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

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.