  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 $Sp(A)$.
