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.
Best Answer
So I think I found the answers myself.
First question
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.
Second question
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 $H_\mathbb{R}^{(\ell)}(S^2)$.