# a question on TITS' note “Reductive groups over local fields”

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.

-
I don't see a question anywhere. –  Igor Rivin Sep 3 '12 at 17:09
@Igor Rivin: the question is how to define $\phi$. Although it might make Alex Trebek unhappy, on MO questions don't need to be stated in the form of a question. :) –  grp Sep 3 '12 at 21:48
@grp: And why do you believe your statement is correct? If you read "how to ask", you will note the request for precision. If you can't even formulate your posting in the form of a question, you are not being precise. –  Igor Rivin Sep 9 '12 at 0:28
@Igor Rivin: My belief was based on my expertise in the relevant math, but I concede it was just an educated guess (which seems to have been correct, but that's beside the point). More importantly, the mixture of English errors in the question and the "name" of the OP suggest that English fluency (as opposed to English proficiency) may be missing here, hence my Jeopardy joke. Math is universal, yet MO practice leaves little choice but to post questions in English, and I think in cases lacking English fluency one should cut the OP some slack if an expert can surmise the likely intent. –  grp Sep 10 '12 at 5:56

Are you sure that $G$ isn't required to be connected? I think this is needed in order to construct the "valued root datum" structure which underlies Bruhat-Tits structure theory. Anyway, the key point is that there is the concept of "valuation" on the root datum, which is really a collection of "valuations" on the possibly non-commutative groups $U_a(K)$ subject to axioms defined in the first big Bruhat-Tits paper in IHES, which I'll call BTI. The existence of this kind of structure on $G(K)$ requires the full power of the theory of the 2nd Bruhat-Tits IHES paper (developed in more "modern" terms in later work of others, such as J-K. Yu), and it requires connectedness of $G$. On the set of such "valuations" there is a natural free action of $V$ and elements of the same equivalence class are called "parallel". The equivalence classes are naturally affine spaces for $V$, and the group $N(K)$ acts naturally on the entire set preserving each equivalence classes through an action by affine transformations (with $Z(K)$ acting through the translation formulas as you have written down). This is all pure group theory formalism (but far from trivial to set up), the definitions of which have nothing to do with any topological structure on $K$. The specification of a valuation on $K$ selects out a preferred equivalence class, and that is the one used to define $\phi$.

(For example, if $G = {\rm{SL}}_2$ and $S$ is the diagonal split maximal torus of $G$ over a field $F$ then the parallelism classes of "valuations" on $G(F)$ in accordance with the root datum for $(G,S)$ correspond exactly to choices of nontrivial non-archimedean $\mathbf{R}$-valued valuations on the abstract field $F$.)

So what you're missing is the (highly non-trivial to develop!) definition of the principal homogeneous space for $V$ which supports the action of $N(K)$. In other words, although one can say in concrete terms that the target of $\phi$ is the group of affine transformations of $V$, this is conceptually misleading: it is really the group of automorphisms of a more intrinsic affine space for $V$ in which there is absolutely no canonical base point (intrinsic to $(G,S,\Phi^+,\omega)$). I suppose there could be a way to make the definition of $\phi$ by bare hands (or at least give formulas, without proving things are well-defined), but my understanding (which could be incomplete) is that using a specific parallelism class of of "valuations" as indicated above provides the only natural way to make the definition. Take a look at section 6 of BTI to learn what a valued root datum is, and the many nontrivial properties of this kind of structure. I think that BTI is more illuminating in certain conceptual respects than the Corvallis paper (though of course it doesn't have the rich supply of interesting examples as in the Corvallis paper, and is a rather challenging paper to read).

-
At the outset of his Corvallis survey, Tits only requires $G$ to be an algebraic group defined over $K$ whose identity component is reductive. But this much generality is bound to create occasional technical problems later on. Anyway, these notes as well as the research papers are definitely challenging, even if the reader first thinks about just a well-behaved low rank group. –  Jim Humphreys Sep 3 '12 at 20:59

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: