The adapted hyper-Kähler structure on the crown domain

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