Abstract
We classify indefinite simply connected hyper-Kahler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m, 4m). We establish a natural 1-1 correspondence between simply connected hyper-Kahler symmetric spaces of dimension 8m and orbits of the group GL(m,H) on the space (S4Cn)τ of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = τS, where τ is the real structure induced by the quaternionic structure of C2m = Hm. We define and classify also complex hyperKahler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.
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