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 Olshanskii semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, that is, in the space of smooth sections of a certain line bundle on the 2n-dimensional torus T2n. We show that operators of the representation are integral operators whose kernels are theta-functions on T4n. We also extend this construction to a functor from a certain category of Lagrangian linear relations between symplectic vector spaces of different dimensions to a category of integral operators acting on sections of line bundles on the tori.