Classification of indefinite hyper-Kaehler symmetric spaces

Abstract

We classify indefinite simply connected hyper-Kaehler 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-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H) over the quaternions on the space (S^4C^n)^{\tau} of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = \tau S, where \tau is the real structure induced by the quaternionic structure of C^{2m} = H^m. We define and classify also complex hyper-Kaehler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.