Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups

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 ${ SO}(3)$, ${ U}(2)$, ${ SO}(4)$.