Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ and let $\tilde W$ be its affine Weyl group. Let $I$ be an Iwahori subgroup of $G$ fixing a chamber in the apartment attached to $T$.

If $\mu$ is the Haar measure on $G$ normalized by $\mu (I)=1$, then we have the formula of Iwahori-Matsumoto: $$ \mu (IwI)= q^{l(w)}, \ w\in {\tilde W} $$ where $l$ is the length function.

My question is the following.

Can one find a formula in the literature giving the volume of $PwP$, where $P$ is a standard parahoric subgroup and $w\in {\tilde W}$?


After Paul Garrett's comments, I'd like to make my question a bit more precise. Let us give an example. Take $G=\mathrm{GL}_n (F)$, with $n=ef$. Let $I$ be the standard Iwahori subgroup formed of matrices with coefficients in ${\mathfrak o}_F$ that are upper triangular modulo ${\mathfrak p}_F$ (${\mathfrak o}_F$ is the ring of integers of $F$, and ${\mathfrak p}_F$ its maximal ideal). Let $P$ be the parahoric subgroup of $G$ formed of $f\times f$ block matrices, which have coefficients in ${\mathfrak o}_F$, and are upper triangular by blocks modulo ${\mathfrak p}_F$. Let $w$ be a permutation matrix in $G$ which is monomial by $f\times f$ blocks. Such a matrix may be seen as an element of the permutation group ${\mathfrak S}_e$.

I know how to prove the following formula: $$ \mu (PwP) = (q^{f^2})^{l_{{\mathfrak S}_e}(w)}\, \mu (P)\ . $$ where $l_{{\mathfrak S}_e}$ is the usual length function in ${\mathfrak S}_e$, attached to the set of generators $\{ (12), (23),...,(e-1\, e)\}$.

For proving that, I did not use the axioms of BN-pairs as Paul suggests. I tried to do so but I got stuck with technical difficulties.

I'm looking for generalizations of this formula. For instance as follows.

Let $W\subset {\tilde W}$ be the Coxeter-group part of the generalized Weyl group $\tilde W$. Let $S$ be the system of generating involutions. Fix a parahoric subgroup $P=I\langle T\rangle I$, with $T\subset S$. Assume that $T=T_1 \sqcup \cdots \sqcup T_e$, where $T_i$ commutes element-wise with $T_j$ for $i \ne j$. Let $w\in W$ be an element such that there exists $\sigma\in {\mathfrak S}_e$ such that $wT_i w^{-1}=T_{\sigma (i)}$ for all $i=1,\dotsc,e$. Assume moreover that, if $wT_i w^{-1}=T_i$, then $w$ commutes with each element of $T_i$. Can one then find a closed formula for $\mu (PwP)$?

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    $\begingroup$ I realized that there is no tag for "Tits systems" or "BN-pairs". $\endgroup$ – Paul Broussous Nov 13 '18 at 15:18
  • $\begingroup$ I gather you mean that $P$ is (most precisely) a parahoric, not parabolic, as in your title? $\endgroup$ – paul garrett Nov 13 '18 at 18:57
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    $\begingroup$ Indeed... But I figured that it's better to use the less ambiguous term, especially since you already had it in your title. $\endgroup$ – paul garrett Nov 13 '18 at 19:56
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    $\begingroup$ I suppose you know as well as I that the cell multiplication rule(s), $BwB\cdot BsB = BwsB$ for length $ws$ greater than that of $w$, and with an addition term $\sqcup BwB$ when $ws$ is not longer, and the Bruhat decomposition of each $P$, give some sort of description of the volume/index. Likewise, I'd bet you know that $P\backslash G/Q\approx W_P\backslash W/W_Q$ for "parabolics/parahorics" $P,Q$, so the issue of systematic description partly devolves into existence of a "nice" choice of reps for $W_P\backslash W/W_Q$... which I do not know, and don't off-hand know a reference for. $\endgroup$ – paul garrett Nov 13 '18 at 22:50
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    $\begingroup$ @paulgarrett, Casselman gives a distinguished choice of representatives for the double cosets you describe in Proposition 1.1.3 of the p-adic notes; for example, one can take the minimal-length elements in each coset. (I guess that doesn't make it terribly easy to count them.) $\endgroup$ – LSpice Nov 14 '18 at 16:40

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