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

Let N(\Gamma,G) be the number of homomorphisms from \Gamma to G up to conjugation by G . The physics of four-dimensional \mathcal{N}{=}4 supersymmetric gauge theories predicts that N(\Gamma,G)=N(\Gamma, \widetilde{G}) when \Gamma is a finite subgroup of \mathrm{SU}(2) , G is a connected compact simple Lie group and \widetilde{G} is its Langlands dual. This statement is known to be true when \Gamma=\mathbb{Z}_{n} , but the statement for non-Abelian \Gamma is new, to the knowledge of the authors. To lend credence to this conjecture, we prove this equality in a few cases, namely (G,\widetilde{G})=(\mathrm{SU}(n),\mathrm{PU}(n)) and (\mathrm{Sp}(n),\mathrm{SO}(2n+1)) for arbitrary \Gamma , and (\mathrm{PSp}(n),\mathrm{Spin}(2n+1)) for exceptional \Gamma . 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.