Thanks to the works of Howe, Yu, Kim and Hakim-Murnaghan, we have a
parametrization and construction of all supercuspidal representations of
a reductive group over a p-adic field in terms of supercuspidal data, provided
p is sufficiently large compared to the rank of the reductive group.
In this talk, I will define and explain the theta lifts of supercuspidal data via
moment maps. The main result is that local theta correspondences between
supercuspidal representations are completely described by these theta lifts.

This is joint project with Jiajun Ma.