Theta liftings for $(\mathrm{GL}_n,\mathrm{GL}_n)$ type dual pairs of loop groups

TL;DR

This paper proves that theta liftings of cusp forms on loop groups associated with the dual pair $(\mathrm{GL}_n, \mathrm{GL}_n)$ produce Eisenstein series, revealing a deep connection in the representation theory of loop groups.

Contribution

It establishes that theta liftings in the context of loop groups for the dual pair $(\mathrm{GL}_n, \mathrm{GL}_n)$ result in Eisenstein series, extending classical results to the infinite-dimensional setting.

Findings

Theta liftings of cusp forms are Eisenstein series.

The result extends classical dual pair theory to loop groups.

Provides new insights into automorphic forms on infinite-dimensional groups.

Abstract

In this article we prove the theta liftings of a cusp form on the loop group $\mathrm{GL}_n$ induced from a classical cusp form for the loop group ``dual pair'' $(\mathrm{GL}_n,\mathrm{GL}_n)$ is an Eisenstein series.