An Explicit Formula for the Dirac Multiplicities on Lens Spaces

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• 作者：Sebastian Boldt ; Emilio A. Lauret
• 关键词：Dirac spectrum ; Lens spaces ; Isospectrality ; Affine lattices
• 刊名：The Journal of Geometric Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：27
• 期：1
• 页码：689-725
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds;
• 出版者：Springer US
• ISSN：1559-002X
• 卷排序：27

文摘

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.