We consider a reductive dual pair (G, G') in the stable range with G' the smaller member and of Hermitian symmetric type. We study the theta lifting of (holomorphic) nilpotent K'ℂ-orbits in relation to the theta lifting of unitary lowest weight representations of G'. We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under the theta lifting. We also develop a theory for the lifting of covariants arising from double fibrations by affine quotient maps.
Kyo Nishiyama. Chen-Bo Zhu. "Theta lifting of unitary lowest weight modules and their associated cycles." Duke Math. J. 125 (3) 415 - 465, 1 December 2004. https://doi.org/10.1215/S0012-7094-04-12531-X