Group theory aspects of spectral problems on spherical factors

TL;DR

This paper explores spectral problems on spherical factors using group theory, applying the Ray-Singer isospectral theorem to relate spectral functions of twisted forms to cyclic subgroups, and deriving the Sunada construction via Artin's theorem.

Contribution

It provides explicit formulas for spectral quantities on spherical factors and connects the Ray-Singer theorem with the McKay correspondence and Sunada construction.

Findings

Derived formulas for spectral functions of twisted forms on spherical factors.

Connected Ray-Singer isospectral theorem with McKay correspondence.

Showed equivalence of Sunada construction with Pesce's approach.

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.