I am interested in structures of the eigenspaces of the Laplace operator on the $n$-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal group of dimension $n$ (denoted by $SO(n)$). My question is: are all of these eigenspaces irreducible representations of $SO(n)$? If yes, how to prove it? Or where can I find a proof?
It is known that the problem of finding eigenspaces of the Laplace operator on the $n$-dimensional unit ball can be translated (by standard separation of variables) to the equation $-\Delta_{S^{n-1}}u=\mu_n u$, where $-\Delta_{S^{n-1}}$ is the Laplace--Beltrami operator on the $(n-1)$-dimensional unit sphere and the radial part $r^{3-n}\frac{\partial}{\partial r}(r^{n-1}\frac{\partial}{\partial r}f(r))-(\lambda-\mu_n) f(r)=0$ (for example if $n=2$ and we consider the Dirichlet condition, then this is a Bessel equation). The problem $-\Delta_{S^{n-1}}u=\mu_n u$ is an eigenvalue problem for the Laplace--Beltrami operator on the sphere and has eigenvalues $\mu_n=m(m+n-2)$, $m=0,1,\ldots$ and the eigenspaces $H_m$ of this equation are spherical harmonics and they are irreducible $SO(n)$-representations of the Laplace--Beltrami operator, see for example D. Gurarie, Symmetries and Laplacians, Introduction to Harmonic Analysis, Group Representations and Applications, North-Holland Mathematics Studies 174, North-Holland, Amsterdam, 1992. From the second equation we get eigenvalues $\lambda$ of the equation on the ball. If we denote by $f_m(\lambda,r)$ the solution of this equation and by $A_m$ the sets of $\lambda$ such that $f_m(\lambda,r)=0$, then with $\lambda\in A_m$ is associated the eigenspace $H_m$. So now it is easy to see that an eigenspace of the Laplace operator on the ball, associated with $\lambda$, is an irreducible $SO(n)$-representation iff $\lambda\in A_m$ implies $\lambda\notin A_{m'}$ for $m\neq m'$. Therefore my question can be reformulated in the following way: are the sets $A_m$ and $A_{m'}$ always disjoint?
