{"title":"关于同质空间上拉普拉斯频谱一般不可还原性的说明","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":49887,"journal":{"name":"Manuscripta Mathematica","volume":"101 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":49887,\"journal\":{\"name\":\"Manuscripta Mathematica\",\"volume\":\"101 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-06-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Manuscripta Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00229-024-01567-x\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Manuscripta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01567-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
Petrecca 和 Röser (Mathematische Zeitschrift 291:395-419, 2018)以及 Schueth (Ann Global Anal Anal Geom 52:187-200, 2017)曾证明,对于某些紧凑均质空间 \(M=G/K\) 上的泛 G 不变度量 g(包括秩 1 的对称空间和一些李群),拉普拉斯-贝尔特拉米算子 \(\Delta _g\)的谱是实 G 简单的。当存在复数或四元数类型的表示时,复数版的\(\Δ _g\)就不是这样了。我们证明了这些类型的表示会诱导一个与拉普拉卡相乘的 \(Q_8\)-action ,这样一来,算子的实数版本的 G 特性就必须被理解为其相应复数版本上的\((Q_8 \times G)\)-特性。
A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces
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.
期刊介绍:
manuscripta mathematica was founded in 1969 to provide a forum for the rapid communication of advances in mathematical research. Edited by an international board whose members represent a wide spectrum of research interests, manuscripta mathematica is now recognized as a leading source of information on the latest mathematical results.