Coxeter groups, symmetries, and rooted representations

Let $(W,S)$ be a Coxeter system, let $G$ be a group of symmetries of $(W,S)$ and let $f:W\to \text{GL}(V)$ be the linear representation associated with a root basis $(V,⟨\mathrm{.},\mathrm{.}⟩,\mathrm{\Pi })$.We assume that $G\subset \text{GL}(V)$, and that $G$ leaves invariant $\mathrm{\Pi }$ and $⟨\mathrm{.},\mathrm{.}⟩$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde{\mathrm{\Pi }}\subset V^G$ so that $(V^G,⟨\mathrm{.},\mathrm{.}⟩,\tilde{\mathrm{\Pi }})$ is a root basis of $W^G$, and we show that the induced representation $f^G:W^G\to \text{GL}(V^G)$ is the linear representation associated with $(V^G,⟨\mathrm{.},\mathrm{.}⟩,\tilde{\mathrm{\Pi }})$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to M"uhlherr and H'ee, but also follows directly from the proof of the other results.