On the convergence of Kac-Moody Eisenstein series

TL;DR

This paper proves the almost everywhere and full absolute convergence of Kac-Moody Eisenstein series inside the Tits cone for certain spectral parameters, extending classical results to infinite-dimensional Kac-Moody groups.

Contribution

It establishes convergence properties of Kac-Moody Eisenstein series, including for rank-2 hyperbolic groups, using new techniques adapted to infinite-dimensional groups.

Findings

Proved almost everywhere convergence inside the Tits cone.

Established full absolute convergence for specific Kac-Moody groups.

Extended classical Eisenstein series convergence results to Kac-Moody groups.

Abstract

Let $G$ be a representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We first consider Kac-Moody analogs of Borel Eisenstein series (induced from quasicharacters on the Borel), and prove they converge almost everywhere inside the Tits cone for arbitrary spectral parameters in the Godement range. We then use this result to show the full absolute convergence everywhere inside the Tits cone (again for spectral parameters in the Godement range) for a class of Kac-Moody groups satisfying a certain combinatorial property, in particular for rank-2 hyperbolic groups.