algebraic groups - Iwasawa and Cartan Decompositions.

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)