A spectral identity for second moments of Eisenstein series of O(n, 1)

TL;DR

This paper establishes a spectral identity for the second moments of Eisenstein series on O(n, 1), linking periods against forms on a subgroup to fundamental solutions of the Laplacian using advanced functional analysis.

Contribution

It introduces a novel spectral identity for second moments of Eisenstein series on O(n, 1), utilizing Levi-Sobolev spaces and Poincaré series as fundamental solutions.

Findings

Derived a spectral identity relating second moments to Laplacian solutions.

Connected Eisenstein series periods to Poincaré series via spectral analysis.

Provided a new analytical framework for studying automorphic forms on O(n, 1).

Abstract

Let H=O(n)xO(1) be an anisotropic subgroup of G=O(n, 1) and let A be the adele ring of k=Q. Consider the periods of an Eisenstein series on G against a form F on H. Relying on a variant of Levi-Sobolev spaces, we describe certain Poincar\'e series as fundamental solutions for the laplacian, and use them to establish a spectral identity concerning the second moments (in F-aspect) of the Eisenstein series.