In the classical setting, we can define automorphic forms on $\text{SL}_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups.
I have encountered the lifting of automorphic forms to the adeles (so to automorphic representations) for $\Gamma$ being a congruence subgroup or, more generally, the unit group of an order in a quaternion algebra. I am wondering what are the precise conditions that make this lift possible in general.
Can we "lift" any classical arithmetic subgroup to a compact open subgroup over the adeles? I understand how this might happen when the subgroup is associated to an order in an algebra (the matrix algebra or a division algebra). But there are other constructions of lattices $\Gamma$, especially in higher rank. What I am maybe asking is whether simply arithmeticity of the subgroup is enough to make full use of adelic lifts (if they even exist).
Best Answer
For a subgroup to have a meaningful lift to the adeles, it is necessary and sufficient for the subgroup to be a congruence subgroup in the sense that for some $N$, the subgroup contains all elements congruent to the identity mod $N$.
Given an element of $SL_n(\mathbb A_{\mathbb Q})$ (or the same thing for the norm 1 unit group of a central simple algebra)that is integral at every place, it makes sense to consider its congruence class mod $N$ (by ignoring all places not dividing $N$ and modding out the local ring by $n$ at each place dividing $n$) and by the Chinese remainder theorem this defines a congruence class in $SL_n(\mathbb Z/N)$ or the appropriate analogue. Thus we can define the subgroup of adelic elements that are congruent mod $N$ to an element of the group, and this gives an adelic interpretation for automorphic forms.
The property of every finite index subgroup being a congruence group is called the congruence subgroup property, and it is known for non-anisotropic arithmetic groups of real rank $\geq 2$, i.e. every arithmetic lattice in $SL_n(\mathbb R)$ except for unit groups of division algebras or $n=2$. For $n=2$ it is known to fail badly and for unit groups of division algebras it is (according to Wikipedia) open.