  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.
