# gr.group idea – Volume of double cosets \$BwB\$ Answer

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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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