Zak transform, Weil representation, and integral operators with theta-kernels

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 $T^{4n}$.