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I've recently needed to learn the basics of automorphic forms and automorphic representations. I've seen two apparently different definitions of automorphic forms, and I'm wondering which is more standard, or if they are in fact equivalent.

In Bump's book, he defines an automorphic form for $\mathrm{GL}_n/F$, $F$ a number field, as a measurable function $\varphi:\mathrm{GL}_n(F)\setminus \mathrm{GL}_n(\mathbf{A}_F)\rightarrow\mathbf{C}$ which, among lots of other conditions, is square-integrable modulo the center of $\mathrm{GL}_n(\mathbf{A}_F)$, in the sense that

$\int_{Z(\mathbf{A}_F)\mathrm{GL}_n(F)\setminus\mathrm{GL}_n(\mathbf{A}_F)} \vert \varphi(g)\vert^2dg<\infty$

(the absolute value of $\varphi$ is well-defined modulo the center because $\varphi$ is assumed to have a unitary central character).

On the other hand, in the books of Goldfeld-Hundley, no such condition of square-integrability is imposed in the definition of an automorphic form.

So, which of these (apparently) different conditions are more standard? Or do the other conditions in the definition of an automorphic form imply the square-integrability modulo the center?

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2 Answers 2

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There are various adjectives that may or may not apply to "automorphic forms". "Square-integrable modulo the center" may or may not apply... and definitely does not apply to any Eisenstein series, which are eigenfunctions, etc. ($K$-finite) Eisenstein series are "of moderate growth", which is sometimes a sufficient growth/integrability condition. Note that $L^2$ (without $\mathfrak z$-finiteness) does not imply moderate growth.

Also, with the requirements of $K$-finiteness and $\mathfrak z$-finiteness and $L^2$-ness, we cannot take $L^2$ limits, because the $\mathfrak z$-finiteness would be lost (even within a fixed $K$-type).

For convenience, in various contexts, authors declare that "automorphic..." entails various not-obvious further conditions, whose implications and inter-relationships are often non-trivial. But I think the sanest broader context is that "automorphic" (without further modifiers) only means "invariant by discrete subgroup" (and, even then, in a classical setting, there is the tradition to have the "cocycle condition" for "automorphic forms" on "domains"; transporting the automorphic forms to the relevant Lie or adele group converts this to left invariance, pleasantly-enough.)

Edit: as to useful implications: the "theory of the constant term" shows that a moderate-growth $\mathfrak z$-finite, $K$-finite automorophic form's asymptotic behavior is dominated by its constant term(s). Thus, e.g., moderate growth (and finiteness conditions) and eventually-vanishing constant term (a so-called "pseudo-cuspform") implies $L^2$.

$L^2$ and sufficiently many derivatives $L^2$ implies (by unsurprising Levi-Sobolev ideas) moderate-growth.

Without $K$-finiteness, Eisenstein series behave quite wildly.

Without "moderate growth", $\mathfrak z$-eigenfunctions can blow up exponentially (although this has been put to good use by Zwegers and others).

It is pretty easy to ask random hard-to-answer questions about inter-relationships among the various properties/adjectives, but most of these issues are not essential to doing things. It is true that many introductory sources do not directly address such comparisons, leaving the reader to wonder ... but mostly the point is just to understand the context of whatever source is at hand. Sometimes it's a bit too implicit, which does create confusion for a novice.

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Dear @paul, Thank you! This is very helpful. Regarding the first paragraph of your edit, is it necessarily the case that a cuspidal form (with the usual finiteness conditions, and moderate growth) is $L^2$? This might be a silly question, but since something called a pseudo-cuspidal form is $L^2$, I was wondering if something that is actually cuspidal is also $L^2$. –  Keenan Kidwell Dec 31 '12 at 20:21
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@KeenanKidwell: Yes, as you surmise, with the Gelfand condition that the integrals along unipotent radicals vanish, and the z-finiteness and K-finiteness, and moderate growth, "the theory of the constant term" proves rapid decay, which gives L^2-ness. –  paul garrett Dec 31 '12 at 21:03

When I was trying to sort this stuff out and learn all the adjectives, I found the Borel--Jacquet article in Corvalis incredibly useful.

Thus: an automorphic form (on $\Gamma\backslash G$ for a congruence subgroup of a real group $G$; the adelic case is a mild modification which Borel--Jacque also explains) is a smooth function of moderate growth which is is $Z(g)$-finite.

It can also be helpful to impose $K$-finiteness. This is okay because Harish-Chandra proved that automorphic forms (without $K$-finiteness) are an admissible $G$-rep'n (the different $K$-types appear with finite multiplicity), so that passing back and forth between closed $G$-subreps. and $(g,K)$-submodules of the $K$-finite vectors (by taking $K$-finite vectors or taking closures, resp.) is a bijective process.

One can then speak of $L^p$-automorphic forms, etc., just by imposing the relevant condition (on top of smoothness and $Z(g)$-finiteness). Cuspforms are automatically $L^2$.

Focusing on $L^2$ is useful for analytic considerations like developing the theory of Eisenstein series. It also shows that cuspforms are semisimple as a $G$-rep. (using the Hilbert space structure), while more general automorphic forms need not be. To see how the $L^2$ theory can be leveraged to apply to the general (i.e. not nec. $L^2$) theory, you can look at Langlands's article following Borel--Jacquet's.

Franke's artilce on weighted $L^2$ spaces (proving the general form of Eichler--Shimura, for arbitrary groups $G$) also illustrates how to relate $L^2$ contexts to the context of general automorphic forms.

As Paul Garret says, it is probably good not to be too doctrinaire about all this, but having at least one set of defintions straight can be very helpful, and I found Borel--Jacquet invaluable for doing this. (And I now feel slightly uneasy whenever someone defines automorphic forms to be $L^2$.)

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Dear @Matt, Thank you for this! At some point after I asked this question I got hold of the Corvallis volumes and indeed I found Borel-Jacquet's article to be very helpful. I haven't looked at Langlands's article but I definitely will. –  Keenan Kidwell Jan 9 at 1:57
    
@KeenanKidwell: Dear Keenan, Another thing I realized at some point, and found helpful, is that continuous spectrum in the $L^2$ theory is reflect in non-semisimplicity in the space of automorphic forms. (So Eisenstein series are not actually in $L^2$, but the continuous spectrum in $L^2$ has a direct integral decomposition labelled by Eisenstein series. Eisenstein series are automorphic forms, and the continuous variation in the parameter which occurs in the $L^2$-continuous spectrum gets reflected in the non-semisimple nature of the Eisenstein part of automorphic forms. Regards, –  Matt E Jan 9 at 2:48

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