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The lattice model of the Weil representation over non-archimedean local field$F$ of odd residual characteristic has been known for decades, and is used toprove the Howe duality conjecture for unramified dual pairs when the residuecharacteristic of $F$ is odd. In this paper, we will modify the lattice modelof the Weil representation so that it is defined independently of the residuecharacteristic. Although to define the lattice model alone is not enough toprove the Howe duality conjecture for even residual characteristic, we willpropose a couple of conjectural lemmas which imply the Howe duality conjecturefor unramified dual pairs for even residual characteristic. Also we will give aproof of those lemmas for certain cases, which allow us to prove (a version of)the Howe duality conjecture for even residual characteristic for a certainclass of representations for the dual pair $({\rm O}(2n), {\rm Sp}(2n))$, where${\rm O}(2n)$ is unramified. We hope this paper serves as a first step toward aproof of the Howe duality conjecture for even residual characteristic.

On the lattice model of the Weil representation and the Howe duality conjecture