To give a bit of background about my question, let $R$ be a rotation that sends a unit vector $r$ to $r'$ and let $Y_{\ell,m}$ be a spherical harmonic of degree $\ell$ and order $m$ (i.e. $\ell \geq 0$ and $-\ell \leq m \leq \ell$). We have the relation $$Y_{\ell }^{m}(r')=\sum_{m'=-\ell }^{\ell }[D_{mm'}^{(\ell )}(R)]^{*}Y_{\ell }^{m'}(r),$$ where $D_{mm'}^{(\ell )}$ is an element of the Wigner D-matrix.
My question is: does this relation generalize to hyperspherical harmonics? If I were to choose a basis of $N_\ell$ elements in $L^2(\mathbb{S}^{n-1})$, let's say $Y_{\ell, m}$ with $m = 1, \ldots, N_\ell$, would I be able to say that $$Y_{\ell }^{m}(r')=\sum_{m'=1}^{N_{\ell}}A_{mm'} Y_{\ell }^{m'}(r)$$ for some matrix with $N_\ell^2$ entries $A_{mm'}$ dependent on $\ell$ and $R$? If anyone has a good reference with more info, that would be very helpful too. Sorry if the tags are wrong for this by the way.
Thanks again!
If $V_\ell\subset L^2(S^{n-1})$ consists of the homogeneous degree $\ell$ functions with basis $\mathcal{B}=\{Y_\ell^m\mid 1\le m\le N_\ell\}$ then $\mathcal{B}'=\{Y_\ell^m\circ R\mid 1\le m\le N_\ell\}$ is also a basis (for any $f\in V_\ell$, write $f\circ R^{-1}$ wrt $\mathcal{B}$, then precompose with $R$ to know $f$ wrt $\mathcal{B}'$), so there exists a change-of-basis matrix.