I met such a rather concrete measure theoretic problem while dealing with induced representations of $p$-adic groups.
So let $G$ be a unimodular locally compact group, with $P$ its closed subgroup (what we think about them are just $p$-adic groups such as $G=GL(n,\mathbb{Q}_p)$ and $P$ its standard parabolic subgroups). Let $\delta_P$ be the module function of $P$ (coming from the difference of the left and right Haar measure on $P$). Let $C_c(P\backslash G,\delta_P^{-1})$ be the space of continuous functions $f:G\rightarrow \mathbb{C}$, such that the support of $f$ is compact modulo $P$, and $$f(pg)=\delta_P(p)^{-1} f(g),~\forall p\in P,\forall g\in G.$$ Then we let $G$ act on this function space by right regular representation: $$(R(g)f)(x):=f(xg).$$
O.K. So this is just a general construction in induced representations. By a very general theorem on Bushnell-Henniart's book, Local Langlands Conjecture for $GL(2)$, there is a $G$-invariant linear functional $\alpha:C_c(P\backslash G,\delta_P^{-1})\rightarrow \mathbb{C}$, namely $\forall f\in C_c(P\backslash G,\delta_P^{-1})$, we have $\alpha(R(g)f)=\alpha(f),\forall g\in G$.
What I want to ask is: suppose we have a compact open subgroup $K$ of $G$, such that $G=PK$ (this is the case for $GL(n)$ by Iwasawa decomposition), then how to prove $$C_c(P\backslash G,\delta_P^{-1})\rightarrow \mathbb{C}, f\mapsto \int_K f(k)dk$$ is such a functional in that general theorem?
This seems to be very striking, because in the proof of the general cases, we don’t have any idea how to write down such thing explicitly. I met this fact in Godement and Jacquet's classic "Zeta functions of simple algebras" (page 21), but I have no idea how to prove it. Can anyone give me some help? Thanks a lot in advance!
Ok, now I understand this crucial fact. Such a compact subgroup (with $G=PK$) is called a ‘good compact’. It’s proved in Casselman’s famous notes (introduction to admissible representations of $p$ adic groups, subsection 3.1), for any such good compact, we have the above property. And this is very helpful in proving normalized induction preserves unitaricity.