Automorphic form - Knowlpedia

Let $F$ be a global field and $G$ a connected reductive $F$ -group. An automorphic form on $G$ is a complex-valued function on

G(F)\backslash G(\mathbb A_F)

where $\mathbb A_F$ is the adele ring, and that is smooth, of moderate growth, right finite under a maximal compact group at the archimedean places, right fixed by some compact open subgroup at the finite places, and finite under the center of the archimedean universal enveloping algebra. A central character may be prescribed.

Why the finiteness conditions appear

At a finite place, smoothness means local constancy, so a single automorphic form is fixed by a sufficiently small compact open subgroup. At an archimedean place, compact finiteness and infinitesimal-character finiteness place the function in the algebraic representation-theoretic category of $(\mathfrak g,K)$ -modules. Moderate growth controls its behavior on the noncompact quotient.

Some authors build uniform moderate growth into the definition, while others obtain it from the remaining conditions in their chosen setting. The precise space should therefore be named when analytic estimates matter.

Central and split-center conventions

If $\omega$ is a character of $Z_G(F)\backslash Z_G(\mathbb A_F)$ , one usually imposes

f(zg)=\omega(z)f(g).

Square-integrability, with the quotient measure compatibly normalized (often from Tamagawa measure), is then measured either modulo the center or on the kernel $G(\mathbb A_F)^1$ of all adelic absolute-value characters. These equivalent-looking presentations encode a real convention and should not be silently interchanged.

Constant terms

For a parabolic subgroup $P=MN$ , with Levi subgroup $M$ and unipotent radical $N$ , the constant term of $f$ along $P$ is

f_P(g)= \int_{N(F)\backslash N(\mathbb A_F)} f(ng)\,dn.

Its vanishing for every proper parabolic defines a cuspidal automorphic form. Nonzero constant terms lead to Eisenstein series and the continuous or residual spectrum.

From forms to representations

Right translation by $G(\mathbb A_F)$ acts on automorphic forms. Irreducible subquotients of this action are automorphic representations. Thus a form is a vector, not an irreducible representation; Hecke eigenforms often generate automorphic representations.

References

A. Borel and H. Jacquet, “Automorphic forms and automorphic representations,” in Automorphic Forms, Representations and $L$ -Functions, Proc. Sympos. Pure Math. 33, part 1, 1979, pp. 189–207.
Jayce Getz and Heekyoung Hahn, An Introduction to Automorphic Representations, Springer, 2024. DOI.