I am studying (co)-induced representations of topological groups and I came across the following situation:
$G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, where $W$ is a (complex) vector space (in fact, it is a vector space on which $H$ acts but for the moment let us ignore this). We say that $f$ is compactly supported modulo $H$ if $supp(f) \subseteq HC$ for some compact set $C$ in $G$.
My question: is this equivalent with saying that the image of $supp(f)$ is compact in $H\backslash G$ (where the quotient space is endowed with the usual topology)?
Now I add some further assumptions: $G$ is a locally profinite group (= locally compact + totally disconnected or equivalently: Hausdorff + the unit $1\in G$ has a fundamental system of compact open subgroups), $H$ a closed subgroup of $G$ acting on $W$, and $f$ a map satisfying
$f(hg) = \sigma(h)f(g)$
for all $h\in H,g\in G$ (where $\sigma$ is the map describing the action of $H$). Is the previous statement true now? Thanks in advance!