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 𝔽 q {\mathbb{F}_{q}} . Let X = X q + 1 {X=X_{q+1}} be the Tits building of G , the ( q + 1 ) {(q+1)} -homogeneous tree, and let Γ be a non-uniform lattice in G . When Γ is a standard parabolic subgroup for the negative BN -pair, we define Eisenstein series on Γ \ X {\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.