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

TL;DR

This paper studies $L^2$-holomorphic automorphic functions on complex spaces associated with isotropic discrete subgroups, revealing their structure as tensor products of theta Fock-Bargmann spaces and providing explicit bases and kernels.

Contribution

It characterizes the space of automorphic functions as a tensor product of theta Fock-Bargmann and classical spaces, with explicit bases and kernel functions expressed via Riemann theta functions.

Findings

The automorphic functions form an infinite reproducing kernel Hilbert space.

Explicit orthonormal basis constructed using Fourier series.

Reproducing kernel expressed in terms of multi-variable Riemann theta functions.

Abstract

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.