Question

Consider a function $f$ on $S^2,$ and its spherical transform $\hat{f}.$ Let $r$ be a rotation by some $\rho \in SO(3).$ Is there some nice formula for $\widehat{f \circ r}?$ I have found some allusions to "Wigner D-matrices"... I am sure there are good references...

Answer 1

Suppose your function is $f(s)$ and its spherical transform, as you call it, is $\hat{f}_{lm}=\int_{S^2} Y_{lm}(s)^* f(s)\,\mathrm{d}s$, where the $Y_{lm}(S)$ are spherical harmonics. Then for fixed $l$, the components of $\hat{f}$ will transform under a rotation $r\in SO(3)$ in an irreducible representation of $SO(3)$ of spin $l$: $$ \hat{f\circ r}_{lm} = \sum_{n} D(r)^l_{m,n} \hat{f}_{ln}. $$ The representation maps $D({\cdot})^l_{m,n}$ are presumably what you are looking for and are precisely the Wigner D-matrices (they are matrices in the $m,n$ indices).

An explicit expression for these matrices depends on how you choose to parametrize your rotations $r\in SO(3)$. If you use Euler angles, then explicit expressions are given on the Wikipedia page for Wigner D-matrices. If you prefer a different parametrization of $SO(3)$, then another reference might be more convenient. In that case, please refine your question.