Suppose that $G$ is a compact, connected Lie group. Such a $G$ possess a bi-invariant Riemannian metric. With the bi-invariant metric, I think it's true that the Laplacian $\Delta$ on $L^2(G)$ is a linear operator which commutes with the action of $G$ (i.e. $g \cdot \phi(h):=\phi(g^{-1}h)$ for $\phi \in L^2(G)$). The equivariance implies that the irreducible representations are summands of the eigenspaces of the Laplacian.
Question 1: Can we say anything more about this relationship?
For example, on the circle, the $k$th Fourier mode $e^{ik\theta}$ is assigned the eigenvalue $k^2$ by $\Delta$. Then, for $k \neq 0$, the $k$th eigenspace has multiplicity $2$. It seems very nice that the Laplacian organizes the irreducible representations in this way, it's nearly the case that each irreducible is assigned a unique eigenvalue (we are only off by $\pm k$).
Also, the fact that eigenvalues grow as $k^2$ is interesting.
Question 2: Are there results concerning the growth rate of eigenvalues?