I'm reading Casselman's notes "[Introduction to the theory of admissible representations of p-adic reductive groups](https://personal.math.ubc.ca/%7Ecass/research/pdf/p-adic-book.pdf)". 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](https://personal.math.ubc.ca/%7Ecass/research/pdf/Macdonald.pdf)", 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](https://mathoverflow.net/questions/90367/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.



Asymptotic behavior of matrix coefficients
For the $p$-adic case, the idea is as follows (thanks to [Elad](https://mathoverflow.net/users/103908/darkl) for pointing out the direction).
Recall (from the unpublished notes by Casselman, [*Introduction to the theory of admissible representations of p-adic reductive groups*](https://personal.math.ubc.ca/%7Ecass/research/pdf/p-adic-book.pdf)) 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.*](https://doi.org/10.1007/BFb0058988) 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](https://arxiv.org/abs/2211.15822).


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)](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1978.2/ICM1978.2.ocr.pdf)).


