Standard parabolic subgroup ($W_J$)

For $J\subseteq I$, the standard parabolic subgroup is $W_J=⟨s_j:j\in J⟩$. Its cosets index faces of a fixed type in the Coxeter complex.

A subset $T\subseteq S$ is spherical when its standard parabolic subgroup $W_T$ is finite.

The Davis complex is the geometric realization of the poset of cosets $w{W}_T$ for spherical subsets $T\subseteq S$. Equivalently, it is the basic construction of a Coxeter group obtained by gluing copies of the Davis chamber along mirrors. It is a contractible proper $W$-CW complex.

For a mirrored chamber $K$, the basic construction iswhere $(w,x)\sim (w^{\prime },x)$ when $w^{-1}{w}^{\prime }$ belongs to the subgroup generated by the mirrors containing $x$.