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gr.group idea – Volume of double cosets $BwB$
In Macdonald’s bespeak “Spherical functions on a group of $p$-adic type”, Prop. (3.1.7), it’s said that if $w=w_1dots w_r$ is a lowered phrase for $earn W$ (the affine Weyl group), and if $q(w)=(BwB:B)$ (right here $B$ is the benchmark Iwahori subgroup), then $q(w)=q(w_1)dots q(w_r)$.
I do not grasp the proof. The proof is by induction on the size of $w$: If $alpha$ is a unostentatious (affine) root, and $earn W$, are such that $l(w_alpha w)>l(w)$ then $Bw_alpha wB=Bw_alpha BBwB$. Since $w_alphane w$ the cosets $BwB$ and $Bw_alpha B$ are disjoint, and due to this fact $q(w_alpha w)=q(w_alpha)q(w)$.
My query is how the truth that the cosets $BwB$ and $Bw_alpha B$ are disjoint implies $q(w_alpha w)=q(w_alpha)q(w)$.
Edit: Note that it isn’t undoubted in common that if $G$ is a bunch and $B$ is a subgroup, and $BxB$ and $ByB$ are disjoint, then $(BxByB:B)=(BxB:B)(ByB:B)$. It is simple to behold that if $BxB=cup x_iB$ and $ByB=cup y_iB$ (disjoint unions) then $BxByB=cup x_iy_jB$, however it may breathe that this union just isn’t a disjoint union.
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