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We classify indefinite simply connected hyper-Kahler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4ra). 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 (5 4 C n ). r of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = rS } where r is the real structure induced by the quaternionicdoi:10.4310/ajm.2001.v5.n4.a4 fatcat:hgxuofcpkjfwjdraboxm3rhva4