Infinite-dimensional hyperkaehler manifolds associated with Hermitian-symmetric affine coadjoint orbits

Abstract

This paper is devoted to the construction of a hyperkaehler structure on the complexification of any Hermitian-symmetric affine coadjoint orbit O of a semi-simple L*-group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kaehler structure of O. By the identification of the complexification of O with the cotangent space of O induced by Mostow's Decomposition Theorem, this leads to the existence of a hyperkaehler structure on the cotangent space of O compatible with Liouville's symplectic form and whose restriction to the zero section is the Kaehler structure of O. Explicit formulas of the metric in terms of the complexified orbit and of the cotangent space are given.