Remark 2

Denote by $G_X \backslash G / G_Y$ the set of subsets of $G$ of the form $G_X g G_Y$, $g \in G$; and define $W_X \backslash W / W_Y$ analogously. The preceding proposition shows that the canonical bijection $w \mapsto C(w)$ from $W$ to $B \backslash G / B$ defines by passage to the quotient a bijection $W_X \backslash W / W_Y \to G_X \backslash G / G_Y$.