Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, the condition of usual automorphic forms "$K$-finite" is replaced by "$K_f$-finite and of uniform moderate growth". The space of smooth automorphic forms is equipped with limit Fréchet topology In particular, the space of usual automorphic forms is dense in the space of smooth automorphic forms in this topology.
And in Prop.5.4, some Eisenstein series is considered. If $g$ is in $ GL_n(\mathbb{A})$, $s$ is a complex number with sufficiently large real part, $\Phi$ is a Schwartz-Bruhat function on $\mathbb{A}^n$, and $\eta$ is a unitary idele class character, then the Eisenstein series
$$E(g,s;\Phi,\eta)=|\det g|^s\int_{k^\times\setminus\mathbb{A}^\times}\eta(a)|a|^{ns}\sum_{\xi\in k^n\setminus\{0\}}\Phi(a\xi g)\mathrm{d}^\times a $$
is convergent and it is a smooth automorphic form.
I also want to recall a theorem in the book of Moeglin and Waldspurger (MW I.3.2)
If $P=NM$ is a parabolic subgroup of a reductive group $G$, then $\mathcal{A}_P$, the space of automorphic forms of automorphic forms of level $P$, has a decomposition: every $\phi$ in $\mathcal{A}_P$ is a finite sum of functions of the form: $$Q(H_P(g))\psi(g)$$ where $Q$ is a polynomial function on $\mathfrak{a}_P$, $\psi$ is in $\mathcal{A}_P$, satisfying that for all $a$ in the center of $M(\mathbb{A})$, $g$ in $G(\mathbb{A})$, $$\psi(ag)=e^{\langle \lambda+\rho_P,H_P(a)\rangle}\psi(g)$$ for some $\lambda\in\mathfrak{a}_P^* $.
We can remark that in the view of Langlands decomposition $G(\mathbb{A})=N(\mathbb{A})A_PM(\mathbb{A})^1K$, $\phi$ is $N(\mathbb{A})$- left invariant of course and then the action of $A_P$ can be "separated" from $M(\mathbb{A})^1K$.
I am considering this question: will there be an analogous structural result for smooth ($P$-level) automorphic forms? Given a smooth $G$-level automorphic forms, could we consider its constant terms along different parabolic subgroups and describe how the center $A_P$ acts according to the result?
At least, consider the Eisenstein series mentioned above. I tried to compute the constant term of $E(g,s;\Phi,\eta)$ along an arbitrary standard parabolic subgroup of $GL_n$ but did not get a good result. I guess the problem is that the Schwartz-Bruhat function lacks of some invariance, and some analysis technique is needed.
