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

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