for $G$ and $\check G$, specifically how their conjectural Langlands correspondence fits within a 2-categorical Fourier transform. Our key tool to make various ideas precise is higher sheaf theory and its microlocalization, specifically a theory of ind-coherent sheaves of categories on stacks. The constructions are inspired by the physics of 3d mirror symmetry and S-duality on the one hand, and the theory of double affine Hecke algebras on the other. We also highlight further conjectures related to ongoing programs in and around geometric representation theory.

Theorems & Definitions (23)

• Example 2.1.3
• Definition 2.1.4
• Remark 2.2.2
• Definition 2.2.4
• Example 2.2.5
• Example 2.2.7
• Remark 2.2.8
• Remark 2.3.2
• Theorem 2.3.3
• Proposition 2.3.5