Title: Some results of theta correspondence over finite fields
报告人：马家骏 副教授 (上海交通大学)
Abstract: Recently there are tremendous progresses concerning the properties and explicit descriptions of theta correspondence over local and global fields. We list some breakthroughs: 0. The proof of Howe conjecture over p-adic fields (Gan-Takeda, Gan-Sun). 1. The proof of the conservation relation (Sun-Zhu); 2. Descriptions of the Whittaker model and associated cycles of theta lifts, 3. Descriptions of theta correspondence for tempered representations (Atobe-Gan-Ichino); 4. Descriptions of the correspondence between supercuspidal representation (Loke-Ma-Savin,Loke-Ma), etc.
On the other hand, the study of theta correspondence over finite fields is lagging behind. Here we list some milestones: 1. Srinivasan (1979) and Pan (2016) expressed the uniform part of the Weil representation in terms of Deligne-Lusztig characters; 2. Adams-Moy (1993) and Liu-Wang (2018), established the explicit correspondence between unipotent cuspidal representations; 3. Aubert-Michel-Rouquier (1996) and Chavez (2017)
On the explicit description of unipotent representations; 4. Pan (2002) on conservation relations of cuspidal representations,
5. Pan (2016) on the explicit description of the correspondence between cuspidal representations for unitary dual pairs.
In this talk, I will present some results in an ongoing joint work with Binyong Sun:
1. Certain degenerate principle series, Rallies quotients and conservation relation;
2. Correspondence between cuspidal representations
3. A conjecture on the theta lift of a general representation (if time permits).