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 $Γ$ 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 $V^{r}_{\mathbb{C}}=Span_{\mathbb{C}}(Γ)$ 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.

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