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For the finite dimensional irreducible representations of $SU(2)$ we have that the group characters $\chi_n(g)$ for the $n^{th}$ representation are eigenfunctions of the Laplacian over the group manifold $S^3$ with the eigenvalue being proportional to the quadratic Casimir of the representation $\sum_a X_aX_a=\mathbb{I} C_2(n)$ (where $X_a$ are the generators, $\mathbb{I}$ is the identity operator and $C_2(n)$ is the quadratic Casimir) \begin{equation} \nabla^2\chi_n=\frac{C_2(n)}{4}\chi_n \end{equation} My questions are:

(1) Is there a nice geometrical way of seeing why this must be the case?

(2) Is this specific to $SU(2)$ or is the group character of a Lie group always an eigenfunction of the Laplacian with an eigenvalue proportional to the quadratic Casimir?

Skip the following: My guess for question (1) is that the algebra elements $X_a$ can be viewed as pushing points on the manifold to nearby points and hence we could probably use this to produce a linear map on the space of functions on the group manifold, hopefully forming a (infinite dimensional) representation of the Lie group where the generators are represented as derivative operators. Then the Laplacian begins to resemble (loosely) the quadratic Casimir as $\sum_a X_a X_a\mapsto \partial_a\partial_a$ (how do we incorporate the metric?). So this motivates the relation between $\nabla^2$ and $C_2$, but why specifically is it the group character function which is an eigenfunction of the Laplacian?

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