题目：Towards an explicit description of local theta correspondence over p-adic fields
摘要：In a joint work with Hung Yean Loke, we described the local theta correspondence between tamely ramified supercuspidal representations using Yu-Kim-Hakim-Murnaghan's parameter when p is sufficiently large. On the other hand, when p not equal to 2, Stevens constructed all supercuspidal representations for classical groups. Furthermore, Miyauchi-Stevens defined the notion of self-dual semisimple types so that Bernstein compoents could be explicitly studied.
In this talk, I will discuss an ongoing work (joint with Loke): We extended our previous result on tamely ramified supercuspidal types to self-dual semisimple types. As a consequence, the correspondence between supercuspidal representations can be explicitly described. The next step is to establish a correspondence between the Hecke algebras associated to these types. We showed that the Hecke algebras related to the positive depth parts are isomorphic. The last missing piece is the depth zero part, which seems could be reduced to the correspondence of unipotent representations.