Spectra of orbifolds with cyclic fundamental groups

Abstract

Abstract. We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamental group. The differential operators considered are Laplace-Beltrami operators twisted by characters of the corresponding fundamental group. To prove the characterization, we first give an explicit description of their spectra by using generating functions. We also include many isospectral examples.

Recent results on the spectra of lens spaces

“…In this section, we first give a description of the spectrum of a lens space by using Ehrhart series, as in [MH17] (see also [La16,Thm. 3.9]), and we end the section by giving a connection with toric varieties, developed by Mohades and Honari in [MH17,§5].…”

“…The last section contains brief discussions on several recent articles related to the spectral theory of lens spaces. Namely, the work on the spectrum of the Dirac operator on spin lens spaces by Boldt [Bo17] and Boldt-Lauret [BL17]; the computational studies on p-isospectral lens spaces ( [GM06,La19]), the work by Bari and Hunsicker ([Ba11, BH19]) on the spectra of lens orbifolds, the article [La16] extending the one-norm method to other compact symmetric spaces of real rank one, and the harmonic counting measure introduced in [MH16].…”

Weight multiplicity formulas for bivariate representations of classical Lie algebras

“…Remark 3.17. In [LR17, Section 7] there is a detailed account of some applications of weight multiplicity formulas in spectral geometry (see [LMR16], [BL17], [La16], [La18]). These expressions for the weight multiplicities are used to determine explicitly the spectra of certain natural differential operators on a manifold (or a good orbifold) of the form Γ\G/K, where G is a semisimple compact Lie group, K is a closed subgroup of G and Γ is a finite subgroup of the maximal torus T of G.…”

The spectrum on p-forms of a lens space

“…Hence, the above mentioned characterization for 0-isospectral lens spaces can be stated as follows: L and L ′ are 0-isospectral if and only if L and L ′ are · 1 -isospectral. (See [BL17], [LMR16b], [La16], [MH17], [MH16] for related results. )…”