Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands

Abstract

We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag manifold and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E_6/(SU(4) x SU(2) x U (1) x U (1)) and E_7/(U(1) x U(6)) we find explicitely all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2\ell +1)/(U(1) x U (p) x SO(2(\ell -p-1)+1)) and SO(2\ell)/(U(1) x U (p) x SO(2(\ell -p-1))) we prove existence of at least two non Kähler-Einstein metrics. For small values of $\ell$ and $p$ we give the precise number of invariant Einstein metrics.

33 pages (Some misprints in the text and Table 3, were corrected)

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• Spin structures on compact homogeneous pseudo-Riemannian manifolds
• Invariant Generalized Complex Structures on Partial Flag Manifolds
• Complex invariant Einstein metrics on $S{O}_{2(n_1+n_2+n_3)+1}/U_{n_1}\times U_{n_2}\times S{O}_{2n_3+1}$ and Ricci-flat manifolds
• On the embeddability of the homogeneous Ricci flow and its collapses
• The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds