Let $G$ be a connected reductive group over a number field $F$ and fix a minimal parabolic subgroup $P\_0$ of $G$. Let $K$ be a fixed good maximal compact subgroup of $G(\mathbb{A}\_F)$ such that $G=PK$ for all standard parabolic subgroup $P=UM$. (here $U,M$ are the unipotent radical and Levi of $P$.


Then Harish-chandra defined a height function $H\_G:G(\mathbb{A}\_F) \to \mathcal{a}\_G$. (here $\mathcal{a}\_G$ is the real vector space spanned by coroots of $G$. For the precise definition of Height function, please see 16 page of the Arthur's book 'An introduction to the trace formula'.)


For a standard parabolic subgroup $P \subset G$, Harish-Chandra height function $H\_P : G(\mathbb{A\_F}) \to \mathbb{a}\_P$ is defined by
$$H\_P(g)=H\_{M\_P}(m)$$ where $g=umk$ where $u \in N\_P, m\in M\_P, k\in K$ and $G=N\_PM\_PK$ is the Iwasawa decomposition of $P$.


Let $G^1(\mathbb{A}\_F)=\{g \in G(\mathbb{A}\_F) \ | \ H\_G(g)=0\}$.


Then I have two questions;


1. Is this true that $U\_P(\mathbb{A}\_F) \subset G^1(\mathbb{A}\_F)$ for all standard parabolic subgroups $P \subset G$?
2. For an element $g \in G(\mathbb{A}\_F)^1$, we decompose $g=umk$ using Iwasawa decomposition with respect to $P$ as above. Then $u$ should be $1$ and $m \in M\_P(\mathbb{A}\_F)^1$?


I think that both are true but I can't prove it.


Any comments are greatly welcome!



Some question on Harish-Chandra height function
1. Is it true that $U\_P(\mathbb A) \subseteq G(\mathbb A\_F)^1$ for standard parabolic subgroups $P \subseteq G$?


Yes. Let $x \in U\_P(\mathbb A)$ be written as $x = umk$ where $u \in U\_P(\mathbb A), m \in M\_P(\mathbb A)$ and $k \in K$. Clearly $u = x, m = 1$ and $k = 1$. Then $H\_P(x) = H\_{M\_P}(m) = 0$ so $x \in G(\mathbb A)^1$. 


**Edit:** If $H\_P(g) = 0$ then $H\_G(g) = 0$. 


Observe that if $P\_1 \subseteq P\_2$ then $M\_{P\_1} \subseteq M\_{P\_2}$ and the restriction homomorphism $X(M\_{P\_2}) \to X(M\_{P\_1})$ is injective [Clay notes $\S 5$]. Thus $H\_{P\_2}(g) = 0$, which is equivalent to $\chi(g) = 0 \ \forall \chi \in X(M\_{P\_2})$ is true whenever $\chi(g) = 0 \ \forall \chi \in X(M\_{P\_1})$, which is equivalent to $H\_{P\_1}(g) = 0$. 


2. On the other hand, the second question is not true. For a quick counterexample, take $G = SL(2), P$ to be the minimal parabolic subgroup of upper triangular matrices and $g = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Note that $G(\mathbb A) = G(\mathbb A)^1$ and $g \in U\_P(\mathbb A)$ but $g \neq 1$.


