Let‘s take $G=GL_n$ over a number field $F$ for example.
It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\sigma_1,\cdots,\sigma_r)$, where $P$ is the standard parabolic subgroup of $G$ with respect to a partition $n=n_1+\cdots+n_r$, and $\sigma_i$ is a cuspidal automorphic representation of $GL_{n_i}$. (The reference is Automorphic forms and automorphic representations by Borel and Jacquet in "Corvallis" Part 1).
As an application, one can define principal $L$- functions for a not-necessarily-cuspidal automorphic representation. (The references are Principal $L$- functions for the linear group by Jacquet in "Corvallis" Part 2, and chap 13 of the book The Genesis of the Langlands Program) However, it seems that the progress is done all in the language of representations.
My question is that, could we express elements in $\pi$ as automorphic forms, i.e. functions on $G(\mathbb{A})$?
More explicitly, given $\sigma$ as above, should we consider the space spaned by the Eisenstein series $E(\phi,\lambda)$, where $\phi$ in $\sigma$ and $\lambda$ fixed? And what's the relation between that space and the Langlands quotient with respect to $\sigma$ ?
