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We investigate the multiplicity-freeness property of holomorphic multiplierrepresentations of an affine transformation group on a Siegel domain of thesecond kind. We first deal with a generalized Heisenberg group. Next, for acertain smaller group, we give some necessary and sufficient conditions for therepresentation defined by the trivial multiplier to be multiplicity-free. Werelate the multiplicity-freeness property to the coisotropicity and thevisibility of the group action and to the commutativity of the algebra ofinvariant differential operators.

Multiplicity-free representations of certain nilpotent Lie groups over Siegel domains of the second kind