Eisenstein series twisted with non-expanding cusp monodromies

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.