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

TL;DR

This work extends the theory of Eisenstein series to rank-2 affine or hyperbolic Kac–Moody groups over finite fields by constructing combinatorial Eisenstein series on the Tits building $X$, a $(q+1)$-regular tree, and analyzing their convergence and meromorphic continuation. The authors leverage a finite spherical Weyl picture via Iwasawa decompositions, define precise Iwasawa cells, and relate them to the tree’s vertices to study spectral data through an adjacency operator with explicit eigenfunctions $\Psi_{i,s}$ and eigenvalues $q^{1-s}+q^{s}$. A robust functional-analytic framework is built, including integral operators on $X$, truncation in the sense of Arthur, and a Bernstein-style continuation principle, to prove meromorphic continuation of the Eisenstein series $E_s$ and to describe the constant term $C_{\mathcal{U}}E_s$. The results generalize classical function-field Eisenstein theory to rank-2 Kac–Moody groups, enabling a spectral decomposition akin to Langlands theory on a combinatorial object, with potential connections to trace formulas and string-theoretic Eisenstein series. The paper also discusses open questions about poles, poles’ locations, and the full spectral picture, highlighting the novelty of rank-2 hyperbolic cases where spherical BN-pairs and a tree structure enable the harmonic-analytic approach used here.

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 $Γ$ be a non-uniform lattice in $G$. When $Γ$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $Γ\backslash X$ 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.

Figures (5)

• Figure 1: Tree of $G$ with coset labels. The 'dots' indicate that the tree is constructed over the field with $q$ elements.
• Figure 2: Local Picture
• Figure 3: The labeling of the standard apartment with its positive and negative halves
• Figure 4: The tree labelled by Iwasawa Cells. The Iwasawa labels contain Weyl group elements of even lengths.
• Figure 5: Local picture of adjacent vertices

Theorems & Definitions (107)

• Lemma 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Corollary 3.3
• Lemma 4.1
• proof : Proof of Lemma \ref{['adjverloc']}
• Lemma 5.1: CG
• Corollary 5.2: CG