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D. V. Alekseevsky The University of Edinburgh and Maxwell Institute for Mathematical Sciences Edinburgh, United Kingdom D.Aleksee@ed.ac.uk

Para-KЁ ahler-Einstein homogeneous manifolds of semisimple Lie group
A 2n-dimensional pseudo-Riemannian manifold (M , g ) is called para-KЁ ahler if it admits a parallel para-complex structure K that is an involutive field of endomorphisms or, equivalently, two complementary n-dimensional isotropic parallel distributions L± . Para-KЁ ahler manifold can be also described as a symplectic manifold with symplectic form = g K and two complementary integrable Lagrangian distributions L± . We give a description of homogeneous para-KЁ ahler manifolds of real semisimple Lie group G in terms of its crossed Satake diagrams and invariant symplectic structures. Using para-holomorphic geometry, we generalize some classical results of KЁ ahler geometry to the para-KЁ ahler case, in particular, derive a formula for the Ricci tensor in terms of paraholomorphic coordinates and para-KЁ ahler potential. We give a classification of invariant para-KЁ ahler-Einstein metrics on homogeneous manifolds M = G/H of semisimple Lie group in terms of Koszul forms. The talk is based on joint works with C. Medori and A. Tomassini (Parma).

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