I am not having the best luck getting this, so any advice is appreciated. I apologize in advance if the question is too low-level. Let $G$ be a locally compact group $\sigma$-compact group, and $H\leq G$ a closed subgroup such that $G/H$ carries a finite invariant measure $\mu$. Let $(\pi,\cal H\_\pi)$ be a unitary representation of $H$, and let $\rho={\rm Ind}\_H^G(\pi)$, and call the Hilbert space it acts on by left translations ${\cal H}\_\rho$. This space can be realized as the Hilbert space completion of the space of functions $G\to\cal H\_\pi$ which are $H$-equivariant, with support compact modulo $H$, with respect to the inner product
$$
\langle f\_1,f\_2\rangle=\int\_{G/H} \langle f\_1(g),f\_2(g)\rangle d\mu(gH).
$$
This is the standard simplest construction. My question is: **how can I define a bounded linear map $\Phi:{\cal H}\_\pi\to{\cal H}\_\rho$ which is an isometry and commutes with orthogonal projections onto subspaces of invariant vectors.**


What I tried is to let $\xi\in{\cal H}\_\pi$ and define the function $f\_\xi$ by
$$f\_\xi(x)=\begin{cases}\pi(x^{-1})\xi&\text{if}~x\in H,\\0&\text{otherwise.}\end{cases}
$$
This defines a map $\Phi:\cal H\_\pi\to\cal H\_\rho$, $\xi\mapsto f\_\xi$. But then, what I obtain is
$$ \langle \Phi(\xi),\Phi(\eta)\rangle\_{\cal H\_\rho}=\mu(H)\langle \xi,\eta\rangle\_{\cal H\_\pi},
$$
with $\mu(H)$ the measure of the identity coset in $G/H$, since $f\_\xi,\,f\_\eta$ are non-trivial only on $H$. My map seemed fairly natural, is there something I am overlooking? Is there some twist to apply to this to make it work?



Induced representation of locally compact groups
Such a $\Phi$ is not going to exist — nor is $H$ going to have nonzero $\mu(H)$ — unless $H$ is *open* as well as closed in $G$. This is the very case treated in Mackey [(1951, Part II)](http://www.ams.org/mathscinet-getitem?mr=42420): then $G/H$ is discrete, $\mu$ can be normalized to be counting measure (i.e. $\mu(eH)=1$), and then your $\Phi$ is precisely the isometric embedding $\mathcal H\_\pi\hookrightarrow\mathcal H\_\rho$ already discussed and denoted $v\mapsto f\_{e,v}$ by Mackey (p. 585).


(When $H$ is not open one resorts to more sophisticated maps, e.g. the $\epsilon:\mathrm C\_0(G)\times\mathcal H\_\pi\to\mathcal H\_\rho$ of Blattner [(1962)](http://www.ams.org/mathscinet-getitem?mr=166621):
$$
\epsilon(\phi,\xi)(g)=\int\_H\phi(hg)\pi(h^{-1})\xi\,dh,
$$
assuming here, as you do, that $G/H$ has a $G$-invariant measure, i.e. the modular functions of $G$ and $H$ coincide on $H$.)


