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 Recall (from the unpublished notes by Casselman, Introduction to the theory of admissible representations of p-adic reductive groups) that $k$ is a non-Archimedean locally compact field,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 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|a_i\right|^{p'_i}\log_q ^{p_i}\left|a_i\right|, \end{equation} where $r$ is such that $A_M\cong k^r$, $\left(p'_{1},\ldots,p'_{r}\right)\in \mathbb{R}^r,\ \left(p_{1},\ldots,p_{r}\right)\in \mathbb{Z}^r_{\ge 0}$, 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The Jacquet module is a smooth $G$-module of finite length (See Theorems 3.3.1 and 6.3.10 in the abovementioned unpublished notes by Casselman).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 $\left\{ \pi_{N}\left(b_{1}\right)u,\ldots, \pi_{N}\left(b_{\ell}\right)u\right\}$ 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 here.

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)).

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)).

Thanks to Elad, I was able to write an answer. Comments are welcome.

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 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|a_i\right|^{p'_i}\log_q ^{p_i}\left|a_i\right|, \end{equation} where $r$ is such that $A_M\cong k^r$, $\left(p'_{1},\ldots,p'_{r}\right)\in \mathbb{R}^r,\ \left(p_{1},\ldots,p_{r}\right)\in \mathbb{Z}^r_{\ge 0}$, 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 $\left\{ \pi_{N}\left(b_{1}\right)u,\ldots, \pi_{N}\left(b_{\ell}\right)u\right\}$ 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 here.

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)).