Eisenstein series twisted with non-expanding cusp monodromies

TL;DR

This paper studies twisted Eisenstein series for geometrically finite Fuchsian groups with non-expanding cusp monodromies, proving their convergence and developing Fourier expansions.

Contribution

It establishes convergence and Fourier expansions for twisted Eisenstein series with non-expanding cusp monodromies, extending previous understanding.

Findings

Eisenstein series converge on some half-plane

Fourier-type expansions are developed for these series

Results apply to geometrically finite Fuchsian groups

Abstract

Let $\Gamma$ be a geometrically finite Fuchsian group and suppose that $\chi\colon\Gamma\to\mathrm{GL}(V)$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $\Gamma$ with twist $\chi$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.