The adapted hyper-K\"ahler structure on the crown domain

TL;DR

This paper explicitly describes the unique G-invariant hyper-Kähler structure on the crown domain of a non-compact hermitian symmetric space, including invariant potentials and moment maps.

Contribution

It provides an explicit description of the adapted hyper-Kähler structure on the crown domain, including potentials and moment maps, which was previously not detailed.

Findings

Explicit formulas for the hyper-Kähler structure

Invariant potentials of the Kähler metrics computed

Moment maps associated with the structure derived

Abstract

Let $\,\Xi\,$ be the crown domain associated with a non-compact irreducible hermitian symmetric space $\,G/K$. We give an explicit description of the unique $\,G$-invariant adapted hyper-K\"ahler structure on $\,\Xi$,$\ $i.$\,$e.$\ $compatible with the adapted complex structure $\,J_{ad}\,$ and with the $\,G$-invariant K\"ahler structure of $\,G/K$. We also compute invariant potentials of the involved K\"ahler metrics and the associated moment maps.