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

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.