Concrete description for rank one \((\Gamma, \chi)\)-theta Fock-Bargmann space in high dimension

We deal with the \((\mathbb{Z}\omega, \chi)\)-theta Fock-Bargmann space consisting of holomorphic automorphic functions associated to given discrete subgroup in \(\mathbb{C}^g\) of rank one and given character \(\chi\). We give concrete description of its elements, it looks like a tensor product of a theta Fock-Bargmann space on the complex plane \(\mathbb{C}\) and the classical Fock-Bargmann space on \(\mathbb{C}^{g-1}\). Moreover, we construct an orthonormal basis and we give explicit expression of its reproducing kernel function in terms of Riemann theta function.