Construction g\'eometrique de representations de Weil sur un corps fini

We construct, by contraction of a suitable complex vector bundle, the Weil representation of the finite symplectic group $Sp(A)$. We give an explicit description of the space of all lagrangian subspaces, which we use to compute the cocycle of our representation in terms of a geometric Gauss sum. We recover in this way previously constructed generalized Weil representations (see \cite{ast,cor}) by restriction of our representation to an appropiate embedding of $SL(n) $ into $Sp(A)$.