Geometric construction of Heisenberg–Weil representations for finite unitary groups and Howe correspondences

Abstract

We give a geometric construction of the Heisenberg–Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for (textrm{Sp}_{2n},textrm{O}_2^-) over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.