For $\tau>0$ , let $d(\tau)\in H$ be uniquely defined by the conditions $(d(\tau))^{\alpha_{i}}=\tau$ , for all simple roots $\alpha_{i}\,$ . Then $d(\tau)x_{i}(a)d(\tau)^{-1}=x_{i}(\tau a)$ for any $i$ . For the type $A_{n-1}\,$ ,

(5.1)

$$d(\tau)=\tau^{-(n-1)/2}\left[\begin{array}[]{cccc}\tau^{n-1}&0&\cdots&0\\ 0&\tau^{n-2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{array}\right]\,,$$

and the automorphism $x\mapsto d(\tau)xd(\tau)^{-1}$ of the group $N$ multiplies each matrix entry $x_{ij}$ of $x$ by $\tau^{j-i}$ .