Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases

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.