Relation between eigenvalue and representation of compact Lie group I read that the Laplacian has eigenvalues that have explicit formulas for all classical groups, for instance for $SO(2n)$ they are of the form
$$\sum_{i=1}^n (x_i+n-j)^2 - \frac{1}{6}n(n-1)(2n-1)$$
with also explicit multiplicities that are
$$\prod_{i=0}^{n-1} \prod_{j=i+1}^{n-1} \left(\frac{(x_{n-j}+j)^2 - (x_{n-i}+i)^2}{j^2 - i^2}\right)^2$$
Apparently this can be obtained directly from the theory of representation of compact groups, but I do not understand the relation. Can we deduce this by knowing the irreducible representations of $SO(2n)$, the corresponding weights, etc.?
 A: I don't know the precise details but this is how it should go in broad strokes. If $G$ is a compact semisimple Lie group it can be equipped with a canonical Riemannian metric given by translating the negative of the Killing form on $\mathfrak{g}$; this gives us a Laplacian $\Delta$ acting on $L^2(G, \mu)$ where $\mu$ is Haar measure. By the Peter-Weyl theorem $L^2(G)$ decomposes under the action of $G$ as a direct sum
$$L^2(G) \cong \bigoplus_i d_i V_i$$
of $d_i = \dim V_i$ copies of each irreducible representation $V_i$, as in the finite group case, and the Laplacian commutes with the action of $G$ and hence acts on each $V_i$ as a scalar by Schur's lemma; this accounts for the large multiplicity $d_i^2$ of each eigenvalue. The dimensions can be computed by the Weyl dimension formula.
The action of $\Delta$ on each $V_i$ can now be identified with the action of the Casimir element (maybe up to some fixed multiple?), and I believe formulas for this should be known in terms of weights etc. although I'm not sure where to find them. But this at least provides a search term; you can try looking up tables of Casimir eigenvalues.
