Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\pi|_M$ on $V/V(N)$, where $V(N)$ is the linear span of $v - \pi(n)v : v \in V, n \in N$.
I am reading Casselman's notes on representation theory, Theorem 3.3.1/3.3.3, where he shows that $\pi_N$ is admissible. I am confused on a small detail of the proof which I'll explain.
Let $\overline{N}$ be an opposite unipotent radical of $N$ relative to some minimal parabolic. Let $K_0$ be a compact open subgroup of $G$ with an Iwahori factorization, $K_0 = \overline{N}_0 \times M_0 \times N_0$, where $M_0 = M \cap K_0$, $N_0 = N \cap K_0$ etc.
Casselman starts the proof with the following (page 36):
"Let $\bar{U}$ be any finite dimensional subspace of $V_N^{M_0}$, and let $U \subset V^{M_0}$ be any finite dimensional subspace of $V$ mapping onto $\bar{U}$."
It is not clear to me why this is possible. It is possible if the projection $V^{M_0} \rightarrow V_N^{M_0}$ is surjective. Is this the case? If $v + V(N) \in V_N$ is in $V_N^{M_0}$, this means that $v - \pi(k)v \in V(N)$ for every $k \in M_0$. It might be the case $v + V(N)$ has a representative in $V$ which is actually fixed by $M_0$, but I haven't been able to show this.
