Categorical relations between Langlands dual quantum affine algebras: Exceptional cases

TL;DR

This paper computes denominator formulas for exceptional quantum affine algebras and establishes isomorphisms among their Grothendieck rings, confirming conjectures and extending Langlands duality insights.

Contribution

It provides the first denominator formulas for all exceptional types and proves key isomorphisms and conjectures related to Langlands duality in quantum affine algebras.

Findings

Denominator formulas for all exceptional types computed

Isomorphisms among Grothendieck rings established

Confirmed conjectures of Kashiwara-Kang-Kim and Kashiwara-Oh

Abstract

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, $\mathscr{C}_{\mathscr{Q}}^{(1)}$ and $\mathscr{C}_{\mathfrak{Q}}^{(1)}$. These results give Dorey's rule for all exceptional affine types, prove the conjectures of Kashiwara-Kang-Kim and Kashiwara-Oh, and provides the partial answers of Frenkel-Hernandez on Langlands duality for finite dimensional representations of quantum affine algebras of exceptional types.