coset poset of reflection subgroup

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.

• What examples have you tried? Does it have nice properties for rank 2 for example? Other small finite examples? Sep 17, 2021 at 19:06
• @NathanReading I edited this question again and explain the motivation Sep 18, 2021 at 2:12
• Thanks for adding this motivation. I'm more asking about which small examples you have tried. There are lots of small examples where you can draw this poset and see what its topology is. For example, does the finite reflection group of type B_2 (the symmetry group of a square) have topological properties that you find interesting? More generally, symmetry groups of other regular polygons, or the symmetric groups, etc. Sep 18, 2021 at 15:16