Recent results on the spectra of lens spaces

Abstract

In this paper we report on recent results by several authors, on the spectral theory of lens spaces and orbifolds and similar locally symmetric spaces of rank one. Most of these results are related to those obtained by the authors in [IMRN (2016), 1054--1089], where the spectra of lens spaces were described in terms of the one-norm spectrum of a naturally associated congruence lattice. As a consequence, the first examples of Riemannian manifolds isospectral on $p$-forms for all $p$ but not strongly isospectral were constructed. We also give a new elementary proof in the case of the spectrum on functions. In this proof, representation theory of compact Lie groups is avoided and replaced by the use of Molien's formula and a manipulation of the one-norm generating function associated to a congruence lattice. In the last four sections we present several recent results, open problems and conjectures on the subject.

References in corpus (8)

• Spectra of orbifolds with cyclic fundamental groups
• Spectra of lens spaces from 1-norm spectra of congruence lattices
• Representation equivalence and p-Spectrum of constant curvature space forms
• The spectrum on $p$-forms of a lens space
• Non-strongly isospectral spherical space forms
• Multiplicity formulas for fundamental strings of representations of classical Lie algebras
• A computational study on lens spaces isospectral on forms
• Weight multiplicity formulas for bivariate representations of classical Lie algebras

Cited by in corpus (2)

• Strong representation equivalence for compact symmetric spaces of real rank one
• Isospectral CR manifolds with respect to the Kohn Laplacian