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, Flath in Corvallis 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.