Classical complexified hyperplane complement

PDF p. 5: Notation 2.1; (2.2)–(2.3)

Let $(W,S)$ be a Coxeter system with $S$ finite. Use the finite-dimensional real dual space $\mathrm{\Theta }$, its topology, the Tits cone $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}$ and its interior ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }$, and the set $\mathcal{H}$ of distinct reflecting hyperplanes from Reflecting hyperplanes, fundamental chamber and Tits cone. The classical complexified hyperplane complement is

${\mathcal{X}}^{\mathbb{C}}=({\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }\times {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ })\setminus {\bigcup }_{H\in \mathcal{H}}(H\times H)\mathrm{.}$

It has the subspace topology inherited from the product topology on ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }\times {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }$. In particular, a point $(x,y)$ belongs to ${\mathcal{X}}^{\mathbb{C}}$ exactly when $x,y\in {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }$ and no reflecting hyperplane contains both $x$ and $y$.

When $W$ is finite, identify $(x,y)\in \mathrm{\Theta }\times \mathrm{\Theta }$ with $x\otimes 1+y\otimes i\in \mathrm{\Theta }{\otimes }_{\mathbb{R}}\mathbb{C}$. Under this identification,

${\mathcal{X}}^{\mathbb{C}}=(\mathrm{\Theta }{\otimes }_{\mathbb{R}}\mathbb{C})\setminus {\bigcup }_{H\in \mathcal{H}}(H{\otimes }_{\mathbb{R}}\mathbb{C})\mathrm{.}$

Here $H{\otimes }_{\mathbb{R}}\mathbb{C}$ denotes the complexification of the real hyperplane $H$, viewed as a subspace of the complexification of $\mathrm{\Theta }$.

The cited arrangement definition supplies $\mathrm{\Theta }$, its topology, ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }$ and the set of hyperplanes. The displayed set difference is therefore a well-defined subset of the product; the product and subspace topology conventions are those of Complexified complements in an open convex cone. The pointwise membership condition is the set-theoretic meaning of removing the union.

If $W$ is finite, Basic Tits-cone properties gives $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}=\mathrm{\Theta }$, hence ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }=\mathrm{\Theta }$. The complexification identification in Complexified complements in an open convex cone sends $H\times H$ to $H{\otimes }_{\mathbb{R}}\mathbb{C}$ for each real hyperplane $H$. It consequently sends the removed union and its complement to the corresponding union and complement displayed above. This justifies the finite-group identification.