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

TL;DR

This paper modifies the lattice model of the Weil representation to be independent of residue characteristic and proposes conjectural lemmas to extend the Howe duality conjecture to even residual characteristic cases.

Contribution

It introduces a residue characteristic independent lattice model of the Weil representation and suggests conjectural lemmas to prove Howe duality for even residual characteristic.

Findings

Proposed conjectural lemmas imply Howe duality for unramified dual pairs in even residual characteristic.

Proved the lemmas for certain cases, establishing Howe duality for specific representations.

Extended the understanding of Weil representation models to include even residual characteristic cases.

Abstract

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 to prove the Howe duality conjecture for unramified dual pairs when the residue characteristic of $F$ is odd. In this paper, we will modify the lattice model of the Weil representation so that it is defined independently of the residue characteristic. Although to define the lattice model alone is not enough to prove the Howe duality conjecture for even residual characteristic, we will propose a couple of conjectural lemmas which imply the Howe duality conjecture for unramified dual pairs for even residual characteristic. Also we will give a proof of those lemmas for certain cases, which allow us to prove (a version of) the Howe duality conjecture for even residual characteristic for a certain class 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 a proof of the Howe duality conjecture for even residual characteristic.