Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher dimensional manifolds) whose spectrum I have seen evaluated explicitly have highly repeated eigenvalues. They also happen to be quite symmetric (non-trivial isometry group), which was instrumental in the calculation of the spectrum in the first place. I have heard many people quote this heuristic: the more symmetric a manifold is, the higher the chances of spectral multiplicity. I was wondering
Is there a way to make this heuristic precise?
What about the converse? If a manifold has high spectral multiplicity, does it need to have a nontrivial isometry group?
If the answer to 2. is negative, and there are large classes of examples of surfaces/manifolds that are not symmetric at all, but have high spectral multiplicity, is there some other property shared by these examples which is causing the multiplicity? In other words, are there other criteria apart from symmetry that would make one suspect spectral multiplicity?