# Questions on norms on Adelic group

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 $$\|g\| for all $$g \in N(\mathbb{A})$$?

2. There exists a constant $$c>0$$ such that $$\|g\| for all $$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.