Résumé
We obtain a necessary and sufficient condition of existence of a Kahler-Einstein metric on a GxG-equivariant Fano compactification of a complex connected reductive group G in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real Monge-Ampere equation, using the invariance under the action of a maximal compact subgroup K x K.