Entirety of certain cuspidal Eisenstein series on Kac--Moody groups

TL;DR

This paper proves that certain cuspidal Eisenstein series on infinite-dimensional Kac--Moody groups are entire functions on the complex plane, extending classical results to a broader algebraic setting.

Contribution

It establishes the entireness of cuspidal Eisenstein series on Kac--Moody groups under specific conditions, a significant generalization of known finite-dimensional cases.

Findings

Cuspidal Eisenstein series are entire on the complex plane for Kac--Moody groups.

The result applies under a natural condition on maximal parabolic subgroups.

Extends classical Eisenstein series theory to infinite-dimensional Kac--Moody groups.

Abstract

Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under a natural condition on maximal parabolic subgroups, we prove that the cuspidal Eisenstein series are entire on the full complex plane.