For $n=2$, the special orthogonal group $SO(n)$ is identified with $S^1$. it is known that, the Fourier Transform on the circle $S^1$ is given by: $$\hat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}f(\theta)e^{-in\theta}d\theta$$ Now, I would like to know the expression of the Fourier transform on the special orthogonal group $SO(n)$ (for n>2)?
@paul garrett, according to N. J. Vilenkin, p: 441, the irreducible unitary representation of $SO(n)$ denoted by $T^{nl}(g)$ are the representation given by $L^{nl}(g)$ in the factor space $\mathfrak{U}^{nl}=\mathfrak{R}^{nl}/r^2 \mathfrak{R}^{n,l-2}$ , where $L^{nl}(g)$ is the reducible representation in $\mathfrak{R}^{nl}$ the space of homogeneous polynomials of degree $l$ in $n$ variables, given by $L^{nl}(g)f(x)=f(g^{-1}x); f(x) \in \mathfrak{R}^{nl}$, and where $r^2 \mathfrak{R}^{n,l-2}$ the subspace of polynomials of the form $r^2f(x); f(x) \in \mathfrak{R}^{n,l-2}$, which is invariant for the operator $L^{nl}(g)$.
Now, according to the general theory; the Fourier transform on $SO(n)$ will be written as follows $\hat{f}(h)=\int_{SO(n)} f(k) L^{nl}(k) dk, f\in \mathfrak{U}^{nl}$ or something like that and that's my problem, I'd like to find explicit writing for the the Fourier transform on $SO(n)$ (as the case of $n=2$)?
Thank you in advance
From Weyl and Peter-Weyl, it is known that the decomposition of the left and right regular representation (meaning that $K=SO(n,\mathbb R)$ acts by translation on both the left and the right on functions in $L^2(K)$, with Haar measure) is that $L^2(K)$ is the $L^2$ completion of $\bigoplus_\pi \pi\otimes \pi^{\vee}$, where $\pi$ runs through isomorphism classes of irreducibles of $K$ (which are all finite-dimensional). These irreducibles are indexed (and described) by their "highest weights".
(On spheres, as quotients of orthogonal groups, only relatively special irreducibles occur, and these are tangibly given by spaces of harmonic polynomials. This also does specialize to the circle, since in that case $SO(2,\mathbb R)$ is the circle.)