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 $\left<\pi(a)v,\tilde{v}\right>$ equlas to $\left<\pi_N(a)u,\tilde{u}\right>_N$, where $\pi$ is an admissible representation, $\pi_N$ is its Jacquet module with respect to a radical unipotent $N$, $v\in V_\pi,\ \tilde{v}\in \widetilde{V}_\pi$, and $u$ ($\tilde{u}$ resp.) is the image of $v$ ($\tilde{v}$ resp.) in $V_{\pi,N}$ ($\widetilde{V}_{\pi,N}$ resp.).

In his notes "Remarks on Macdonald’s book on p-adic spherical functions", Casselman's 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 $\left<\pi_N(a)u,\tilde{u}\right>_N$ is a sum of $A$-finite functions. I can't figure out why this is true.

This question is related to this old question. I would very much appreciate a rather detailed explanation or a good reference for both the p-adic case and the Archimedean case.

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.