On Theta and Eta Correspondences for Finite Unitary Dual Pairs

It is known that the $\mathrm{\Theta }$-correspondence for a finite reductive dual pair is not one-to-one in general. In this paper, we propose two maximal one-to-one sub-relations $\underset{‾}{\theta },\bar{\theta }$ of the $\mathrm{\Theta }$-correspondences for a finite dual pair of two unitary groups. These two correspondences are extensions of the analogue of the $\eta$-correspondence proposed by Gurevich-Howe.