K\"ahler-Einstein metrics on group compactifications

Abstract

We obtain a necessary and sufficient condition of existence of a K{\"a}hler-Einstein metric on a $G\times G$-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-Amp{\`e}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.