Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups

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 = S U ( 2 ) × ⋯ × S U ( 2 ) × 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 S O ( 3 ) , U ( 2 ) , S O ( 4 ) .