Geometric characterization of the group law in the Weyl group

Abstract: Let $G$ be a reductive group with Borel $B$ and Weyl group $W$. Then $B$-double cosets in $G$ are indexed by the Weyl group, say $O(w)$ for $w\in W$. Then we prove the minimal $B$-double coset in the convolution $O(w_1)*O(w_2)$ is $O(w_1w_2)$, which gives a geometric characterization of multiplication in $W$. This defines the abstract Weyl group $\mathbf W$ which is a Coxeter group acting on the abstract Cartan $\mathbf T$.