On Theta and Eta Correspondences for Finite Symplectic/Orthogonal Dual Pairs

In this paper, we propose two maximal one-to-one sub-relations $\underset{‾}{\theta },\bar{\theta }$ of the Howe correspondence $\mathrm{\Theta }$ for a finite reductive dual pair consisting of a symplectic group and an orthogonal group. Moreover, we show that both the correspondences $\underset{‾}{\theta }$ and $\bar{\theta }$ are extensions beyond the stable range of the $\eta$-correspondence defined by Gurevich-Howe.