Spherical harmonics as representations of SO(3)

Example

Let $G = \mathrm{SO}(3)$ (rotation group) act on the unit sphere $S^2 \subset \R^3$, and let $\sigma$ be the rotation-invariant surface measure. The stabilizer of the north pole $e_3$ is the subgroup $K \cong \mathrm{SO}(2)$ of rotations about the $z$-axis, so $S^2 = G / K$. The group acts unitarily on $L^2(S^2, \sigma)$ by $(g f)(x) = f(g^{-1} x)$.

For $\ell \ge 0$, let $\mathcal{H}_\ell$ be the space of restrictions to $S^2$ of harmonic polynomials on $\R^3$ that are homogeneous of degree $\ell$. Its elements are the spherical harmonics of degree $\ell$. Then:

1. $\dim \mathcal{H}_\ell = 2\ell + 1$, and $\mathcal{H}_\ell$ is invariant under $G$.
2. Each $\mathcal{H}_\ell$ is an irreducible representation of $G$, and the $\mathcal{H}_\ell$ are pairwise non-isomorphic. Every irreducible continuous representation of $\mathrm{SO}(3)$ is isomorphic to one of them.
3. $L^2(S^2) = \widehat{\bigoplus}_{\ell \ge 0} \mathcal{H}_\ell$, an orthogonal Hilbert direct sum.
4. The spherical Laplacian acts on $\mathcal{H}_\ell$ as multiplication by $-\ell(\ell + 1)$.

A check of the multiplicity one in 3: the functions in $\mathcal{H}_\ell$ fixed by $K$ form a line. This line is spanned by the zonal harmonic $x \mapsto P_\ell(x_3)$, where $P_\ell$ is the Legendre polynomial. For example, for $\ell = 1$ the space $\mathcal{H}_1$ is spanned by $x_1, x_2, x_3$, and it is the standard representation of $\mathrm{SO}(3)$ on $\C^3$. Its $K$-fixed line is spanned by $x_3 = P_1(x_3)$. The spherical functions of the pair $(\mathrm{SO}(3), \mathrm{SO}(2))$ are $g \mapsto P_\ell(\ip{g e_3}{e_3})$, one for each $\ell$.