This might be stupid question to some experts who works in the realm of automorphic form.


Let $K$ be a number field and $\mathbb{A}$ is a adele ring of $K$. Let $G$ be a connected reductive group defined over $K$. Moeglin and Waldspurger defined the norm function on $G(\mathbb{A})$ as follows:


For $GL\_{2n}(\mathbb{A})$, its norm function is defined by
$||g||=\prod\_v \text{sup} \{|g\_{rs}|\_v : r,x=1,...,2n\}$, where we write $g=(g\_{rs})$.
(See Section 2.2 of the book 'Spectral decomposition and Eisenstein series' by Moeglin and Waldspurger.)


Fix an embedding $i' :G \to GL\_n$ and let $i':G \to SL\_{2n}$ be the embedding defined by


$i(g)=\begin{pmatrix} i(g) & 0 \\ 0 & i(g)^{-1} \end{pmatrix}$.


Define the norm function $\| \cdot \|$ on $G(\mathbb{A})$ as $||g||=|i(g)|$.


Then I am wondering whether the followings are true.


1. Let $N$ be a unipotent subgroup of $G$. There exists a constant $c>0$ such that $\|g\|<c$ for all $g \in N(\mathbb{A})$?
2. There exists a constant $c>0$ such that $\|g\|<c$ for all $g \in G(F)$.


I think that they are both true but I can't find any reference which proves these.


Does this hold for some simple reason? Or if there is a reference dealing with this, I would appreciate if you let me know it.


Thank you in advance.



Questions on norms on Adelic group
For the first question,(and hence for the second,) it is false.


For example, you can take elements that are upper triangular unipotent and have trivial non-archimedean components as its counterexamples.


However, it is true that the norm is lower bounded. It is easily proven if you use the fact that the norm is inverse-invariant, and the triangle inequality.


