Repr\'esentations de r\'eflexion de groupes de Coxeter -- Quatri\`eme partie: La repr\'esentation $R$ est r\'eductible. G\'en\'eralit\'es

In this fourth part, (with the notations of the preceding parts) we make the following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical and $S$ is finite. Let $R:W\to GL(M)$ be a reducible reflection representation of $W$. Let $G:=ImR$. Each sub-space of $M$ $(\mathrm{\ne }M)$ stabilize by $G$ is contained in $C_M(G)$. Let $M^{\mathrm{\prime }}:=M\mathrm{/}{C}_M(G)$ and $N(G):={g|g\in G,g, $\text{acts trivially on}$,M’$\mathrm{.}Wecall$N(G)$thetranslationsub-groupof$G$. One of the goals of this part is to study $M^{\mathrm{\prime }}$ and $N(G)$.