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.



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


