  We investigate the multiplicity-freeness property of holomorphic multiplier
representations of an affine transformation group on a Siegel domain of the
second kind. We first deal with a generalized Heisenberg group. Next, for a
certain smaller group, we give some necessary and sufficient conditions for the
representation defined by the trivial multiplier to be multiplicity-free. We
relate the multiplicity-freeness property to the coisotropicity and the
visibility of the group action and to the commutativity of the algebra of
invariant differential operators.
