Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions

Abstract

There is a known hyperkähler structure on any complexified Hermitian symmetric space G/K, whose construction relies on identifying G/K with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space Gu/K0. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on G/K; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on \(T^{*}\left (G_{u}/K_{0}\right )\). We highlight the intermediate Kähler structures, which share a holomorphic action of G related to the one on G/K, but moment geometry related to that of \(T^{*}\left (G_{u}/K_{0}\right )\). As an application, for the real form G0 ⊂ G corresponding to G0/K0, the Hermitian symmetric space of noncompact type, we give a strategy for study of the action on G/K using the moment-critical subsets for the intermediate structures. We give explicit computations for SL(2).