Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard parabolic proper subgroup generated by a subset $J \subset S$. It is well known that the poset of cosets $\{xW_J\}$ ordered by reverse inclusion is face poset of Coxeter complex.
Consider the poset of cosets $\{xH\}$, ordered by reverse inclusion, $H$ run over proper reflection subgroup of $W$(subgroup generated by a collection of reflections in $W$). This poset may not be face poset of a complex. Do we know any topological property of this poset, is this poset spherical ?
Edit 2021/9/18
This question comes from our consideration about orbit configuration space $$F_G(X,n)= \{(x_1,x_2,...,x_n)\in X^n | Gx_i \cap Gx_j = \emptyset \text{ for } i\neq j\}$$ for a space $X$ with $G$-action, we have calculated the cohomology ring of orbit configuration space of standard $\mathbb{Z}_k^m$-action (a standard $\mathbb{Z}_k^m$-action of $\mathbb{C}^m$ is a map $\varphi : \mathbb{Z}_k^m \times \mathbb{C}^m \rightarrow \mathbb{C}^m$ that $\varphi((z_1,z_2,...), (x_1,x_2,...))=(e^{2\pi iz_1/k}x_1, e^{2\pi iz_2/k}x_2,...)$). The result is beautiful and I will put the paper on arxiv recently.
Furthermore, it seems that I need deep knowledge about reflection group if I want to consider orbit configuration space of general reflection group on $\mathbb{R}^m$ or $\mathbb{C}^m$. The poset in the question appears naturally for two point orbit configuration space, i.e., $n=2$ and observe the intersection lattice (all possible intersection) of these subspaces $x_1=gx_2$, where $g$ runs over elements of a reflection group.
I am not an expert of reflection group, so I wonder may be there is some known results about this question and I haven't found it.
