Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields, with an appendix by Paul Garrett

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. Let $\Gamma$ be a nonuniform lattice in $G$. When $\Gamma=P_i^-$, $i=1,2$, the standard parabolic subgroup for the negative $BN$--pair, the quotient graph $P^-_i\backslash X$ is the positive half of the fundamental apartment of $X$, a semi-infinite ray. We define Eisenstein series on $P_1^-\backslash X$. We prove convergence of Eisenstein series in a half space. This uses Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells, which we give. We prove meromorphic continuation of Eisenstein series using the Selberg--Bernstein continuation principle. This requires an analog of integral operators on the Tits building and the classical truncation operator for Eisenstein series, which we construct.