Clarification about Tits' article in the Corvallis

Question

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ (Section 1.4, page 32); in particular the following sentences:

Let $\alpha(a,u)$ denote the affine function on $A$ whose vector part is $a$ and whose vanishing hyperplane is the fixed point set of $r(u)$ and let $\Phi'$ be the set of all affine functions whose vector part belongs to $\Phi$. For $\alpha \in \Phi'$, we set $X_\alpha = \{u \in U_a(K) : u=1 \text{ or } \alpha(a,u)\geq \alpha\}$.

Here, $a$ is a root and $u$ is a non-identity element of $U_a(K)$. Also, $r(u)$ is the value (additive valuation) of the unique element in the intersection $U_{-a}uU_{-a} \cap N_G(S(K))$. I am attaching the Google Books link (p.32) below.

http://books.google.com/books?id=ob-oRBICenMC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=Filtration%20of%20the%20groups&f=false

I understand he wants to index subgroups in a systematic way but the exposition seems a bit convoluted.

Answer 1

I think Tits could use better notation and define terminology more accurately. This Corvallis article, being the standard reference must have, and will continue to confuse many. I finally understand what's going on so let me elaborate.

Notations as in Tits' article. One has an exact sequence

$\require{AMScd} \begin{CD} 0 @>>> \Lambda = Z(K)/Z_c @>>> \tilde{W}=N(K)/Z_c @>>> {}^\nu\tilde{W} = N(K)/Z(K) @>>> 0 \\ \end{CD} $

which gives rise to an action of the normalizer $N(K)$ on the affine space $A$ (which is $V$ as a set) is given from the following commuting diagram (which can be found in Landvogt's book ``A compactification of the Bruhat-Tits building").

$\require{AMScd} \begin{CD} 0 @>>> \Lambda @>>> \tilde{W} @>>> {}^\nu\tilde{W} @>>> 0 \\ & @VV{\nu}V @VVV @VV\text{reflection}V & \\ 0 @>>> V @>>> \text{Aff}(A) @>>>GL(V) @>>> 1 \end{CD} $

Any element $n \in N(K)$ gives an affine action on the apartment $A$ as follows. Using the top exact sequence, one writes $n = a.z$ with $a \in {}^\nu\tilde{W}$ and $z \in Z(K)$. What Tits calls the "vector part" $r_a$ (notation abused as $a$ sometimes) is the reflection on $V$ induced by the the Weyl group element $a$. The translation is then $\nu(z)$. The map $\nu$, initially defined for $S(K)$ extends uniquely to $Z(K)$ because their character groups are commensurable. (I recommend the example of $G=SU(3)$ to see that $S(K)$ and $Z(K)$ need not be equal.) The affine action of $N(K)$ on $A$ can be computed knowing the reflection (vector part) and translation $\nu(z)$. (The example of $GL_{n,D}$ computed on the last line on p.39 is instructive.)

Choosing a non-identity element $u\in U_a(K)$ gives an element $m(u) \in N(K)$ acting on $A$ as above. How do we define the affine function (functional, rather) $$ \alpha(a,u) : A \to \mathbb R?$$

Well, $\alpha(a,u) = \langle a, \circ \rangle + c$ for some real number $c$. If only we could find $c$!

The map $r(u) = \nu(m(u)) : A \to A$ has a fixed point set which is a hyperplane, say $H_u$. Solving the equation

$$ \langle a, x \rangle + c = 0 $$ for some (every) $x \in H_u$ gives the value of $c$. I highly recommend the example 1.14 of $GL_{n,D}$.