Flag manifolds, symmetric tt-triples and Einstein metrics

Let G   be a compact connected simple Lie group and let M=GC/P=G/KM=GC/P=G/K be a generalized flag manifold. In this article we focus on an important invariant of G/KG/K, the so-called tt-root system RtRt, and we introduce the notion of symmetric tt-triples, that is triples of tt-roots ξ,ζ,η∈Rtξ,ζ,η∈Rt such that ξ+η+ζ=0ξ+η+ζ=0. We describe their properties and we present an interesting application on the structure constants of G/KG/K, quantities which are straightforward related to the construction of the homogeneous Einstein metric on G/KG/K. We classify symmetric tt-triples for generalized flag manifolds G/KG/K with second Betti number b2(G/K)=1b2(G/K)=1, and next we treat the case of full flag manifolds G/TG/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/KG/K with five isotropy summands, determined by the simple Lie group G=SO(7)G=SO(7). By solving the corresponding algebraic system we classify all SO(7)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.