The number of invariant Einstein metrics on a homogeneous space, Newton polytopes and contractions of Lie algebras

Abstract

To every homogeneous space of a Lie group with a compact isotropy group , where the isotropy representation consists of irreducible components of multiplicity , we assign a compact convex polytope in , namely, the Newton polytope of the rational function defined to be the scalar curvature of the invariant metric on . If is a compact semisimple group, then the ratio of the volume of to the volume of the standard -simplex is a positive integer . We note that in many cases, coincides with the number of isolated invariant holomorphic Einstein metrics (up to homothety) on . We deduce from results of Kushnirenko and Bernshtein that in all cases, . To every proper face of we assign a non-compact homogeneous space with Newton polytope that is a contraction of . The appearance of a “defect” is explained by the fact that there is a Ricci-flat holomorphic invariant metric on the complexification of at least one of the .