Title: 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.