Invariant hyperkahler structures on the cotangent bundles of Hermitian symmetric spaces

Abstract

Let $G/K$ be an irreducible Hermitian symmetric spaces of compact type with the standard homogeneous complex structure. Then the real symplectic manifold $(T^{\ast }(G/K),\Omega )$ has the natural complex structure $J^{-}$. We construct all $G$-invariant Kähler structures $(J,\Omega )$ on homogeneous domains in $T^{\ast }(G/K)$ anticommuting with $J^{-}$. Each such a hypercomplex structure, together with a suitable metric, defines a hyperkähler structure. As an application, we obtain a new proof of the Harish-Chandra and Moore theorem.

arXiv comment24 pages, AMSTEX,some offprints and the proof of Lemma 4.10 are corrected