Theta lifting of holomorphic discrete series: The case of $U(n, n) \times U(p, q)$

Let $( G, G' ) = ( U( n, n ), U( p, q ) ) \; ( p + q \leq n )$ be a reductive dual pair in the stable range. We investigate theta lifts to $G$ of unitary characters and holomorphic discrete series representations of $G'$, in relation to the geometry of nilpotent orbits. We give explicit formulas for their $K$-type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme $K'$-type, we show that the $K$ structure of the resulting representations of $G$is almost identical to the $K_{\mathbb{C} }$-module structure of the regular function rings on the closure of the associated nilpotent $K_{\mathbb{C} }$-orbits in $\mathfrak{s}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{s}$ is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.