Repr\'esentations de r\'eflexion de groupes de Coxeter -- Troisi\`eme partie: les groupes di\'edraux affines

Abstract

In this third part, we make the following hypothesis: representation $R=R(\alpha,\beta,\gamma ;l)$ of $W(p,q,r)$ is reducible and there exist a $G$-invariant non-nulle bilinear form where $G=Im R$. With those conditions, we know the structure of $G$: $G'=G/N(G)$ is isomorphic to a finite dihedral group and $N(G)$ is given explicitly as well as the action of $G$ on $N(G)$. We begin by giving conditions on $p,q,r$ as well on $\alpha,\beta,\gamma$ and we proove them in the appendix. The general case will be studied in the next part.