# Volume of a double class of a parahoric subgroup

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 $$\mu (PwP)$$?

• I realized that there is no tag for "Tits systems" or "BN-pairs". – Paul Broussous Nov 13 '18 at 15:18
• I gather you mean that $P$ is (most precisely) a parahoric, not parabolic, as in your title? – paul garrett Nov 13 '18 at 18:57
• Indeed... But I figured that it's better to use the less ambiguous term, especially since you already had it in your title. – paul garrett Nov 13 '18 at 19:56
• 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. – paul garrett Nov 13 '18 at 22:50
• @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.) – LSpice Nov 14 '18 at 16:40