On Homomorphisms from Finite Subgroups of SU(2) to Langlands Dual Pairs of Groups

Let N(Γ,G) be the number of homomorphisms from Γ to G up to conjugation by G . The physics of four-dimensional N{=}4 supersymmetric gauge theories predicts that N(Γ,G)=N(Γ, {G}) when Γ is a finite subgroup of SU(2) , G is a connected compact simple Lie group and {G} is its Langlands dual. This statement is known to be true when Γ=Zₙ , but the statement for non-Abelian Γ is new, to the knowledge of the authors. To lend credence to this conjecture, we prove this equality in a few cases, namely (G,{G})=(SU(n),PU(n)) and (Sp(n),SO(2n+1)) for arbitrary Γ , and (PSp(n),Spin(2n+1)) for exceptional Γ . A more refined version of the conjecture, together with proofs of some concrete cases, will also be presented. The authors would like to ask mathematicians to provide a more uniform proof applicable to all cases.