About height function in the Arthur trace formula

Question

I am reading a book of Arthur's book "Introduction to the trace formula".

In page 24, Arthur defined height function $H_P:G(\mathbb{A}) \to \mathcal{a}_P$ by setting $H_p(nmk)=H_{M_p}(m)$ where $n\in N(\mathbb{A}), m\in M(\mathbb{A}), k\in K$. (here, $P$ is a standard subgroup of $G$ and $NMK$ is the Iwasawa decomposition of $G$, $a_P$ is Lie algebra of $P$.)

If $P_1 \subset P$ is a standard subgroup of $G$, I am wondering whether $p \circ H_{P_1}(x)=H_P(x)$ where $p:a_{P_1} \to a_P$ is the projection map.

It seems true because Arthur used it implicitly in many parts. But I cannot prove it.

Does it really true? If so, how to prove it?

I am sorry for not explaining all the notation on my question. It is too long!

Any comments will be great helpful!

Answer 1

This is natural because $H_{G}$ for reductive $G$ is defined by $\langle H_{G}(x),\chi\rangle=|\log(x^{\chi})|$ for $x\in G(\mathbb{A})$ and $\chi\in X(G)_{\mathbb{Q}}$, and because the projection $p:\mathfrak{a}_{P_{1}}\rightarrow\mathfrak{a}_{P}$ is the dual of the injection $X(M_{P})_{\mathbb{Q}}\rightarrow X(M_{P_{1}})_{\mathbb{Q}}$ which is just the restriction. Namely for $\chi\in X(M_{P})_{\mathbb{Q}}$, $\langle p\circ H_{P_{1}}(x),\chi\rangle=\langle H_{P_{1}}(x),\chi\rangle$ where $\chi$ in the RHS is seen as a character of $M_{P_{1}}$ so the RHS becomes $|\log(x^{\chi})|$.