Journal of the Ramanujan Mathematical Society
Volume 29, Issue 3, September 2014 pp. 321–378.
On the lattice model of the Weil representation and the Howe duality
conjecture
Authors:
Shuichiro Takeda
Author institution:Department of Mathematics, University of Missouri 202 Math Science Bldg 65211
Summary:
The lattice model of the Weil representation over a
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 (O(2n),
Sp(2n)), where 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.
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