Recent results on the spectra of lens spaces

• Published in: São Paulo Journal of Mathematical Sciences Vol. 15; no. 1; pp. 240 - 267
• Main Authors: Lauret, Emilio A., Miatello, Roberto J., Rossetti, Juan Pablo
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2021; Springer Nature B.V
• Subjects: Lenses; Lie groups; Mathematics; Mathematics and Statistics; Riemann manifold; Special Section: An Homage to Manfredo P. do Carmo; Spectra; Spectral theory; 58J53; Lens space; Isospectral; Spherical harmonics; 58J50; Spectrum
• ISSN: 1982-6907; 2316-9028; 2306-9028
• DOI: 10.1007/s40863-019-00154-3

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 Lauret et al. (IMRN 2016(4):1054–1089, 2016. https://doi.org/10.1093/imrn/rnv159 ), 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.