Classification of indefinite hyper-Kaehler symmetric spaces

TL;DR

This paper classifies indefinite hyper-Kaehler symmetric spaces, establishing a correspondence with quaternionic linear group orbits on certain polynomial spaces, and extends the classification to complex hyper-Kaehler symmetric spaces.

Contribution

It provides a complete classification of indefinite simply connected hyper-Kaehler symmetric spaces and introduces a correspondence with quaternionic linear group orbits on polynomial spaces.

Findings

Spaces without flat factors have commutative holonomy and signature (4m,4m).

A 1-1 correspondence with GL(m,H) orbits on quartic polynomials is established.

Complex hyper-Kaehler symmetric spaces exist in all dimensions divisible by 4.

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.