ABSTRACT
The cotangent bundle of a complex manifold Σ is a holomorphic-symplectic manifold. If Σ is a generalized flag manifold, then this holomorphic-symplectic structure underlies a hyperkahler metric, whose restriction to Σ is the given homogeneous metric. This chapter establishes the equations for hyperkahler metrics on cotangent spaces of Hermitian symmetric spaces, that are invariant under the isometry group of the base. It proves a formula for the potential in symplectic quotients. The formula contains a term involving a character of the group: this term is needed when taking the inverse image by a normalized moment map of non-zero vectors in the dual of the Lie algebra. The chapter reviews some root theory for Hermitian symmetric spaces. It provides a general formula for the Kahler potential of Kahler quotients and can be used for hyperkahler quotients when applied to the zero set of the complex part of the hyperkahler moment map.
