I consider this as a differential geometry problem. I have asked some of my classmates who are more interested in that, and also looked into some literature, but none of what I've found seems to help.
Context
Thanks to the Peter-Weyl theorem, on any compact Lie group there's a set of nice smooth functions that span the functions spaces of interest, and form an orthonormal basis in suitable sense. The simplest case is the circle group, where the nice functions are the trigonometric functions.
On any compact Lie group, there's also a natural metric, so we can define operators like the Laplacian or the Hodge star naturally which depend on a choice of metric.
I notice one thing on the circle group: the nice functions are also nice with respect to the Laplacian. They are the eigenvectors, whose eigenvalues are known.
Question
I wonder if this is also the case for an compact Lie group (or more generally homogeneous spaces): does the set of good functions tell us a lot about the Laplacian or even more general operators? How much can we know from them in particular for the spectrum?
If 1. is true, what happen then when we perturb the natural metric on the compact Lie group a bit (for example, by some Ricci flow)? How will the eigenvalues and eigenvectors change correspondingly? Til what extent does this description (if any) breaks down, for example when two first distinct eigenspaces collapse to each other?
Any related pointers or suggestions will be highly appreciated. I'm very curious about what happen: just think of you are poking your finger into a 2-sphere (as a homogeneous space). How will the nice functions change? Thank you very much.
