| Affiliation: | Faculty of Integrated Human Studies, Kyoto University, Sakyo, Kyoto 606-8501, Japan ; Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 |
| Abstract: | Let  be a reductive dual pair in the stable range. We investigate theta lifts to  of unitary characters and holomorphic discrete series representations of  , in relation to the geometry of nilpotent orbits. We give explicit formulas for their  -type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme  -type, we show that the  structure of the resulting representations of  is almost identical to the  -module structure of the regular function rings on the closure of the associated nilpotent  -orbits in  , where  is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free. |
