Renormalization of integrals of Eisenstein series and analytic continuation of representations

TL;DR

This paper develops a method combining renormalization of automorphic functions and analytic continuation of group representations to analyze triple products of Eisenstein series on hyperbolic surfaces, including non-arithmetic cases.

Contribution

It introduces a novel approach integrating Zagier's renormalization with Bernstein and Reznikov's representation theory for studying Eisenstein series.

Findings

Extended the analysis of Eisenstein series to non-arithmetic hyperbolic surfaces.

Provided new insights into the structure of triple products of Eisenstein series.

Established a framework for future research on automorphic forms and their integrals.

Abstract

We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of $\mathrm{SL}(2,\mathbb{R})$ due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.