Volume of a double class of a parahoric subgroup

Question

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 $w\in {\tilde W}$?

Edit

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 $\mu (PwP)$?

Answer 1

Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

Edit: for such $\sigma, w$, $l(\sigma w) = l(\sigma) + l(w) $.