GL_n( \mathbb{\mathbb{Z}_p})$?

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Dec 18, 2011 at 18:22	comment	added	paul garrett		@pm: Yes, in this situation, $B$ can be described as the inverse image in $GL(n,o)$ of the Borel in $GL(n,o/p)$. Yes, the affine Weyl group is normalizer of diagonals, modulo diagonal unit matrices. In my book/notes I verify (as did Bruhat and Tits ages ago, maybe it's in Bourbaki Lie ch. iv-...?) that the affine building for $SL(n,k)$ can be constructed via homothety classes of lattices. Then the building properties prove an abstracted Bruhat decomposition. One can also prove, following Tits, that the Coxeter-group property follows from the building set-up.
Dec 18, 2011 at 16:53	comment	added	Marc Palm		So just to confirm that I understand what you are saying notationwise: In your notation $B$ is the pullback of the Borel subrgoup $B'$ in $GL(n, o/p)$ along the projection $GL(n,o) \rightarrow \GL(n, o/p)$ and $w$ runs through the normalizer of all diagonal matrices $M$ (=affine Weyl group $MW$?). This is called affine Cartan decomposition? What is $N$ in the $BN$ pair, if $B$ is the Iwahori? ($N=MW$?) I am rather fine with using axioms, but it usually gives me a headache to verify that they hold for specific examples.
Dec 18, 2011 at 15:01	comment	added	paul garrett		@pm, In my "Buildings and Classical Groups" book, (math.umn.edu/~garrett/m/buildings/book.pdf) chapter 5 exactly treats this: it is simply the BN-pair (with corresponding Bruhat decomposition, etc) attached to the affine building. Probably any book on buildings mentions this at some point. In some sources, the cell multiplication rule is an "axiom". In my chapter 5 I prove that it follows from the building properties.
Dec 18, 2011 at 10:04	comment	added	Marc Palm		I checked the other answer, because it directly answers my question, but your suggestion seems a lot more suitable for my purposes than the ``naive'' KAK decomposition I was aking about. Do you have a reference for this? Thanks a lot for mentioning this.
Dec 17, 2011 at 20:45	history	answered	paul garrett	CC BY-SA 3.0