Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$.

Then for an automorphic form, $\varphi : \mathrm{G}(F)\backslash\mathrm{G}(\mathbb{A}) \to \mathbb{C}$ is said to be cuspidal if for each parabolic subgroup $\mathrm{P}=\mathrm{MU}$,

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_{\mathrm{U}(F)\backslash\mathrm{U}(\mathbb{A})} \varphi(ug)du = 0$,

where $\mathrm{U}$ is the unipotent radical of $\mathrm{P}$.

In the case that $\mathrm{G}=\mathrm{GL_2}$, then we can unravel the adelic language to the more classical setting, where this becomes

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_0^1f(x+iy)dx = 0$, (ie: the constant term vanishes)

where $f$ is a Maass form which for simplicity I will assume has level 1. In this setting, we view $f$ as a function on the upper half plane,

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad H \cong Z\backslash\mathrm{GL_2}/\mathrm{SO}(2)$,

and $H$ comes equipped with a "canonical" hyperbolic metric $ds^2 = \frac{dxdy}{y^2}$. Passing to the quotient, $\mathrm{SL_2}(\mathbb{Z})\backslash H$, we get a punctured sphere with one geometric cusp at infinity, and the requirement that $f$ be a cusp form reduces to $f$ "vanishing at the cusp".

For more general $\mathrm{G}$ (eg: $\mathrm{G}=\mathrm{GL_3}$), I am curious about analogous geometric understanding of unipotent radicals of various parabolic subgroups corresponding to the "cusp at infinity". I don't know if this is better to think about this on the adelic quotient, $\mathrm{G}(F)\backslash\mathrm{G}(\mathbb{A})$, or perhaps on a locally symmetric space like $ Z\backslash\mathrm{G}/\mathrm{K}$.

Specifically,

Is there a geometric way of understanding how the unipotent radicals correspond to "cusps" beyond simply as failure of a fundamental domain to be compact, perhaps with respect to some nice metric, for groups beyond $\mathrm{GL_2}$?

I am interested in answers in either the classical or adelic language.

Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$.

Then for an automorphic form, $\varphi : \mathrm{G}(F)\backslash\mathrm{G}(\mathbb{A}) \to \mathbb{C}$ is said to be cuspidal if for each parabolic subgroup $\mathrm{P}=\mathrm{MU}$,

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_{\mathrm{U}(F)\backslash\mathrm{U}(\mathbb{A})} \varphi(ug)du = 0$,

where $\mathrm{U}$ is the unipotent radical of $\mathrm{P}$.

In the case that $\mathrm{G}=\mathrm{GL_2}$, then we can unravel the adelic language to the more classical setting, where this becomes

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_0^1f(x+iy)dx = 0$, (ie: the constant term vanishes)

where $f$ is a Maass form which for simplicity I will assume has level 1. In this setting, we view $f$ as a function on the upper half plane,

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad H \cong Z\backslash\mathrm{GL_2}(\mathbb{R})/\mathrm{SO}(2)$,

and $H$ comes equipped with a "canonical" hyperbolic metric $ds^2 = \frac{dxdy}{y^2}$. Passing to the quotient, $\mathrm{SL_2}(\mathbb{Z})\backslash H$, we get a punctured sphere with one geometric cusp at infinity, and the requirement that $f$ be a cusp form reduces to $f$ "vanishing at the cusp".

For more general $\mathrm{G}$ (eg: $\mathrm{G}=\mathrm{GL_3}$), I am curious about analogous geometric understanding of unipotent radicals of various parabolic subgroups corresponding to the "cusp at infinity". I don't know if this is better to think about this on the adelic quotient, $\mathrm{G}(F)\backslash\mathrm{G}(\mathbb{A})$, or perhaps on a locally symmetric space like $ Z\backslash\mathrm{G}/\mathrm{K}$.

Specifically,

Is there a geometric way of understanding how the unipotent radicals correspond to "cusps" beyond simply as failure of a fundamental domain to be compact, perhaps with respect to some nice metric, for groups beyond $\mathrm{GL_2}$?

I am interested in answers in either the classical or adelic language.