# Iwasawa decomposition and Cartan decomposition

The Iwasawa decomposition and Cartan decomposition for $GL(n)$ is available for local fields. This can be proven for totally disconnected fields and archimedian fields seperatly by hand.

Here is a question, I am asking out of curiosity:

Is there a proof, which does not use the exact structure of the maximal compact, but e.g. only property that it is maximal compact?

You have a proof which uses the fact that maximal compact open subgroups correspond to vertices in the building of ${\rm GL}(n)$. For instance for the Cartan decomposition you want to classify the orbits of $K={\rm GL}(n,{\mathfrak o}_F)$ (${\mathfrak o}_F$ denotes the ring of integers of your p-adic field $F$) in the vertex set of the building. To the aim, you first use the fact that $K$ acts transitively on the apartments containing the vertex fixed by $K$. This way you may send any pair of vertices $(s,t)$ in a fixed apartment $A$. The stabilizer of $s$ acts transitively on the Weyl chambers and you may assume that $t$ lies in a fixed Weyl chamber. From this it's easy to prove that $$t={\rm Diag}(\varpi_F^{k_1},...,\varpi_F^{k_n}).s$$ where e.g. $k_1 \geq k_2 \geq ...\geq k_n$ and where $\varpi_F$ is a uniformizer of $F$.
Similarly the Iwasawa decomposition $G=KB$ corresponds to the fact that $K$ acts transitively on the germs of "quartiers" (you may prove this using the fondamental properties of the building stated in Bruhat-Tits, IHES, volume 1).
• In the archimedean case, you need to replace the building by the symmetric space $GL(n,{\mathbb R})/O(n)$, that you may identify with the set of positive definite quadratic forms on ${\mathbb R}^n$. – Paul Broussous Nov 2 '11 at 10:36