Regularized theta lifts for orthogonal groups over totally real fields

TL;DR

This paper introduces a new regularized theta lift for orthogonal groups over totally real fields, generalizing Borcherds' automorphic product construction and connecting harmonic forms to automorphic Green functions.

Contribution

It defines a novel regularized theta lift from SL_2 to orthogonal groups over totally real fields, extending previous automorphic product theories.

Findings

Constructs a lift mapping harmonic Whittaker forms to Green functions.

Shows weakly holomorphic forms correspond to meromorphic modular forms with divisors.

Establishes the relationship between the lift and Kudla-Millson cohomological theta lift.

Abstract

We define a regularized theta lift from SL_2 to orthogonal groups over totally real fields. It takes harmonic `Whittaker forms' to automorphic Green functions and weakly holomorphic Whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.