This talk was part of the Workshop on "Minimal Representations and Theta Correspondence" held at the ESI April 11 to 15, 2022.
This talk will consider theta correspondence of type I dual pairs over a finite field F_q. Aubert-Michel-Rouquier established explicit formula on theta correspondence between unipotent representations of unitary groups. They also made a conjecture for the symplectic group-even orthogonal group pair, which was proved by Shu-Yen Pan recently. These works are based on Srinivasan's formula on the uniform projection of the Weil representation. We will present an alternative poof by studying the natural Hecke algebra bi-modules in the theta correspondence. We will show that the normalization of the Hecke algebra is related to the first occurrence index, which leads to yet another proof of the conservation relation. We will compute the specialization at q=1 of the natural generic Hecke bimodule, which recovers AMR and Pan's results. The work is joint with Jialiang Zhou and Congling Qiu.
