The adapted hyper-Kähler structure on the crown domain

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-Kähler structure on $\Xi$, i.e. compatible with the adapted complex structure $J_{ad}$ and with the $G$-invariant Kähler structure of $G/K$. We also compute invariant potentials of the involved Kähler metrics and the associated moment maps.