Representation theory and massive spectral zeta functions on spherical space forms

Abstract

We study the spectral theory of the massive scalar kinetic operator on spherical space forms $Γ\backslash S^{3}$, where $Γ$ is a finite subgroup of $\mathrm{SO}(4)$ acting freely on $S^{3}$. Using the Peter-Weyl theorem and Molien series we compute the spectrum of the Laplace-Beltrami operator on $Γ\backslash S^3$ directly from representation theory, obtaining the eigenvalues and their multiplicities for all lens spaces $L(p,q)$ and for each ADE-class of finite subgroups of $\mathrm{SU}(2)$, which includes homogeneous lens spaces, prism spaces and quotients by the binary tetrahedral, octahedral and icosahedral groups, the last also known as the Poincaré homology sphere. For homogeneous quotients by subgroups of even order, we construct the meromorphic continuation of the spectral zeta function of the massive kinetic operator and obtain the general zeta-regularized determinant in closed form. The determinant is expressed as a finite sum of polylogarithms, valid for all real mass, and depends on $Γ$ only through its order and a finite set of parameters which we call twist coefficients fixed by the dimensions of the $Γ$-invariant subspaces. As a complementary approach, we express the heat trace of the kinetic operator on spherical space forms in terms of Jacobi theta functions, whose modular properties relate the spectral zeta function to geometric data, the volume and closed geodesic lengths of $Γ\backslash S^3$. We find that the spectral zeta function admits a rapidly convergent expansion in the heavy mass sector in terms of modified Bessel functions, and provides an independent check on the determinant. This framework extends to quotients of higher-dimensional manifolds directly, which we outline for future work.