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 $(T^*(G/K),Ω)$ has the natural complex structure $J^-$. We construct all $G$-invariant Kähler structures $(J,Ω)$ on homogeneous domains in $T^*(G/K)$ 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.