Orthogonal Eisenstein Series and Theta Lifts

TL;DR

This paper demonstrates that certain non-holomorphic Eisenstein series are orthogonal and explores their properties, including meromorphic continuation, Fourier expansion, and functional equations, with implications for the Borcherds lift's image and surjectivity.

Contribution

It provides a new proof of the properties of Eisenstein series via orthogonality and investigates the surjectivity of the Borcherds lift.

Findings

Eisenstein series are orthogonal non-holomorphic Eisenstein series for O(2,l).

They have meromorphic continuation and explicit Fourier expansions.

The paper establishes a functional equation similar to the classical case.

Abstract

We show that the additive Borcherds lift of vector-valued non-holomorphic Eisenstein series are orthogonal non-holomorphic Eisenstein series for $O(2, l)$. Using this we give another proof that they have a meromorphic continuation, calculate their Fourier expansion and show that they have a functional equation analogous to the classical case. Moreover, we will investigate the image of Borcherds lift and give a sufficient condition for surjectivity.