This is a joint work with Hung Yean Loke. By the works of Yu, Kim, Hakim-Murnaghan, we have a parameterization and construction of all supercuspidal representations for a reductive p-adic group when p is sufficiently large. In this talk, I will explicitly describe the local theta correspondences between supercuspidal representations in terms of Yu/Kim’s parameters. In fact, a notion of lifting of supercuspidal data will be defined. The definition is via moment maps and theta correspondences over finite fields. As a corollary, the occurrence of a supercuspidal representation becomes an easy problem, modulo the part of theta correspondence over finite fields (which may not always enter the picture).