TITLE: High multiplicity eigenvalue implies symmetry?
QUESTION [17 upvotes]: 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.

REPLY [9 votes]: One situation where people have thought hard about this issue is the multiplicity of the spectrum of the Laplacian on the modular surface. This is a notoriously difficult problem. 
Conjecturally, the (discrete) spectrum is simple, but as far as I know, the best known bounds  on the multiplicity of a Maas cusp form of eigenvalue $\lambda$ are somewhere in the neighborhood of  $O(\lambda^{1/2}/\log \lambda).$
(Weyl's law in dimension $2$ says that the number of eigenvalues of size at most $N$ is $O(N)$). 
See e.g. section 4 of the following survey of Peter Sarnak: http://www.math.princeton.edu/sarnak/baltimore.pdf
So I would say that some of the stronger versions of your conjecture are almost certainly out of reach.