Convergence of Kac--Moody Eisenstein Series over a function field

TL;DR

This paper proves the convergence of Eisenstein series on symmetrizable Kac--Moody groups over function fields, extending previous results and methods beyond affine cases and real number analogs.

Contribution

It establishes the first convergence results for these series over function fields, using a novel approach that differs from affine case methods.

Findings

Eisenstein series converge everywhere on symmetrizable Kac--Moody groups over function fields.

The convergence proof does not require additional root system conditions.

Method extends beyond affine cases, applicable over function fields.

Abstract

We establish everywhere convergence in a natural domain for Eisenstein series on a symmetrizable Kac--Moody group over a function field. Our method is different from that of the affine case which does not directly generalize. In comparison with the analogous result over the real numbers, everywhere convergence is achieved without any additional condition on the root system.