A spectral identity for second moments of Eisenstein series of $\mathrm{O}(n,1)$

Abstract

Let $H=\mathrm{O}(n)\times\mathrm{O}(1)$ be an anisotropic subgroup of $G=\mathrm{O} (n,1)$ and let $\mathbb{A} $ be the adele ring of $k=\mathbb{Q}$. Consider the periods

\[(E_{\varphi },F)_{H}=\int_{H_{k}\backslash H_\mathbb{A}}E_{\varphi}\cdot{\overline {F}},\]

of an Eisenstein series $E_{\varphi}$ on $G$ against a form $F$ on $H$. Relying on a variant of Levi–Sobolev spaces, we describe certain Poincaré series as fundamental solutions for the Laplacian, and use them to establish a spectral identity concerning the second moments (in $F$-aspect) of $E_{\varphi }$.