{"title":"A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces","authors":"Diego S. de Oliveira, Marcus A. M. Marrocos","doi":"10.1007/s00229-024-01567-x","DOIUrl":null,"url":null,"abstract":"<p>Petrecca and Röser (Mathematische Zeitschrift 291:395–419, 2018) and Schueth (Ann Global Anal Geom 52:187–200, 2017) had shown that for a generic <i>G</i>-invariant metric <i>g</i> on certain compact homogeneous spaces <span>\\(M=G/K\\)</span> (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator <span>\\(\\Delta _g\\)</span> was real <i>G</i>-simple. The same is not true for the complex version of <span>\\(\\Delta _g\\)</span> when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a <span>\\(Q_8\\)</span>-action that commutes with the Laplacian in such way that <i>G</i>-properties of the real version of the operator have to be understood as <span>\\((Q_8 \\times G)\\)</span>-properties on its corresponding complex version.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01567-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Petrecca and Röser (Mathematische Zeitschrift 291:395–419, 2018) and Schueth (Ann Global Anal Geom 52:187–200, 2017) had shown that for a generic G-invariant metric g on certain compact homogeneous spaces \(M=G/K\) (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator \(\Delta _g\) was real G-simple. The same is not true for the complex version of \(\Delta _g\) when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a \(Q_8\)-action that commutes with the Laplacian in such way that G-properties of the real version of the operator have to be understood as \((Q_8 \times G)\)-properties on its corresponding complex version.