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It is known that the $\Theta$-correspondence for a finite reductive dual pairis not one-to-one in general. In this paper, we propose two maximal one-to-onesub-relations $\underline\theta,\overline\theta$ of the$\Theta$-correspondences for a finite dual pair of two unitary groups. Thesetwo correspondences are extensions of the analogue of the $\eta$-correspondenceproposed by Gurevich-Howe.

On Theta and Eta Correspondences for Finite Unitary Dual Pairs