A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces

Pub Date : 2024-06-03 DOI:10.1007/s00229-024-01567-x
Diego S. de Oliveira, Marcus A. M. Marrocos
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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.

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关于同质空间上拉普拉斯频谱一般不可还原性的说明
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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