Likewise theta functions of rank $r$ on ${\mathbb{R}}^d$: analytic properties and associated Segal-Bargmann transform

Abstract

We introduce and study the Hilbert space of $(L^2,\Gamma ,\chi )$-likewise theta functions on ${\mathbb{R}}^d$ with respect to a given discrete subgroup $\Gamma$ of arbitrary rank and a character $\chi$ of $\Gamma$. A concrete description is given and an orthonormal basis is then constructed. Its range by the classical Segal-Bargmann transform is also characterized and leads to the so-called theta-Bargmann Fock space.

14 pages