An explicit formula for the Dirac multiplicities on lens spaces

TL;DR

This paper derives explicit formulas for the eigenvalue multiplicities of the Dirac operator on lens spaces, enabling analysis of their spectral properties and revealing that spin structures and isometry classes are not spectrally unique.

Contribution

It provides a new explicit formula for Dirac eigenvalue multiplicities on lens spaces using representation theory, and demonstrates non-uniqueness in spectral geometry.

Findings

Explicit formulas for Dirac eigenvalue multiplicities

Conditions for lens spaces to be Dirac isospectral

Existence of infinite families of Dirac isospectral lens spaces

Abstract

We present a new description of the spectrum of the (spin-) Dirac operator $D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space $\Gamma\backslash \operatorname{Spin}(2m)/\operatorname{Spin}(2m-1)$ and exploiting the representation theory of the $\operatorname{Spin}$ groups, we obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in terms of finitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.