Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields

TL;DR

This paper develops Eisenstein series on rank 2 Kac--Moody groups over finite fields, proving their convergence and meromorphic continuation using harmonic analysis and geometric descriptions of the associated Tits building.

Contribution

It introduces Eisenstein series on these groups, establishes their convergence in a half space, and proves their meromorphic continuation through new integral and truncation operators.

Findings

Eisenstein series converge in a half space

Meromorphic continuation of Eisenstein series established

Construction of integral and truncation operators for analysis

Abstract

Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $\Gamma \backslash X$ and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building $X$ and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein's Continuation Principle.