Group theory aspects of spectral problems on spherical factors

Abstract

The Ray-Singer isospectral theorem (1971) is applied to a general spectral function for Laplacians of twisted p-forms (say) on homogeneous Clifford-Klein factors of the three-sphere. The inducing formulae necessary to express any spectral quantity for any twisting in terms of those for cyclic subgroups of the tetrahedral, octahedral and icosahedral deck groups are detailed. Further, Artin's theorem allows the McKay correspondence to be obtained. The isospectral theorem is shown to yield a derivation of the Sunada construction which is equivalent to the later one by Pesce.