Refined equivalence of V_Z(Γ,G) and V_{Z^∧}(Γ,\tilde H) under the swap isomorphism

Establish that, for a finite subgroup Γ of SU(2) and a central extension 0→Z→G→H→0 with Langlands dual extension 0→Z^∧→\tilde H→\tilde G→0, the representation V_Z(Γ,G) of F(Γ;Z)=H^1(BΓ;Z)×H^2(BΓ;Z)^∧ is equivalent to the representation V_{Z^∧}(Γ,\tilde H) of F(Γ;Z^∧), after identifying F(Γ;Z) with F(Γ;Z^∧) via the swap isomorphism s constructed from Poincaré duality on S^3/Γ.

Background

Beyond counting homomorphisms, the authors refine the problem by organizing twisted homomorphisms classified by H^2(BΓ;Z) into a representation V_Z(Γ,G) of a finite abelian group F(Γ;Z)=H^1(BΓ;Z)×H^2(BΓ;Z)^∧. For Langlands dual data, they construct an analogous V_{Z^∧}(Γ,\tilde H).

Using a natural swap isomorphism s that interchanges H^1 and H^2 via Poincaré duality for S^3/Γ, the refined conjecture predicts an equivalence of these representations after identifying F(Γ;Z) and F(Γ;Z^∧) through s. The authors prove this refined conjecture in several cases but leave the general statement open.