Spectra, Group Representations and Twisted Laplacians

Abstract

AbstractIn this chapter, we review basic notions about spectra, group representations, and twisted Laplace operators. We first recall how to define the spectrum and the spectral zeta function for a general symmetric second order elliptic differential operator acting on smooth sections of a Hermitian line bundle. We prove that the non-zero spectrum (i.e., the spectral zeta function) determines the entire spectrum on an odd-dimensional manifold, but also give an example showing that this is not always true for even-dimensional manifolds; the example is obstructed by the non-vanishing of some topological genus. After setting up some notation from representation theory, we discuss G-sets and weak conjugacy (“Gaßmann equivalence”) of subgroups of a group, explaining the interrelations. In the final sections, we introduce twisted Laplacians, corresponding to unitary representations of the fundamental group. After this, we focus on the case of a twisted Laplacian arising from a finite Galois cover of manifolds and we relate the spectrum on the top manifold to that of the induced representation on the bottom manifold. We relate the multiplicity of zero in the spectrum to the multiplicity of the trivial representation in the given representation, and finally we show that, contrary to the general case, the multiplicity of zero in the spectrum of a twisted Laplacian is determined from the non-zero spectrum, provided one also knows the usual Laplace spectrum of the manifold.