Representation Theory and Noncommutative Geometry, Paris

The Weil representation for a finite field of characteristic two

12 mars 2025, 15:45

1h

Amphitheater Darboux (IHP)

Orateur

Aurélie Paull (Université de Lorraine)

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 ${\mathbb{F}}_2$.