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

Abstract

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