Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A\_f} \times \mathbb{A\_{\infty}}$.


The standard definition of an *automorphic representation* $(\pi,V)$ for $\textrm{GL}\_n(\mathbb{A})$ is its realization as (irreducible) subquotient of the space of automorphic forms $\mathcal{A}(\textrm{GL}\_n(k) \backslash \textrm{GL}\_n(\mathbb{A})), \omega)$, where $\omega$ is some central character. This is for example in Bump's Automorphic Forms and Representations on p.300.


This implies that $(\pi,V)$ is a representation for the finite part $\textrm{GL}\_n(\mathbb{A\_f})$ and a $(\mathfrak{g}\_{\infty}, K\_{\infty})$-module for $\textrm{GL}\_n(\mathbb{A\_{\infty}})$ with the property that each two actions commute. By abuse of notation I write $\pi$ for 'all' actions.
Then there is the notion of *admissibility* of such a representation, which is fairly standard too.


However, in the Corvallis proceedings, Flath gives the notion of an (admissible) automorphic representation in a purely algebraic way, i.e. he abstracts the properties from above, but it is not clear that his definition is 'embeddable' into the space of automorphic forms, i.e. that the 2 definitions are equivalent.


What is specially not clear to my; if there is a notion of **cuspidality** in the pure algebraic description of Flath? (I am working over $\mathbb{C}$, so cuspidality = supercuspidality).


It is I think due to Jacquet that cuspidality at local non-archimedean places has equivalent meanings; by a vanishing integral and that one is *not* (properly) parabolically induced. I could therefore imagine that one can define cuspidality for $(\pi,V)$ as a representation of $\textrm{GL}\_n(\mathbb{A\_f})$ in an analogue way. But I do not know what should be the notion of cuspidality for a $(\mathfrak{g}, K)$-module.


I am concerned with construction of $L$-functions attached to cuspidal representations. I imagine that the best way to introduce cuspidal representations is how Bump does it. However, I wondered if I can skip all the 'analytic conditions' on automorphic forms.



On the notion of cuspidality
To every local admissible representation $\pi\_v$ of $\mathrm{GL}\_n(k\_v)$, there is a local $L$-function $L(s,\pi\_v)$. For a global admissible representation $\pi=\otimes\_v \pi\_v$ of $\mathrm{GL}\_n(\mathbb{A}\_k)$, the corresponding global $L$-function is defined as $L(s,\pi)=\prod\_v L(s,\pi\_v)$. However, unless $\pi$ is automorphic, this global $L$-function will not have the usual analytic properties (cf. [converse theorems](https://en.wikipedia.org/wiki/Converse_theorem)). So it is very special for a global admissible representation to be automorphic. Automorphicity of $\pi=\otimes\_v \pi\_v$ connects the various local factors $\pi\_v$. In particular, if you change finitely many local factors $\pi\_v$ in a given cuspidal automorphic representation $\pi=\otimes\_v \pi\_v$, the resulting global admissible representation will no longer be cuspidal automorphic (cf. [multiplicity one theorems](https://en.wikipedia.org/wiki/Multiplicity-one_theorem)).


BTW I recommend Goldfeld-Hundley's two-volume textbook on automorphic forms (in the same series as Bump's textbook). Also, there is this [related earlier post of mine](https://mathoverflow.net/questions/349122/are-the-l-functions-of-a-normalized-newform-and-the-corresponding-cuspidal-repre/349124#349124) that you might find useful.


