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

TL;DR

This paper explores the hyperk"ahler structures on complexified Hermitian symmetric spaces, showing their equivalence under diffeomorphisms, and investigates the associated geometric and group action properties with explicit examples.

Contribution

It introduces explicit diffeomorphisms relating complex structures and symplectic forms on Hermitian symmetric spaces, revealing intermediate K"ahler structures and their group actions.

Findings

Almost all complex structures are equivalent via explicit diffeomorphisms.

Symplectic structures on the cotangent bundle are shown to be equivalent.

Explicit computations are provided for the case of SL(2).

Abstract

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 $SL(2)$.