Groupes Quantiques d'Interpolation de Langlands de Rang 1

TL;DR

This paper introduces a family of double deformations of U(sl2) parameterized by g, which simultaneously deform two rank 1 quantum groups and explain Langlands duality in their representations.

Contribution

It proves the existence of representations that deform two Langlands dual representations simultaneously for all g, confirming a conjecture and exploring finite rank representation theory.

Findings

Each double deformation deforms two rank 1 quantum groups.

The interpolating property explains Langlands duality for quantum group representations.

Existence of dual-deforming representations for all g confirms the conjecture.

Abstract

Interpolating Langlands Quantum Groups of Rank 1 -- We study a certain family, parameterized by an positive integer g, of double deformations of the envelopping algebra U(sl2), in the spirit of arXiv:0809.4453. We prove that each of these double deformations simultaneously deforms two rank 1 quantum groups. We show this interpolating property explains the Langlands duality for the representations of the quantum groups in rank 1. Hence we prove a conjecture of arXiv:0809.4453 in this case : we prove for all g the existence of representations which simultaneously deform two Langlands dual representations. We also study more generaly the finite rank representation theory of this family of double deformations. ----- On \'etudie une certaine famille, param\'etr\'ee par un entier g strictement positif, de doubles d\'eformations de l'alg\'ebre enveloppante U(sl2), dans l'esprit de arXiv:0809.4453. On prouve que chacune de ces doubles d\'eformations d\'eforme simultan\'ement deux groupes quantiques de rang 1. On montre que cette propri\'et\'e d'interpolation explique la dualit\'e de Langlands pour les repr\'esentations des groupes quantiques en rang 1. On r\'esout ainsi une conjecture de arXiv:0809.4453 dans ce cas : on prouve pour tout g l'existence de repr\'esentations qui d\'eforment simultan\'ement deux repr\'esentations Langlands duales. On \'etudie aussi plus g\'en\'eralement la th\'eorie des repr\'esentations de rang fini de de cette famille de doubles d\'eformations.