Small questions in studying Arthur 's book 'Introduction to the Trace formula'

Question

I am reading Arthur's book "Introductionto the trace formula".

In reading the book, two small question has arised and so I would like to ask it.

1. Let $G$ be a connected reductive group over $\mathbb{Q}$ and $P=M_PN_P$ a standard parabolic subgroup. (here $M_P$ is Levi subgroup and $N_P$ is the unipotent subgroup of $P$)

I am wondering whether $G(\mathbb{Q})$ acts on $N_P(\mathbb{A})$. Because it looks that Arthur used such fact in some argument.

Is it really true?

2. Let $N$ be an arbitrary unipotent group defined over $\mathbb{Q}$ which has $G$-action over $\mathbb{Q}$. Denote $G$-action by $\rho \colon G \to Aut(N)$.

The for arbitrary $g\in G(\mathbb{A})$, let $n'=\rho(g^{-1})(n)$. Then I heard that two measures $dn$ and $dn'$ has the relation $dn'=\delta_{\rho}(g)dn$ for some character $\delta_{\rho} \colon G(\mathbb{A}) \to \mathbb{C}^{\times}$.

I am wondering the explicit formula for the character $\delta_{\rho}$.

For these two questions, any comments will be greatly appreciated.

Answer 1

I will try to answer both your questions in the context of Arthur's notes.

There does not exist a canonical action of a general connected reductive group $G$ on $N_P$ where $P = N_P M_P$ is a parabolic subgroup. One however looks at the exact sequence $$ 1 \to N_P \to P \to M_P \to 1 $$ which gives an action of $M_P$ on $N_P$ by conjugation.

Suppose $\text d n$ is a Haar measure on the unimodular group $N_P(\mathbb A)$. For every $m \in M_P(\mathbb A)$, we have another measure on $N_P(\mathbb A)$, namely $\text d(mnm^{-1})$. By the uniqueness of Haar measures, there is a positive number $\delta_m$ such that $\text d(mnm^{-1}) = \delta_m \text dn$. It is not difficult to show that the map $m \mapsto \delta_m$ is a homomorphism of groups. There must thus be a character $\delta : M_P(\mathbb A) \to \mathbb R^*_{> 0}$ which we call the modulus character.

This modulus character plays an important role in the theory of automorphic forms. For instance, Arthur uses this when making a change of variables involving the truncated kernel. (If I recall correctly, Cartier's notes in the Corvallis proceedings talk about the modulus character).

Given $P = M_P N_P$, there is an explicit formula for this modulus character $\delta = \delta_{M_P}$. It is a good exercise to show that this equals half of the sum of positive roots of $(P, A_P)$ where $A_P$ is the split part of the center of $M_P$. Try this out when $G = SL(2)$. Perhaps look at the Lie algebra picture.