Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where 


What happens in the $p$-adic case? Is there sphere still a compact manifold? By sphere I mean:


$$ S^2 = \{ (x,y,z) \in \mathbb{Q}\_p^3: x^2 + y^2 + z^2 = 1 \}$$


In order to have harmonic analysis, what does it mean to integrate over the 2-sphere in this case? What is an example of an element of $L^2$ ?



harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?
First, the group $G=SO(3,\mathbb Q\_p)$ acts transitively on that quadric surface, by Witt's theorem. For $p\not=2$ this group is non-compact. The isotropy group of a point on the quadric is a subgroup $H$ isomorphic to $O(2)$, split or not depending on $p$ mod $4$. Then the harmonic analysis on $G/H$ can be addressed by fairly standard methods, although it's more complicated since $G$ is non-compact. (The irreducible admissible repns of $G$ containing a vector fixed under a ("special") maximal compact, a.k.a. "spherical repns", are classified by Borel-Casselman-Matsumoto as being unramified principal series.)


There's also the isogeny of $SL\_2$ to (split) $SO(3)$, ... 


