Equality of homomorphism counts for Langlands dual pairs with finite SU(2) subgroups

Establish that for every finite subgroup Γ of SU(2) and every Langlands dual pair (G, \tilde G) of connected compact simple Lie groups, the quantity N(Γ,G), defined as the number of group homomorphisms from Γ to G modulo conjugation by G, equals N(Γ,\tilde G).

Background

The paper studies counts of homomorphisms from finite subgroups Γ of SU(2) into compact simple Lie groups G, modulo conjugation, denoted N(Γ,G). Motivated by S-duality in four-dimensional N=4 supersymmetric quantum field theories, the authors propose that these counts should be invariant under Langlands duality (G ↔ \tilde G).

They verify the conjecture in several families and special cases, but its general validity remains unproven. The conjecture extends previously known results for the cyclic case Γ=Z_n to non-Abelian Γ, which is new to the authors.