Caracteres de rigidite du groupe de Grothendieck-Teichmuller

Abstract

Let $\k$ be a (topological) field of characteristic 0. Using a Drinfeld associator $\Phi$, a representation $\Phi(\rho)$ of the braid group over the field $\k((h))$ of Laurent series can be associated to any representation $\rho$ of a certain Hopf algebra $\mathfrak{B}_n(\k)$. We investigate the dependance in $\Phi$ of $\Phi(\rho)$ 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 $\Q(\mu_{l^{\infty}})$. 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 $\Phi(\rho)$ when $\Phi$ 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.