# On the notion of cuspidality

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.

• I think Flath speaks of general representations, not automorphic representations.
– Echo
Jun 5 at 9:35
• Just to be clear on one point: it is a non-trivial theorem (perhaps declared a definition in the mid-to-late 1970's) that the (irreducible) adele-group repns generated by (suitably-strong-sense) cuspforms are admissible. 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, most admissible adele-group repns are not automorphic. Unsurprising, much like saying that not every real number is algebraic... Jun 5 at 18:17
• @MatyMangoo In general, an irreducible unitary representation of a linear connected reductive Lie group is admissible. See Theorem 8.1 in Knapp: Representation theory in semisimple Lie groups. This result is due to Harish-Chandra (1951) who even treated Banach space representations. Jun 6 at 8:38
• Two threads: unitary repns of reductive groups over local fields, where Harish-Chandra proved admissibility for real, Bernstein for p-adic. These results were 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, and that 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. Jun 6 at 17:00
• Also, by "the theory of the constant term", a not-necessarily-$L^2$, but moderate growth (and sufficient finiteness under center of enveloping algebra, and Hecke algebras at finite places) is in fact of rapid decay, so is in $L^2$. In particular, the repns generated by such cfms are unitary. Jun 6 at 18:51

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). 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).
• Thanks @GH. I see. And what is then the difference between cuspidal automorphic and just automorphic in relation with their $L$-functions? I thought one could attach $L$-fcts to cuspidal reps (as to cuspidal modular forms), as they are generic (i.e. have Whittaker models). A general automorphic form does not neet to be generic. Jun 5 at 12:33
• Oh yes, those two books are quite nice, although they only treat the case $k = \mathbb{Q}$. Jun 5 at 12:38
• @MatyMangoo Also, from a representation theoretic point of view, there is not much difference between $\mathbb{Q}$ and a number field. Jun 5 at 13:22
• What do you exactly mean by "if you change finitely many local factors $\pi_{\nu}$ in a given automorphic representation $\pi=\bigotimes'_{\nu} \pi_{\nu}$ , the resulting global admissible representation will no longer be automorphic"? Due to strong multiplicity I lose 'cuspidality', but not necessarily automorphy (if that word exists) Jun 5 at 14:27