# What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$?

Consider $G = GL_n( \mathbb{Q}_p)$ and $K = GL_n( \mathbb{Z}_p)$, or more favorable replace $\mathbb{Q}_p$ by a non archimedean field and $\mathbb{Z}_p$ by the ring of integers.

Note that the space $G //K$ is discrete, since $K$ is open and closed in $G$.

Given a Haar measure on $G$, I can prove that there exists a unique (discrete) measure on $G//K$, such that $$\int\limits_{G} f(g) d g = \sum\limits_{x \in G//K} w(x) \int\limits_{K \times K} f(k_1 x k_2) d k_1 d k_2 .$$

How can $w$ be expressed, if we pick a representative $x = diag( w^{k_1}, \dots, w^{k_n})$ for a uniformizer $w$?

Perhaps easier, but equivalent what is the ratio: $vol_G (K xK)/ vol_G(K)?$

(More out of curiousity: How is the Plancherel measure related to this?)

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You are dealing with the integral formula for the Haar measure on $G$, associated with the Cartan decomposition $G=KAK$. Since the Plancherel measure lives on the dual object $\hat{G}$, I don't see any obvious relation between the two. –  Alain Valette Dec 17 '11 at 22:05

The measure of $KxK$ is a classical computation that may be found in: Macdonald "Symmetric Functions and Hall Polynomials" (Oxford Mathematical Monographs), more precisely in Chapter V: The Hecke ring of ${\rm GL}(n)$ over a local field.

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This is very useful, thank you. I will delete my response as obsolete. The final answer is (2.9) on p.298 of Macdonald's book. –  GH from MO Dec 17 '11 at 18:05
My copy of Mc Donalds (1979) has it on pg. 162! Thanks to both of you. –  plusepsilon.de Dec 18 '11 at 10:02
Depending on taste, one might also find appealing or helpful the description of this in terms of the Iwahori-Hecke algebra, with affine Weyl group and affine cartan decomposition $G=\bigcup_w BwB$ where $B$ is the Iwahori. Among other features, this does give a way to inductively determine the measure of $BwB$, once the measure of $B$ is normalized, because there is a precise cell-multiplication rule $BwB\cdot BsB=BwsB$ when the length of $ws$ is strictly greater than that of $w$, and $s$ is one of the affine reflections generating $W$. That is, the length in $W$ is equivalent to knowing the measure of the Iwahori coset.
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. –  plusepsilon.de Dec 18 '11 at 10:04
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. –  plusepsilon.de Dec 18 '11 at 16:53
@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. –  paul garrett Dec 18 '11 at 18:22