Let $G$ be a compact Lie group and $H$ a subgroup. In The geometry of inequivalent quantizations (right under equation 2.11) the authors seem to claim that $$ L^2(G) = \bigoplus_{\mu \in \widehat{H}} d_\mu Ind_H^G(\mu). $$ Here $\widehat{H}$ is the set of irreps of $H$, $d_\mu$ is the dimension of the irrep, and $Ind_H^G(\mu)$ is the induced representation (space) of $\mu$.
How do you prove this claim?
Based on @elemelons response I think the following works.
Using properties (1) and (3) from Wolfram, we can write \begin{align} L^2(G) &= \text{Ind}_{ \{e\} }^G( 1 ) \\ &= \text{Ind}_H^G\left( \text{Ind}_{ \{e\} }^H (1 ) \right) \\ &= \text{Ind}_H^G\left( L^2(H) \right) \\ &=\text{Ind}_H^G\left( \bigoplus_{\mu \in \widehat{H}} d_\mu V_\mu \right) \\ &= \bigoplus_{\mu \in \widehat{H}} d_\mu \text{Ind}_H^G\left( V_\mu \right). \end{align}
The penultimate line follows from the fact that the regular representation of a group $H$ can be split into a direct sum over irreps of $H$ with multiplicity equal to the dimension of the irrep.