It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions $\operatorname{Isom}(X)\to \operatorname{SO}(W_\lambda)$ for eigenspaces $W_\lambda$ give many representations of the Lie group $\operatorname{Isom}(X)$.

Suppose one has an unknown compact Riemannian manifold $X$ ($n=\dim X$), but where the eigenspaces of the Laplacian have large dimension (I don't have a precise definition of "large" here; the weakest definition would probably be something like $\dim W_\lambda>1$ for infinitely many $\lambda$. I'd be happy even with a much stronger assumption, say $\dim W_\lambda$ is at least $\epsilon$ times the Weyl bound $\operatorname{const}\cdot\lambda^{(n-1)/2}$ infinitely often). Can one conclude that $X$ is a symmetric space, or close to one in some sense?

EDIT: I would be interested in any result which takes as a hypothesis some assumption of large multiplicity in the Laplace spectrum, and whose conclusion is some sort of symmetry of the underlying manifold.

anyresult which takes some multiplicity assumptions as hypotheses and whose conclusion is some sort of symmetry of the manifold. $\endgroup$ – John Pardon May 23 '11 at 14:53