Eisenstein series on rank 2 hyperbolic Kac--Moody groups

TL;DR

This paper develops the theory of Eisenstein series on rank 2 hyperbolic Kac--Moody groups over the reals, proving convergence properties and analyzing Fourier coefficients, including series induced from cusp forms.

Contribution

It introduces the definition and analysis of Eisenstein series on rank 2 hyperbolic Kac--Moody groups, including convergence proofs and Fourier coefficient calculations.

Findings

Proved convergence of the constant term of Eisenstein series.

Calculated degenerate Fourier coefficients explicitly.

Showed Eisenstein series from cusp forms are entire functions.

Abstract

We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.