The Weil representation in characteristic two

Abstract

In this paper we construct a new variant of the Weil representation, associated with a symplectic vector space (V,ω) defined over a finite field of characteristic two. Our variant is a representation ρ:AMp(V)→GL(H), where the group AMp(V) is the fourth cover of a group ASp(V) which is a non-trivial extension of the symplectic group Sp(V) by the dual group V∗. In particular, the group ASp(V) contains Weil’s pseudo-symplectic group as a strict subgroup. Along the way, we develop the formalism of canonical vector spaces which enables us to realize the group AMp(V) and the Weil representation ρ in a transparent manner and also yields a conceptual explanation for these important objects of representation theory.