TITLE: a question on TITS' note "Reductive groups over local fields"
QUESTION [6 upvotes]: 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.

REPLY [10 votes]: Of course, grp is right that to justify the whole discussion in Tits about the construction of the apartment you need the full strength of Bruhat-Tits. But for the purpose of Tits' article (who assumes you accept facts 1.4.1 and 1.4.2) there is a rather simple construction to get $\nu$ which doesn't require the Bruhat-Tits books. I think it's what Tits had in mind. I learnt this construction from J.K.Yu, who says "This is one of the questions I got asked most often regarding Tits’ article.". You can find it in his very useful notes on buildings, or (as he notes) in Landvogt's book. Unfortunately JK's notes are not available on the web; they are in this book:
http://books.google.ca/books/about/Ottawa_Lectures_on_Admissible_Representa.html?id=7oBZIxZfPQAC&redir_esc=y
On second thought, I assume it's ok with copyright laws to take just the relevant two pages and put them here: http://www.math.toronto.edu/~herzig/apartment_jkyu04.pdf