Langlands duality for representations and quantum groups at a root of unity

TL;DR

This paper provides a representation-theoretic interpretation of Langlands duality for quantum groups at roots of unity, linking it to tensor product multiplicities and tilting modules.

Contribution

It offers a novel interpretation of Langlands duality in terms of representation theory and tensor products for quantum groups and Kac-Moody algebras.

Findings

Langlands branching multiplicities equal tensor product multiplicities

Connection between Langlands duality and tilting modules in quantum groups

Representation-theoretic interpretation of Langlands character duality

Abstract

We give a representation-theoretic interpretation of the Langlands character duality of Frenkel and Hernandez, and show that the "Langlands branching multiplicities" for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.