  By the works of Yu, Kim and Hakim-Murnaghan, we have a parameterization and
construction of all supercuspidal representations of a reductive $p$-adic group
in terms of supercuspidal data, when $p$ is sufficiently large. In this paper,
we will define a correspondence of supercuspidal data via moment maps and theta
correspondences over finite fields. Then we will show that local theta
correspondences between supercuspidal representations are completely described
by this notion. In Appendix B, we give a short proof of a result of Pan on
"depth preservation".
