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

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