Let $G$ be a compact Lie group. Part of the Peter-Weyl theorem is a decomposition result: $$ L^2(G) = \bigoplus_\ell V_\ell \otimes V_\ell^*. $$ Here $V_\ell$ is an irrep of $G$ and the sum is over all irreps $\ell$.
Now suppose $G/H$ is a (smooth) manifold. The Peter-Weyl theorem appears to descend as $$ L^2(G/H) = \bigoplus_\ell V_\ell \otimes W_{A,\ell}^*. $$ Here $W_{A,\ell} \subset V_\ell$ is a trivial representation of $H$. See Peter-Weyl Theorem on the Sphere.
The previous two spaces are examples of induced representations $Ind_H^G(V^A)$ where $V^A$ is a trivial irrep of $H$ (where in the first, $H = \{e\}$ is the trivial subgroup). Is there a similar result for induced representation spaces in general? Specifically, can we write $$ Ind_H^G(V^i) = \bigoplus_\ell V_\ell \otimes W_{i,\ell}^*, $$ where $V^i$ is the $i$-th irrep of $H$ and $ W_{i,\ell} \subset V_\ell$ is the $i$-th irrep of $H$? If this is true (or not true) can a reference be provided?
