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.

  • $\begingroup$ Just take any representative and average it over $M_0$ (i.e., apply the canonical projection to the $M_0$-fixed space). $\endgroup$ – LSpice May 10 at 17:38
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    $\begingroup$ I'm not sure of the general perspective, but I think that questions like this, that you can answer yourself using standard tools, eventually have to be considered not research-level, even if they are part of understanding research mathematics. Do you have a nearby community (advisor or whatever) to whom you might be able to pose such questions? I would consider it perfectly reasonable if a student or colleague asked me this question in person or by e-mail, but I don't think of it as appropriate for MO. I'm sorry for a personal communication here, but I don't know how to reach you privately. $\endgroup$ – LSpice May 10 at 17:45
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    $\begingroup$ Thanks for your answer. You're right it's not research level. I don't know where else to seek help on technical questions like this. No one at my university knows the nitty gritty details $\endgroup$ – D_S May 10 at 17:49
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    $\begingroup$ You can find my contact information at mathematics.tcu.edu/faculty-staff/faculty . Please feel free to e-mail me. $\endgroup$ – LSpice May 10 at 17:55
  • $\begingroup$ Thank you $\space$ $\endgroup$ – D_S May 10 at 18:08

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