Asymptotic behavior of matrix coefficients I'm reading Casselman's notes "Introduction to the theory of admissible representations of p-adic reductive groups". In chapter 4 "The asymptotic behavior of matrix coefficients", the main result of this chapter states that for some torus elements $a$ (in a sense, those who are "close to zero"), the matrix coefficients $\left\langle\pi(a)v,\tilde{v}\right\rangle$ equlas to $\left\langle\pi_N(a)u,\tilde{u}\right\rangle_N$, where $\pi$ is an admissible representation, $\pi_N$ is its Jacquet module with respect to the unipotent radical $N$, $v\in V_\pi$, $\tilde{v}\in \widetilde{V}_\pi$, and $u$ (respectively, $\tilde{u}$) is the image of $v$ (respectively, $\tilde{v}$) in $V_{\pi,N}$ (respectively, $\widetilde{V}_{\pi,N}$).
In his notes "Remarks on Macdonald’s book on p-adic spherical functions", Casselman refers to this result (Theorem 6.7) and then makes the following remark: "This result says that any matrix coefficient is asymptotically equal to an $A$-finite expression".
Apparently, Casselman means that $\left\langle\pi_N(a)u,\tilde{u}\right\rangle_N$ is a sum of $A$-finite functions. I can't figure out why this is true.
This question is related to the old question Reference request - Jacquet module and asymptotic of matrix coefficients. I would very much appreciate a rather detailed explanation or a good reference for both the p-adic case and the Archimedean case.
 A: For the $p$-adic case, the idea is as follows (thanks to Elad for pointing out the direction).
Recall (from the unpublished notes by Casselman, Introduction to the theory of admissible representations of p-adic reductive groups) that $k$ is a non-Archimedean locally compact field, $G$ is a group of $k$-rational points of a reductive algebraic group defined over $k$, and $P$ is a parabolic subgroup of $G$ with Levi decomposition $P=MN$.
We note that it is sufficient to show that the function $x\mapsto\left\langle \pi_{N}(x)u,\tilde{u}\right\rangle _{N}$ is an $A_M$-finite function ($A_M$ is the center of the Levi part $M$). Indeed, From Jacquet and Langlands (Lemma 8.1 in Automorphic Forms on $\operatorname{GL}(2)$: Part I, volume 114. Springer, 2006), one can deduce that the space of continuous finite functions on the locally compact abelian group $A_M$ is spanned by functions of the form
\begin{equation}
        \prod_{i=1}^{r}\chi_i(a_i)\left\lvert a_i\right\rvert^{p'_i}\log_q ^{p_i}\left\lvert a_i\right\rvert,
\end{equation}
where $r$ is such that $A_M\cong k^r$,  $\left(p'_{1},\dotsc,p'_{r}\right)\in \mathbb{R}^r$, $\left(p_{1},\dotsc,p_{r}\right)\in \mathbb{Z}^r_{\ge 0}$, and for all $1\leq i\leq r$, $\chi_i:k^\times \to \mathbb{C}^\times$ are unitary characters.
Now, in order to prove that the function $x\mapsto\left\langle \pi_{N}(x)u,\tilde{u}\right\rangle _{N}$ is an $A_M$-finite function, we use the following technical lemma.

*

*Lemma.   Let $R$ be a group with center $Z\left(R\right)\cong K\times\mathbb{Z}^{r}$, where $K$ is a compact group. Let $\left(H,\sigma\right)$ be a (complex) smooth $R$-module of finite length and let $v\in H$. Then, the $Z\left(R\right)$-module generated by $v$ is finite dimensional.

The Jacquet module is a smooth $G$-module of finite length (See Theorems 3.3.1 and 6.3.10 in the abovementioned unpublished notes by Casselman). Hence, we apply the lemma with $R=M$ (with $Z(R)=A_M$), $H=V_N$, $\sigma=\pi_N$, and $v=u\in V_{N}$. This gives that $U:=\left\{ \pi_{N}\left(a\right)u|\ a\in A_{M}\right\} $
is of finite dimension. Let $\left\{  \pi_{N}\left(b_{1}\right)u,\dotsc, \pi_{N}\left(b_{\ell}\right)u\right\}$
be a basis of $U$. Then,
\begin{equation}
\pi_{N}\left(a\right)u=\sum_{i=1}^{\ell}c_{i}(a)\pi_{N}\left(b_i\right)u.
\end{equation}
Therefore,
\begin{equation}
\left\langle \pi_{N}\left(ma\right)u,\tilde{u}\right\rangle _{N}=\sum_{i=1}^{\ell}c_{i}\left(a\right)\left\langle \pi_{N}\left(m b_i\right)u,\tilde{u}\right\rangle _{N}.
\end{equation}
For more details, as well as the proof of the lemma, see Hazan - A Note on the Asymptotic Expansion of Matrix Coefficients over $p$-adic Fields.
For the Archimedean case, one can find an asymptotic expansion of the matrix coefficients, as a finite sum of finite functions, in Casselman's paper Jacquet modules for real reductive groups (see the Lemma in Section 5, Proceedings of the International Congress of Mathematicians (Helsinki, 1978)).
