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Let $G$ be a simple algebraic group defined over $\mathbb Q$. In their Corvallis article (automorphic forms and automorphic representations), Borel and Jacquet define an automorphic form to be a smooth function on $G(\mathbb A)$ which is left-$G(\mathbb Q)$-invariant, satisfies some "finiteness" conditions, and also the following moderate growth condition:

  • For every $y\in G(\mathbb A)$, the map $x\mapsto f(x\cdot y)$ on $G(\mathbb R)$ is slowly increasing.

I have a couple of questions about this condition:

  1. Assume that $f$ appears in the discrete spectrum (but is not necessarily a cusp form). Is it true that the map $x\mapsto f(x\cdot y)$ is bounded on unipotent subgroups of $G(\mathbb R)$?

  2. What can be said about the "growth" of the map $x\mapsto f(x\cdot y)$ for a fixed $y\in G(\mathbb A)$ when $x$ varies in $G(\mathbb Q_p)$? For example, will this map be bounded on unipotent subgroups of $G(\mathbb Q_p)$?

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If the unipotent subgroup is defined over $\mathbb Q$, then its quotient by the congruence subgroup will be compact, hence the function in question be periodic, hence bounded.

To the second question, lets say the function is fixed by a compact open subgroup of the finite adelic-valued points, which is to say that the real component is fixed by an arithmetic group. Further, this infinite component is an eigenfunction of the algebra of invariant differential operators. Say it is $K$-finite, maybe of a fixed $K$-type, then it satisfies a differential equation of the form $\Delta f=\lambda f$, where $\Delta$ is induced by the Casimir, normalize it to have positive definite principle symbol, then $\Delta+C$ is invertible for some $C>0$ and the operator $(\Delta+C)^{-N}$ has kernel of any order of differentiablility as $N$ increases. So you have $$ f=(\lambda+C)^N(\Delta+C)^{-N}f $$ from which moderate growth of $f$ is concluded as the resolvent kernel shows moderate growth.

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You are right about the rational unipotent subgroups, but what about the other ones? I don't understand your answer to my second question. I am asking about growth of a function when the parameter varies in $G(\mathbb Q_p)$, whereare you are talking about the Laplacian, which is probably related to the real place. –  Valerie Mar 6 '13 at 5:47
    
Yes, but the set $G({\mathbb Q})\backslash G({\mathbb A})/K$ for an open compact subgroup $K$ of the finite adele-group is up to finite index the same as $\Gamma\backslash G({\mathbb R})$, where $\Gamma$ is the arithmetic subgroup of $G({\mathbb R})$ which is given as the intersection of $G({\mathbb Q})$ and $K$. So all growth assertions on automorphic forms can be considered at the infinite place alone. –  anton Mar 6 '13 at 12:52
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