Induced representation I'm doing the problem section of the induced representations chapter by Steinberg, and I'm having problems with the following one:
Let $G$ be a group and $H$ subgroup. Given a representation $\rho:H\rightarrow GL(V)$, let $Hom_H(G,V)$ be the vector space of all functions $f:G\rightarrow V$ such that $f(gh)=\rho(h)^{-1}f(g)$, for all $g\in G$ and $h\in H$, equipped with pointwise operations. Define a representation
$$ \varphi: G \rightarrow GL(Hom_H(G,V))$$ 
by $\varphi_g(f)(g_0)=f(g^{-1}g_0)$. Prove that $\varphi$ is a representation of $G$ equivalent to $Ind_H^G \ \rho$ (Hint: find a basis for $Hom_H(G,V)$ and compute the character).
If we take the whole set of functions from $G$ to $V$ as vector space, it has dimension $|G|\dim(V)$ and we can show a basis. But for $Hom_H (G,V)$ I don't know how to construct a basis. Any help is welcome! 
 A: Fix $g\in G$ arbitrary and ask yourself, given the intertwining condition inherent in the definition of $\hom_H(G,V)$, and knowledge of the representation $\rho$ of $H$, what other values $f(g')$ do you get automatically just by knowing the value of $f(g)$? Consider the collection of all such $g'$s; can you describe this as a subset of $G$ in terms of $g$? See if you can proceed from there.
A: Recall the definition of an induced representation.
Let a representation $\rho$ of a subgroup $H<G$ act in the space of linear operators ${\rm{GL}}(W)$, where $W$ is a linear space (e.g., that of all square-integrable functions):
$$
\sigma(h):\quad H\to {\rm{GL}}(W).
$$
The induced representation of the group $G$, denoted with 
$$
\pi = Ind^{\,G}_{H}\rho\;:\quad G\to{\rm{GL}}(V),
$$
is a homomorphism acting in the space of linear operators ${\rm{GL}}(V)$ on a subspace $V\subset W$. This subspace is defined by 
$$
V = \{
    F \colon G \to W
    \mid
    \text{$F(hg) = [\rho(h)F](g)$, for all $h \in H$, $g \in G$}
    \}.
$$
This definition imposes a restriction on the functions we are to use, and is sometimes called "structural condition".  
Technically, the induced representation $U:\;G \to {\rm{GL}}(V)$ is implemented by 
$$
[U(g) F](g_0) = F(g_0g)\quad\mbox{for all}\quad g,\,g_0\in G
$$
Without loss of generality, we can rewrite the above two formulae as:
$$
[U(g) F](g^{-1}_0) = F(g^{-1}_0)\quad\mbox{for all}\quad g,\,g_0\in G
$$
$$
V = \{
    F \colon G \to W
    \mid
    \text{$F(h^{-1}g^{-1}) = [\rho(h^{-1})F](g^{-1})$, for all $h \in H$, $g \in G$}
    \}.
$$
Now, we introduce an inversion operator $i$:
$$
i(g) = g^{-1}\quad\mbox{for all}\quad g\in G.
$$
It enables us to rewrite the above as
$$
\left[U(g) [Fi]\right](g_0) = [Fi](g^{-1}g_0)\quad\mbox{for all}\quad g,\,g_0\in G
$$
$$
Vi = \{
    [Fi] \colon G \to W
    \mid
    \text{$[Fi](gh) = \left[ \rho(h^{-1}) [Fi] \right](g)$, for all $h \in H$, $g \in G$}
    \},
$$
where $Vi$ is a notation for the space of all functions $Fi$.
Finally, denote $U$ with $\phi$, $Fi$ with $f$, and $Vi$ with $V$:
$$
\left[\phi(g) f\right](g_0) = f(g^{-1}g_0)\quad\mbox{for all}\quad g,\,g_0\in G
$$
$$
V = \{
    f \colon G \to W
    \mid
    \text{$f(gh) = \left[ \rho(h^{-1}) f \right](g)$, for all $h \in H$, $g \in G$}
    \},
$$
which is exactly the representation $Hom_{H}(G,V)$. 
