An Explicit Formula for the Dirac Multiplicities on 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 $$\Gamma \backslash {\text {Spin}}(2m)/{\text {Spin}}(2m-1)$$ and exploiting the representation theory of the $${\text {Spin}}$$ 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 infinitely many examples of finite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.