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.

areadmissible. So proving theorems about factoring, etc., of irreducible admissible repns of adele groups is relevant to the (irreducible) repns generated by (suitable) cuspforms. But, for general and uninteresting reasons,mostadmissible adele-group repns arenotautomorphic. Unsurprising, much like saying that not every real number is algebraic... $\endgroup$anticipated(at least by optimists) because of the theorem (sketched in the Gelfand-PiatetskiShapiro book, treated more carefully by Godement in the Boulder conference) that the induced action of test functions on $L^2$ cuspforms is by compact operators,andthat there's an orthonormal basis $\{f_i\}$ such that there are test functions $\varphi_i$ such that $\varphi_i\cdot f_i=f_i$. This gives the admissibility. $\endgroup$6more comments