Caracteres de rigidite du groupe de Grothendieck-Teichmuller

Abstract

Let $\k$ be a (topological) field of characteristic 0. Using a Drinfeld associator $Φ$, a representation $Φ(ρ)$ of the braid group over the field $\k((h))$ of Laurent series can be associated to any representation $ρ$ of a certain Hopf algebra $\mathfrak{B}_n(\k)$. We investigate the dependance in $Φ$ of $Φ(ρ)$ for a certain class of representations -- so-called GT-rigid representations -- and deduce from it (continuous) projective representations of the Grothendieck-Teichmuller group $GT_1(\k)$, hence for $\k = \Q_l$ representations of the absolute Galois group of $\Q(μ_{l^{\infty}})$. In most situations, these projective representations can be decomposed into linear characters, which we do for the representations of the Iwahori-Hecke algebra of type A. In this case, we moreover express $Φ(ρ)$ when $Φ$ is even, and get unitary matrix models for the representations of the Iwahori-Hecke algebra. With respect to the action of $GT_1(\k)$, the representations of this algebra corresponding to hook diagrams have noticeable properties.