If $\Gamma$ is a countably infinite discrete group and $F$ a finite subgroup of $\Gamma$, then one knows that there is an isomorphism of $F$-representations
$l^2(\Gamma)\cong l^2(F)\otimes l^2(F\backslash\Gamma)$,
where $F$ acts on $l^2(\Gamma)$ and $l^2(F)$ by via the left regular representation and on $l^2(F\backslash\Gamma)$ trivially.
Now suppose $G$ is a Lie group and $K$ a compact subgroup of $G$ (equipped with suitable Haar measures). Is there an isomorphism analogous to the above, that is, an isomorphism of $K$-representations
$L^2(G)\cong L^2(K)\otimes L^2(K\backslash G),$
where $K$ acts on $L^2(G)$ and $L^2(K)$ via the left regular representation and on $L^2(K\backslash G)$ trivially?
Thanks!
Edit: I guess the first question is whether there is a measure-theoretic isomorphism $L^2(G)\cong L^2(K)\otimes L^2(K\backslash G)$. If $K$ were a maximal compact subgroup this would be true since $G$ would be a trivial $K$-bundle over $K\backslash G$, but even for an arbitrary compact subgroup $K$ this may be true at a measure-theoretic level; I'm not entirely sure. Such an isomorphism of $K$-representations seems plausible since $K$ acts only on each fibre of the principal $K$-bundle $G\rightarrow K\backslash G$.