Complexified Hermitian Symmetric Spaces, Hyperk\"ahler Structures, and Real Group Actions

There is a known hyperk\"ahler 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 $G_u/K_0$. 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 K\"ahler 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 $G_0\subset G$ corresponding to $G_0/K_0$, 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 $SL(2)$.