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Does anyone know where I can find a proof of the Iwasawa decomposition for reductive groups? I know that there are a couple of related results that are called the Iwasawa decomposition, but I am interested in the following statement:

Let G be a complex reductive group, let O be Taylor series with complex coefficients, and let K be Laurent series with complex coefficients. The G(K) = G(O) * T(K) * U(K), where T is a maximal torus and U is a maximal unipotent subgroup of G.

I am interesting in finding the proof because this is how one shows that the semi-infinite cells in the affine Grassmannian cover the entire space. I have been using this fact for quite a while now and am becoming uncomfortable about not knowing where to find the proof. A proof in the case where K is a p-adic field and O is its ring of integers would also be great since I am sure a proof would carry over to the above case.

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3 Answers

The original reference for this is the paper of Bruhat-Tits (available on NUMDAM), see Prop. 4.4.3. Another reference, probably easier to read, is the book of Macdonald, "Spherical functions on a group of p-adic type", Theorem 2.6.11.

There is a nice proof of this fact using the geometry of buildings, which goes as follows. You can think only about trees (eg for G=SL(2)) to get the main ideas.

Consider the affine building X associated to G(K). The buildings at infinity of X (which is also the space of flags) can be seen as equivalence classes of sectors in X. Then U(K) is the union of fixators of sectors in such a class \xi. The group T(K) is the group of translation in some apartment A, and B:=T(K)U(K) is the stabiliser of the equivalence class of sector. G(O) is the stabiliser of some vertex o.

The main point is that the building is the union of all apartments containing a sector pointing towards \xi. It follows that, for every x in X, there is an element u of U(K) such that u.x is in A.

Let g in G. Applying this to the element x=g.o, we see that the vertex ug.o is in A. By transitivity of the action of T(K) on vertices of A, there is an element t in T such that tug.o=o. Thus tug is in G(O), which gives the decomposition of g.

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Do you know a good reference for learning about buildings? –  Dinakar Muthiah Oct 27 '09 at 19:18
    
Sure. The new book of Abramenko and Brown, "Buildings", is quite good. There is also a book by Paul Garett, available online at math.umn.edu/~garrett/m/buildings –  Jean Lecureux Oct 27 '09 at 19:38
    
Thanks! . –  Dinakar Muthiah Oct 28 '09 at 2:10
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Is it possible to use the fibration B -> G -> G/B for a chosen Borel subgroup B in G to prove the covering statement? As H^1(K,B) = *, this fibration should say that, G(K)/B(K) = G/B(K). By properness of G/B, G/B(K) = G/B(O). By the previous argument on O, we have G/B(O) = G(O)/B(O). In particular, we get that G(K) = G(O) B(K). Since H^1(K,U) = *, the exact sequence U -> B -> T gives B(K) = T(K) U(K), etc. I have no idea if this actually works!

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Can you elaborate? What do you mean by H^1(K,B) = *? And why does properness give you G/B(K) = G/B(O)? –  Dinakar Muthiah Oct 24 '09 at 2:45
    
First question: G/B is the fppf quotient of G by B. Thus, the obstruction to G(K) -> G/B(K) being surjective is an fppf B-torsor over K i.e., a class in H^1(K,B), and ditto for O. Second question: for any proper complex variety X, X(K) = X(O) by the valuative criterion of properness. –  Bhargav Oct 24 '09 at 3:14
    
Just as an additional clarification, by the cohomology groups I mean the fppf cohomology groups of the corresponding schemes. As the coeffecients are always smooth (we are in char 0!), this is the same as etale cohomology (which vanishes for O, and is the cohomology of Z-hat for K, but this is irrelevant). –  Bhargav Oct 24 '09 at 3:16
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This reverses your logic a bit, but perhaps you can prove directly that the semi-infinite orbits cover the Grassmannian and then deduce the Iwasawa decomposition.

Here is a possible approach to doing this.

First, show that the semi-infinite orbits are the attracting sets for a C^* action on Gr. This C^* action comes from a generic sub-C^* of the maximal torus T(C) of G(C) (for example from the coweight \rho).

The fixed points of this C^* action will be the same as the fixed points of the T(C) action action, namely the points t^\mu, where \mu is a coweight. The semiinfinite orbits are the attracting sets for this C^* action.

To show that the semi-infinite orbits cover Gr, we just need to show that every point in Gr has a limit under this C^* (ie is in a C^* orbit). This would be automatic if Gr was projective. However, the C^* action preserves the \overline{Gr^\lambda}, and every point lives in some \overline{Gr^\lambda}, so every point has a limit under the C^* action.

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How do you show that the fixed points of the torus action are the image of the coweights? I know how to do this for GL_n using the lattice model for the affine Grassmannian, but I don't know a method in general. –  Dinakar Muthiah Oct 25 '09 at 16:00
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