# Fourier expansion of $\mathrm{GL}(n)$ automorphic forms for congruence subgroups

$$\DeclareMathOperator\GL{GL}$$There are several books describing the classical Fourier/Whittaker expansion of automorphic forms on $$\GL_3(\mathbb{R})$$, e.g. Bump - Automorphic forms on $$\GL_3(\mathbb{R})$$, and Goldfeld - Automorphic forms and $$L$$-functions for the group $$\GL_n(\mathbb{R})$$. The expansions presented there are valid for forms invariant under $$\GL_n(\mathbb{Z})$$. Bump mentions on page 66 that there are analogous expansions for congruence subgroups as well, albeit more complicated ("[...] one must also sum over the cusps of the congruence subgroup of $$\operatorname{SL}_2(\mathbb{Z})$$ [...]" ). Probably because of that, he does not consider the problem in the book.

Has the Fourier expansion been worked out for congruence subgroups somewhere since then? The usual proof goes through for the group $$\Gamma_0(N)$$ without any modification, but there are other subgroups one might consider, such as the analogue of $$\Gamma(N)$$ from $$\GL_2$$ (I am assuming that Bump was talking about this group).

• There is a nice trick that often makes it unnecessary to consider congruence subgroups. It goes like this: The finite group $H=\Gamma(1)/\Gamma(N)$ acts on the space of $\Gamma(N)$ automorphic forms. You decompose the space of automorphic forms into the finitely many irreducible components you get and then you consider one component at a time. Many $\Gamma(1)$ computations still go through, only, that the functions are vector-valued and spit out a representation of $H$ instead of being invariant under $\Gamma(1)$.
– user130903
Commented Jun 1, 2021 at 9:26
• One way to think about the Fourier--Whittaker expansion is by adelizing the automorphic form so that there is only one cusp'' to think about. Given the congruence subgroup one may adelize the space via strong approximation and the automorphic form to consider forms on $\mathrm{GL}_n(\mathbb{A})$ invariant under some hyperspecial open compact subgroup $K_f(N)$. Then understanding the local Whittaker functions is enough to obtain the full Fourier expansion adelically which one may re-understand in classical language. Commented Jun 1, 2021 at 10:29