The Weil representation for a finite field of characteristic two
Résumé
We study the Weil representations associated to a finite field $\mathbb{F}$ of characteristic two. Starting from a non-degenerate symplectic form on a finite-dimensional vector space $W$ over $\mathbb{F}$, we consider any associated bilinear form $B$ on $W$ and the corresponding Heisenberg group $\operatorname{H}(B)$. The pseudo-symplectic group $\operatorname{Ps}(B)$ acts on $\operatorname{H}(B)$ by automorphisms leaving its center fixed. Let $W=X\oplus Y$ be a complete polarization and let $\widetilde\chi$ be a character of the abelian subgroup $X\times\mathbb{F}$ of $\operatorname{H}(B)$ having non-trivial restriction $\chi$ to $\mathbb{F}$. From this data, we construct the projective Weil representation of $\operatorname{Ps}(B)$. We linearize this representation and define the Weil representation of a two-fold covering $\widetilde{\operatorname{Ps}(B)}_{\widetilde\chi}$ of $\operatorname{Ps}(B)$. All the formulas we obtain are explicit. In particular, we exhibit explicit formulas for the projective cocycle and the character of the Weil representation. Finally, we use our results to give the complete description of the two-dimensional case $W\simeq\mathbb{F}_2^2$.
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