Iwasawa and Cartan Decompositions.

Question

Consider the tome of Bruhat and Tits: Groupes réductifs sur un corps local : I. Données radicielles valuées. Publications Mathématiques de l'IHÉS, 41 (1972), p. 5-251. (available on NUMDAM). I am interested primarily in the statements of Propositions 4.4.3 and 4.4.4 (and maybe also 7.3.1).

In the swathe of notation and technical conditions present, I find it hard to read exactly what the precise statements of these two propositions are. My question is, can anyone give

(a) a precise version of the statements of 4.4.3 and 4.4.4?

(b) if the request in (a) is too much, some sort of simplified version that is easy to state/comprehend and hopefully still reasonably general.

or

(c) an alternative reference covering at least the case of a split reductive group?

(this question is related to, but more general than Dinakar's question)

Answer 1

I think the key point is the proposition 4.4.2, where "good" subgroups are caracterised geometrically as stabilisers of special subgroups (ie, stabilisers of a point o such that the Weyl group W is the semidirect product of its translations and of the stabiliser of o in W).

Then the group G is the product of B (the stabiliser of a class of sector, a minimal parabolic group for an algebraic group) and the group K. Moreover, the group B itself is the product of B^0 (which is the union of pointwise fixators of sectors) and the group of translations (acting on some apartment containing o and a sector in this class).

The Cartan decomposition is, as usual, the decomposition of an element g in kvk', where k and k' are element of K and v is an element which sends o to a vertice of the sector starting at o in the class defined by B.

The proposition 4.4.4 is meant to explain the relation between the two decompositions (ie, when you know the translation part in the Iwasawa decomposition, can you deduce it in the Cartan decomposition ?)

If you know how to attach a building to a reductive group, then the book "Buildings" by Abramenko and Brown is a good reference (see chap. 11), much easier to read. They treat every building, but construct only the affine building associated to SL(n). Another reference is the small book of Macdonald, "Spherical functions on a group of p-adic type" (chapter II, Theorem 2.6.11)

Answer 2

I figured this exact same stuff a while ago. I will look at my notes sometime soon and post something more precise, but here is what I can say off the top of my head.

• "bon" or "good" is pretty much by definition specifically the capability of $K$ to make true the 'Iwasawa decomposition', which most people nowadays would write $G=P\cdot K$ (not direct) where $K$ is one of these maximal compact-open subgroups and $P$ is a minimal parabolic (a Borel subgroup, essentially by definition, if the group is quasi-split). In fact, let's just say it's a Borel subgroup $B$ and be done with it.
• that horrible symbol $\hat{\mathfrak{B}}^0$ is really the unipotent radical $U$ of the $B$
• that $\hat{V}$ is something like the cocharacter group of the maximal torus $T\subset B$ interpreted as a subgroup of $B$ by evaluation at the uniformizer of the base field, i.e. $X_{*}(T)\rightarrow B : \mu \mapsto \mu(\pi)$

(so "$\hat{G}=\hat{\mathfrak{B}}^0 \hat{V} K$" really just means $G=BK$)

• you have to be very careful in this book with what kind of thing $K$ is. There are four (I think) closely related groups that are differentiated only by cryptic combinations of hats, superscript zeros, and primes. The can be pointwise fixers or merely stabilizers of facets in the whole group or some subgroup, and under various hypotheses on $G$ some of them turn out to be equal to others.

Answer 3

Hi,

As far as I understand the paper, they define a good subgroup as a maximal bounded subgroup which splits into the whole group, that is to say the whole group is the direct product of the maximal bounded one with another subgroup. In that case, 4.4.3 says that if this maximal bounded subgroup K contains the group B defining with N a Bruhat-Tits system, then we can write an Iwasawa and a Cartan decomposition of the whole group with repect to K. 4.4.4 shows the implentation of 4.4.3 in several particular cases.

Hopes this helps,

Eric