Flag manifolds, symmetric [formula omitted]-triples and Einstein metrics

Abstract

Let G be a compact connected simple Lie group and let M=GC/P=G/K be a generalized flag manifold. In this article we focus on an important invariant of G/K, the so-called t-root system Rt, and we introduce the notion of symmetric t-triples, that is triples of t-roots ξ,ζ,η∈Rt such that ξ+η+ζ=0. We describe their properties and we present an interesting application on the structure constants of G/K, quantities which are straightforward related to the construction of the homogeneous Einstein metric on G/K. We classify symmetric t-triples for generalized flag manifolds G/K with second Betti number b2(G/K)=1, and next we treat the case of full flag manifolds G/T, with b2(G/T)=ℓ=rkG, where T is a maximal torus of G. In the last section we construct the homogeneous Einstein equation on flag manifolds G/K with five isotropy summands, determined by the simple Lie group G=SO(7). By solving the corresponding algebraic system we classify all SO(7)-invariant (non-isometric) Einstein metrics, and these are the very first results towards the classification of homogeneous Einstein metrics on flag manifolds with five isotropy summands.