  The Weil representation of a real symplectic group $Sp(2n,R)$ admits a
canonical extension to a holomorphic representation of a certain complex
semigroup consisting of Lagrangian linear relations (this semigroup includes
the Olshanski semigroup). We obtain the explicit realization of the Weil
representation of this semigroup in the Cartier model, i.e., in the space of
smooth sections of a certain line bundle on the $2n$-dimensional torus
$T^{2n}$. We show that operators of the representation are integral operators
whose kernels are theta-functions on $T^{4n}$.
