A regularized theta lift on the symmetric space of $SL_N$

TL;DR

This paper introduces a new regularized theta lift from harmonic weak Maass forms to differential forms on the symmetric space of SL_N, revealing connections to Fourier coefficients of Hilbert-Eisenstein series.

Contribution

It defines a novel regularized lift on SL_N's symmetric space and establishes its relation to theta lift derivatives and Fourier coefficients.

Findings

The lift is shown to be adjoint to the derivative of a theta lift.

Periods over tori relate to Fourier coefficients of Hilbert-Eisenstein series.

The construction extends the theory of theta lifts to higher rank symmetric spaces.

Abstract

We define a regularized lift from harmonic weak Maass forms of weight $2-N$ to differential forms of degree $N-1$ on the symmetric space $\SL_N(\R)/\SO(N)$, that are smooth outside of certain modular symbols. We show that this lift is adjoint to the derivative of a theta lift. We compute periods of the regularized lift over tori and relate them to Fourier coefficients of Hilbert-Eisenstein series.