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

To every homogeneous space $M=G/H$ of a Lie group $G$ with a compact isotropy group $H$, where the isotropy representation consists of $d$ irreducible components of multiplicity $1$, we assign a compact convex polytope $P=P_M$ in $\mathbb R^{d-1}$, namely, the Newton polytope of the rational function $s(t)$ defined to be the scalar curvature of the invariant metric $t$ on $M$. If $G$ is a compact semisimple group, then the ratio of the volume of $P$ to the volume of the standard $(d-1)$-simplex is a positive integer $\nu(M)>0$. We note that in many cases, $\nu(M)$ coincides with the number $\mathcal E(M)$ of isolated invariant holomorphic Einstein metrics (up to homothety) on $M^{\mathbb C}=G^{\mathbb C}/H^{\mathbb C}$. We deduce from results of Kushnirenko and Bernshtein that in all cases, $\delta_M=\nu(M)-\mathcal E(M)\geqslant0$. To every proper face $\gamma$ of $P$ we assign a non-compact homogeneous space $M_\gamma=G_\gamma/H_P$ with Newton polytope $\gamma$ that is a contraction of $M$. The appearance of a “defect” $\delta_M>0$ is explained by the fact that there is a Ricci-flat holomorphic invariant metric on the complexification of at least one of the $M_\gamma$.