Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G \twoheadrightarrow G/H$.

The Peter-Weyl theorem states that the two-sided regular representation of $G$ on $L^2(G)$ is isomorphic to this direct sum: $$\underset{{[\rho] \in \Lambda}}{\bigoplus} \rho^* \otimes \rho$$ $\Lambda$ is the set of isomorphism classes of irreducible unitary representations.

My question now is: What does $L^2(G/H)$ look like? Is there a similar direct sum decomposition? How does it depend on the representations of $G$ and $H$?

This is a very nice, natural question, also with a pleasant answer: recalling that the expression $\bigoplus_\rho \rho^*\otimes \rho$ is the decomposition under the action of $G\times G$ by $(g,h)\cdot f(x)=f(g^{-1}xh)$, a given $\rho^*\otimes\rho$ when restricted to $G\times H$ becomes $\rho^*\otimes \rho|_H$. For such functions to descend to $G/H$ is exactly that $\rho|_H$ contains the trivial repn of $H$, giving $\rho^*\otimes \mathbb C^{m(1,\rho)}$ where $m(1,\rho)$ is the multiplicity of the trivial repn of $1$ in $\rho|_H$. The whole is the direct sum over $\rho$...
• Great! Could we also say that the summand is $\rho^* \otimes \operatorname{Rep}_H(1, \rho|_H)$? (The second factor might be better known as $\operatorname{Hom}_{\operatorname{Rep}_H}(1, \rho|_H)$ to some people, the space of $H$-equivariant maps from the trivial representation to $\rho|_H$) – Turion Dec 10 '13 at 15:39
• Sure, yes, that's essentially the same thing, unless potentially someone cares about a distinction between trivial repns and their duals, or something. That is, the second "factor" is the trivial $H$-repn subspace of $\rho$... A secondary issue... – paul garrett Dec 10 '13 at 15:43