This note appears in "Proceedings of Symposia in pure mathematics" vol.33 1979 part 1 pp. 26-69.
The question will be about materials on page 31-32. 

Let $G$ be a reductive algebraic group (not necessarily connected) defined over a local field $K$. 
We fix a maximal $K$-split torus $S$ of $G$ and take N(resp. Z) to be normalizer 
(resp. the centralizer) of $S$ in $G$. Let $X\_\*=Hom\_K (Mult, S)$ (Resp. $X^\*=Hom\_K (S, Mult)$) be the group of cocharacters (resp. characters) of $S$. Let $V=X\_\*\otimes\_{\mathbb Z}\mathbb R$.
We fix a discrete valuation $\omega: K\to (-\infty, \infty]$. 
Let $\nu: Z(K)\to V$ be the homomorphism defined by 
$$
\chi (\nu (z))=-\omega (\chi (z)) \quad \mbox{for}\ z\in Z(K) \mbox{ and } \chi \in X^\*(\mathbb Z).
$$
Let $Z\_c$ be the kernel of $\nu$. Then we have a short exact sequence of gorups
$$
0\to Z(K)/Z\_c\to N(K)/Z\_c\to N(K)/Z(K)\to 0
$$
where $N(K)/Z(K)$ is a finite group.

Then it is claimed that the map $\nu$ induces a group homomorphism $\phi$ from $N$
to the group of affine transformations of $V$ such that for $z\in Z(K)$ and $x\in V$ one has
$\phi(z)x=x+\nu (z) $. 

I do not understand in which way this function $\phi$ is defined.