# Definition of Automorphic Representation

I would like to think of an automorphic representation as a representation weakly contained in $L^2(G_F\backslash G_A)$ where $G_A$ is the reductive group of rational points in the adeles over $F$, $F$ a number field. I know there is other definitions, I just want to know if this is correct. A representation is automorphic iff is weakly contained in $L^2(G_F\backslash G_A)$. There are references that suggest this but none of them officially makes the claim. A reference that makes this claim explicit will be much appreciated.

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What is the definition of weakly contained in $L^2(G_F\backslash G_A)$? –  Giovanni De Gaetano Jun 20 '13 at 17:40
I do not have 50 on reputation so I can not leave comments on my own question that is why I write it here. The article that Marc Palm is mentioning has been mention to me by other people. However, no body has ever been able to point out the precise definition or theorem or anything concrete. I do not think Langlands article has what I am asking, and if it does, then I would appreciate if you tell me where, page number or theorem. –  Carlos De la Mora Aug 12 '13 at 21:52
Dear Carlos, This is incorrect. I will try to post an answer later explaining why. Regards, –  Matt E Aug 12 '13 at 22:31
May be you can clarify some things for me. It seems to me that you are saying that the reason there are unitary automorphic representations not appearing in any sense in $L^2$, is because there are unitary automorphic representations coming as quotients of non unitary representations. I do not think this is enough to say that they are not contained in $L^2$. I also do not understand why if there are unitary representations not appearing in any sense in $L^2$ why Borel says in his crovallis article that an automorphic representation is the one that appears as a subquotient of the regular repres –  Carlos De la Mora Aug 20 '13 at 15:51

I think the following Corvallis article might help: http://sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/notion-ps.pdf

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In brief: no, there are more automorphic repns than those appearing (in any sense) in $L^2$.

"Weakly contained" is potentially ambiguous, but I presume it is aiming at the idea that in a Hilbert direct-integral decomposition your given representation appears. (Edited:) As the questioner makes precise in comment below, the notion of "appears in" has a precise sense for unitary repns, yes, and if the question is restricted to refer to unitaries, the answer is still "no, there are many more".

(Edit (in light of questioner's comment just below): Yes, if we are sure that everything we care about happens inside a unitary representation, then the von-Neumann or C-star algebra definitions give a good notion of "occurs in", much as we still have a good notion of "continuous spectrum" beyond eigenvalues. (But for automorphic representations things are not this "simple".) In any case, as below, there are non-unitary automorphic representations, there are unitary automorphic representations not occurring in automorphic $L^2$, and some unitary automorphic representations are (in Langlands' description) presented naturally as quotients of non-unitary representations. Someone else may know, but I do not know how to make the nice unitary-repn discussion of decompositions apply in such situations.)

But this disambiguation is not the issue. Rather, no, very many automorphic forms generating irreducible repns of the adele group do not appear in any fashion in a decomposition of automorphic $L^2$. First, Eisenstein series with continuous parameter out of the unitary range are legitimate moderate-growth automorphic forms, and (generically) generate irreducibles (if any cuspidal data does), but do not appear in the decomposition of automorphic $L^2$.

Further, even some Eisenstein series which generate irreducible unitaries do not appear in the decomposition of automorphic $L^2$. For example, in $Sp(n)$ bigger than $SL_2$, the Siegel-type Eisenstein series (with "s") appear in the spectral decomposition for no values of the parameter $s$, even though for $s$ on a certain line they generate a (typically irreducible) unitary. Already for $SL_2$, Eisenstein series $E_s$ with real $s$ in the range ${1\over 2}<s<1$ generate unitaries, but do generically do not appear, since the continuous spectrum decomposition only uses these Eisenstein series with $\Re(s)=1/2$ (and their residue at $1$, namely, constants).

That is, it's not just that some repns appearing in the decomposition of automorphic $L^2$ are moderate growth, rather than being in $L^2$, so are in the "continuous spectrum". Rather, there are automorphic repns that are not unitary, whether or not some automorphic forms/functions in them are or aren't in automorphic $L^2$.

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I am sorry Dr. Garret, but there is no ambiguity of when a representation is weakly contained in a given representation. Let me write the definition, if $\rho$ is a unitary representation of $G(\mathbb{A})$ we can associate a unique unitary representation $\tilde{\rho}$ of the enveloping $C^*$-algebra of $G(\mathbb{A})$. We say that a unitary representation $\pi_1$ is weakly contained in $\pi_2$ if $\ker \tilde{\pi_1}\supset \ker \tilde{\pi_2}$. –  Carlos De la Mora Aug 16 '13 at 2:36