Langlands duality for representations of quantum groups

TL;DR

This paper introduces a duality between representations of quantum groups linked to Langlands dual Lie algebras, using an interpolating quantum group to connect their characters and crystal bases.

Contribution

It presents a novel duality framework for quantum group representations via an interpolating quantum group that bridges Langlands dual pairs.

Findings

Established a duality between characters and crystal bases of dual quantum groups.

Constructed an interpolating quantum group depending on two parameters.

Provided examples of interpolating representations between dual quantum groups.

Abstract

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an "interpolating quantum group" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.