Hello expensive customer to our community We will proffer you an answer to this query gr.group idea – Volume of double cosets $BwB$ ,and the respond will breathe typical by documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

## 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.

we’ll proffer you the answer to gr.group idea – Volume of double cosets $BwB$ query through our community which brings all of the solutions from a number of dependable sources.

## Add comment