On holomorphic theta functions associated to rank \(r\) isotropic discrete subgroups of a \(g\)-dimensional complex space

We are interested in the \(L^2\)-holomorphic automorphic functions on a \(g\)-dimensional complex space \(V^g_{\mathbb{C}}\) endowed with a positive definite hermitian form and associated to isotropic discrete subgroups \(\Gamma\) of rank \(2\leq r \leq g\). We show that they form an infinite reproducing kernel Hilbert space which looks like a tensor product of a theta Fock-Bargmann space on \(V^{r}_{\mathbb{C}}=Span_{\mathbb{C}}(\Gamma)\) and the classical Fock-Bargmann space on \(V^{g-r}_{\mathbb{C}}\). Moreover, we provide an explicit orthonormal basis using Fourier series and we give the expression of its reproducing kernel function in terms of Riemann theta function of several variables with special characteristics.