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

Abstract

We are interested in the L 2 -holomorphic automorphic functions on a gdimensional complex space V g C endowed with a positive definite hermitian form and associated to isotropic discrete subgroups Γ of rank 2 ≤ r ≤ 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 C = Span C (Γ) and the classical Fock-Bargmann space on V g−r 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.