Proposition 2

Let $m \rightarrow \|m\|$ be a norm on the $\mathbf{R}$-vector space $\mathbf{R}\otimes X(T)$ and $\Gamma$ a Casimir element of G. Let $\varphi$ be a function on $\widehat{G}$ with positive real values.

a) The following conditions are equivalent:

(i) There exists an integer $n >0$ such that $\varphi (u)\preccurlyeq (\|\lambda (u)\|+ 1)^n$ (resp. for every integer $n >0$, we have $\varphi (u)\preccurlyeq (\|\lambda (u)\|+ 1)^{-n}$).

(ii) There exists an integer $n >0$ such that $\varphi (u)\preccurlyeq ( \widetilde{\Gamma}(u) + 1)^n$ (resp. for every integer $n >0$, we have $\varphi (u)\preccurlyeq ( \widetilde{\Gamma}(u) + 1)^{-n}$).

b) If G is semi-simple, conditions (i) and (ii) are also equivalent to:

(iii) There exists an integer $n >0$ such that $\varphi (u)\preccurlyeq d(u)^n$ (resp. for every integer $n >0$, we have $\varphi (u)\preccurlyeq d(u)^{-n}$).

Note first of all that condition (i) is clearly independent of the choice of norm. Thus we can use the norm defined by the quadratic form $Q_{\Gamma}$ associated to $\Gamma ($§7, no. 6, Prop. 4). Then

so $\widetilde{\Gamma}(u) + 1\preccurlyeq (\|\lambda (u)\|+ 1)^2\preccurlyeq \widetilde{\Gamma}(u) + 1$, hence a).

Further, if G is semi-simple,

$\|\lambda (u) +\rho \|\preccurlyeq d(u)\preccurlyeq \|\lambda (u) +\rho \|^N$, where $N = 1/$2(dim $G-$ dim T)

(§7, no. 5, Cor. 1 of Th. 3), so $\|\lambda (u)\|+ 1\preccurlyeq d(u)\preccurlyeq (\|\lambda (u)\|+ 1)^N$, hence b).

It follows from Prop. 2 that condition (i) is independent of the choice of maximal torus, chamber, and norm, and that condition (ii) is independent of the choice of Casimir element. A function $\varphi$ satisfying conditions (i) and (ii) is said to be moderately increasing (resp. rapidly decreasing). The product of two moderately increasing functions is moderately increasing; the product of a moderately increasing function and a rapidly decreasing function is rapidly decreasing. If $\varphi$ is rapidly decreasing, the family $(\varphi (u))_{u\in\widehat{G}}$ is summable.