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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.

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  • $\begingroup$ What is the question? The point of Tits's definition is that he has first defined (or at least referred to the existence of) an extension of the natural translation action of "centraliser of a maximal split torus" on the apartment to an affine action of "normaliser of a maximal split torus" on the apartment, which behaves as one would expect upon taking gradients. $\endgroup$
    – LSpice
    Commented Jan 13, 2015 at 20:47
  • $\begingroup$ If your question is, say, "why not just define $X_{a + 0}$ to be $U_a \cap G(\mathcal O)$, and then the other $X_\alpha$ accordingly?", note that choosing an integral model for $G$ (if it even exists!) amounts to choosing a hyperspecial vertex as a base point, at which point all these definitions simplify considerably. Essentially all the convolution is an attempt to figure out where to 'anchor' the filtration; once you've got one term of it, finding the others is easy. $\endgroup$
    – LSpice
    Commented Jan 13, 2015 at 20:50

1 Answer 1

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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 $GL_{n,D}$.

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