# Question on the proof that the Jacquet module preserves admissibility

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.

• Just take any representative and average it over $M_0$ (i.e., apply the canonical projection to the $M_0$-fixed space). – LSpice May 10 at 17:38
• 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. – LSpice May 10 at 17:45
• 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 – D_S May 10 at 17:49
• You can find my contact information at mathematics.tcu.edu/faculty-staff/faculty . Please feel free to e-mail me. – LSpice May 10 at 17:55
• Thank you $\space$ – D_S May 10 at 18:08