Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups

TL;DR

This paper establishes algebraic conditions under which Laplace eigenspaces on certain compact Lie groups are irreducible under group action, showing that generic left invariant metrics on these groups typically have this property.

Contribution

It provides algebraic criteria for irreducibility of Laplace eigenspaces and proves that generic metrics on specific compact Lie groups exhibit this irreducibility.

Findings

Generic metrics on certain Lie groups have irreducible Laplace eigenspaces.

The results apply to quotients of these groups by discrete central subgroups.

Includes classical groups like SO(3), U(2), and SO(4).

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 $G=\operatorname{SU}(2)\times\ldots\times\operatorname{SU}(2)\times T$, 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 $\operatorname{SO}(3)$, $\operatorname{U}(2)$, $\operatorname{SO}(4)$.