某些紧李群上Laplace特征空间的通有不可约性
Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups
摘要：
如果$G$是一个紧李群，赋予它一个左不变度量$g$，则$G$通过等距拉回作用于相应拉普拉斯算子$Δ_g$的每个本征空间。我们建立了$G$上左不变度量$g$存在的代数判据，使得$Δ_g$的每个特征空间，作为对应的实特征函数的实向量空间，在$G$的作用下是不可约的.我们证明了一般的左不变度量的李群$G=\operatorname{SU}(2)\times\ldots\times\operatorname{SU}(2)\times T$，其中$T$是一个（可能是平凡的）环面，有刚才所描述的属性。这同样适用于由离散中心子群构成的这类群$G$的子群。特别地，它也适用于$\operatorname{SO}(3)$、$\operatorname{U}(2)$、$\operatorname{SO}(4)$。
英文摘要：
If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $Δ_g$. We establish algebraic criteria for the existence of left invariant metrics $g$ on $G$ such that each eigenspace of $Δ_g$, regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of $G$. We prove that generic left invariant metrics on the Lie groups $G=\operatorname{SU}(2)\times\ldots\times\operatorname{SU}(2)\times T$, where $T$ is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups $G$ by discrete central subgroups. In particular, it also holds for $\operatorname{SO}(3)$, $\operatorname{U}(2)$, $\operatorname{SO}(4)$.
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