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

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