Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups |
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Authors: | Dorothee Schueth |
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Institution: | 1.Institut für Mathematik,Humboldt-Universit?t zu Berlin,Berlin,Germany |
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Abstract: | 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 \(\Delta _g\). We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of \(\Delta _g\), 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 \(G={ SU}(2)\times \cdots \times { SU}(2)\times T\), 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 \({ SO}(3)\), \({ U}(2)\), \({ SO}(4)\). |
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