# Siegel domains and cuspidal functions

Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.

Do we have a analog of Siegel subset for the quotient $GL_{n}(\mathbb{A})/P_{n}(F)$?

Moreover, if we consider $f:GL_{n}(\mathbb{A})/P_{n}(F)\rightarrow\mathbb{C}$ a cuspidal function, is it a rapidly decreasing function?

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First, whether or not it is commonplace, there might be some danger in using an adorned "P" for mirabolic, since it suggests "parabolic". And, indeed, modulo the center, the mirabolic is parabolic, so maybe it doesn't matter too much.

Then, yes, for any parabolic there is a notion of Siegel set: for a choice of split torus and positive roots, the positive simple roots not among the root spaces appearing in the Levi component (or vice-versa, obviously there's a bijection between subsets of roots and their complements) gives inequalities specifying the corresponding Siegel set. That is, for positive roots $\alpha$ with rootspaces {\it not} in the Levi, the collection of inequalities $|\alpha(a)|\ge y_o>0$, together with the corresponding unipotent radical coordinates lying in a fixed compact, specify a Siegel set.

Depending on what is intended by "cuspidal" on that quotient, such functions may or may not be rapidly decreasing... and the notion of "rapidly decreasing" might need corresponding clarification.

E.g., it is true that a ${\mathfrak z}$-finite and $K$-finite function all whose constant terms (along a connected family of parabolics) vanish is of rapid decay _in_the_directions_ where the corresponding positive simple roots go to $+\infty$. That is, the functions need not be automorphic forms on $G_{\mathbb A}$. Indeed, that point is necessary in the general setting-up of Eisenstein series and automorphic spectral theory.

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as the mirabolic contains the unipotent radical $N$ of $GL_{n}$, the definition is the usual one for a cuspidal function. An other weaker question would be, do we have that $f\in L^{1}(GL_{n}(\mathbb{A})/P_{n}(F)$? –  prochet Mar 2 at 9:27
In some contexts, "cuspidal" might merely mean satisfying the Gelfand condition(s) $\int_{N_k\\N_{\mathbb A}} f(ng(\;dn=0$, without an eigenfunction condition, so I think one should be careful about what is presumed. And, no, $f$ will be well-behaved in a Siegel set, but not "below" a Siegel set, since that "below" part contains infinitely-many copies of a Siegel set (or fundamental domain). –  paul garrett Mar 2 at 14:15
and if we assume that it is an eigenfunction and also right invariant by $K$, it might still not be $L^{1}$? –  prochet Mar 2 at 17:12
If it is also moderate growth or $L^2$ in addition, then it will be of rapid decay in Siegel sets... but not in the part "below" the Siegel set. E.g., just on the upper half-plane, trying to integrate a Bessel function against $dx\,dy/y^2$ on the strip $0\le x\le 1$ is a problem. On the Siegel set it's fine, on the whole strip, not quite. –  paul garrett Mar 2 at 19:02