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Let (G,G′) = (U(n, n), U(p, q)) (p + q ≤ 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 KC-module structure of the regular function rings on the closure of the associated nilpotent KC-orbits in s, where g = k⊕ s is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

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