The Fourier-Jacobi expansion of the singular theta lift

TL;DR

This paper explicitly evaluates the Fourier-Jacobi expansion of a singular theta lift for dual reductive pairs, providing new product expansions related to Borcherds forms, and adapts Kudla's method for this purpose.

Contribution

It offers an explicit Fourier-Jacobi expansion of a singular theta lift for $U(1,1) imes U(p,q)$, extending Kudla's method and deriving new product expansions for Borcherds forms.

Findings

Explicit Fourier-Jacobi expansion derived

New infinite product expansion for Borcherds forms in $U(p,1)$ case

Method adapted from Kudla's approach

Abstract

Recently, Funke and Hofmann constructed a singular theta lift of Borcherds type for the dual reductive pair $U(1,1)\times U(p,q)$, $p,q\geq 1$, the input functions of which are harmonic weak Maass forms of weight $k= 2-p-q$. In the present paper, we give an explicit evaluation of the Fourier-Jacobi expansion of the lift. For this purpose, we adapt a method introduced by Kudla in his paper 'Another product for a Borcherds form'. As an application, in the case $U(p,1)$ we recover a new infinite product expansion associated to a Borcherds form, analogous to the case $O(p,2)$ treated by Kudla.