Orateur
Description
The Weil representation is relatively well understood for local fields or finite fields of odd characteristics. In characteristic two, even in the case of a finite field, it is still not understood.
In this talk, we will present an explicit construction of the Weil representation for a finite field of characteristic two.
After defining the Heisenberg group in this setting, we construct the Weil representation of a two-fold covering of the pseudo-symplectic group.
We obtain explicit formulas for this representation, its character and the associated cocycle.
The pseudo-symplectic group is an extension of the orthogonal group, which is smaller than the symplectic group. Therefore, for the field $\mathbb{F}_2$, using the ring $\mathbb{Z}/4\mathbb{Z}$, we will provide a second construction of the Weil representation, defined on a covering of the affine symplectic group. All along, we will illustrate our results with the example of a two-dimensional vector space over $\mathbb{F}_2$.
