TITLE: Iwasawa and Cartan Decompositions.
QUESTION [7 upvotes]: 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)

REPLY [2 votes]: 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.