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

Let (G, G') =(U(n, n), U(p, q)) (p + q less than or equal to n) be a reduct ive dual pair in the stable range. We investigate theta lifts to G of unita ry characters and holomorphic discrete series representations of G', in rel ation to the geometry of nilpotent orbits. We give explicit formulas for th eir K-type decompositions. In particular, for the theta lifts of unitary ch aracters, or holomorphic discrete series with a scalar extreme K'-type, we show that the K structure of the resulting representations of G is almost i dentical to the K-C-module structure of the regular function rings on the c losure of the associated nilpotent K-C-orbits in s, where g = k + s is a Ca rtan decomposition. As a consequence, their associated cycles are multiplic ity free.