# Group characters are eigenfunctions of the Laplacian with eigenvalue proportional to the quadratic Casimir

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) $$$$\nabla^2\chi_n=\frac{C_2(n)}{4}\chi_n$$$$ 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?