  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.
