# Spherical harmonics and irreducible representations of $SO(2)$ and $SO(3)$

In the Wikipedia article it is mentioned (without source) that the spherical harmonics of degree $$\ell$$ on the $$n$$-sphere are an irreducible (whether real or complex is not mentioned) representations of $$SO(n+1,\mathbb{R})$$.

Spherical harmonics

How I define the space of spherical harmonics: Let $$H^{(\ell)}(\mathbb{R}^{n+1})$$ be the (real or complex) vector space of harmonic homogeneous polynomials of degree $$\ell$$ in $$(n+1)$$ real variables. The (real or complex) vector space of spherical harmonics of degree $$\ell$$ on the $$n$$-sphere is defined by restricting the polynomials $$h(x)$$ in $$H^{(\ell)}(\mathbb{R}^{n+1})$$ to $$x \in S^n \subset \mathbb{R}^{n+1}$$ and is denoted by $$H_\mathbb{R}^{(\ell)}(S^n)$$ and $$H_\mathbb{C}^{(\ell)}(S^n)$$.

First question

For $$n=1$$ the spaces $$H_\mathbb{R}^{(m)}(S^1)$$ and $$H_\mathbb{C}^{(m)}(S^1)$$ have (real or complex) dimension $$2$$ for $$m \ge 1$$. I know that all non-trivial irreducible real representations of $$SO(2, \mathbb{R})$$ have dimension $$2$$ and all irreducible complex representations have dimension $$1$$. I understand that $$H_\mathbb{R}^{(m)}(S^1)$$ is an irreducible real representation of $$SO(2, \mathbb{R})$$, but most certainly is $$H_\mathbb{C}^{(m)}(S^1)$$ not an irreducible complex representation of $$SO(2, \mathbb{R})$$ (by dimensional arguments). So is the above statement false?

Second question

For $$n=2$$ the spaces $$H_\mathbb{R}^{(\ell)}(S^2)$$ and $$H_\mathbb{C}^{(\ell)}(S^2)$$ have (real or complex) dimension $$2\ell+1$$. I understand that $$H_\mathbb{C}^{(\ell)}(S^2)$$ is an irreducible complex representation of $$SO(3, \mathbb{R})$$ and a suitable basis are the spherical harmonics. Is $$H_\mathbb{R}^{(\ell)}(S^2)$$ an irreducible real representation of $$SO(3,\mathbb{R})$$ with a suitable basis the real spherical harmonics? I see no obstruction but it is mentioned here that there might be problems since $$\mathbb{R}$$ is not algebraically closed.

• See the section 3 and other sections of the thesis-pub.math.leidenuniv.nl/~edixhovensj/teaching/2019-2020/CIMPA/…
– MAS
Apr 15, 2020 at 16:39
• As far as I understand the thesis is only concerned with irreducible complex representations of $SO(3)$ and thus does not answer my second question. Apr 15, 2020 at 20:02

The statement that the spherical harmonics of degree $$\ell$$ on the $$n$$-sphere are irreducible complex representations of $$SO(n+1,\mathbb{R})$$ holds for $$n \ge 2$$ which answers the first question. Also I might add that in general these are not all irreducible complex representations, but only those of highest weight $$(\ell, 0, \dots, 0)$$. About irreducible real representations I have not found a similar statement yet.
The universal (double) cover $$SU(2, \mathbb{C})$$ has irreducible real representations in all dimensions which are odd or divisible by $$4$$, see e.g. here. As in the complex case, the irreducible real representations in odd dimensions descend to $$SO(3, \mathbb{R})$$ and are isomorphic to $$H_\mathbb{R}^{(\ell)}(S^2)$$.