On the spectral expansion of hyperbolic Eisenstein series (Q2267763)

Let \(\Gamma\) be a Fuchsian group of the first kind and \(X=\Gamma\backslash \mathbb H^2\) be the induced finite area hyperbolic surface. To any hyperbolic primitive element \(\gamma\in\Gamma\) with invariant geodesic line \(L_\gamma\) in the hyperbolic plane \(\mathbb H ^2\) is associated the hyperbolic Eisenstein series \[ {\mathcal E}_\gamma(s,z)=\sum_{\eta\in \langle\gamma\rangle\backslash\Gamma} \cosh(d_{\mathbb H^2}(\eta z,L_\gamma))^{-s} \] with \(z\in\mathbb H^2\) which is absolutely convergent for \(\mathrm{Re} s>1\) : it is \(\Gamma\)-invariant and the induced function \({\mathcal E}_C(s)\) on the surface \(X\), which depends only on the closed geodesic \(C\) obtained by projection of the line \(L_\gamma\) on \(X\), is proved to be square integrable. The authors analyse the spectral expansion of \({\mathcal E}_C(s)\) with respect to the Laplace-Beltrami operator and deduce its meromorphic continuation with poles and residues description. The hyperbolic Eisenstein series considered here and inducing functions on the surface \(X\) are analogous to the hyperbolic Eisenstein series introduced by \textit{S. Kudla} and \textit{J. Millson} [Invent. Math. 54, 193--211 (1979; Zbl 0429.30038)] inducing 1-forms on the hyperbolic space form \(X\); parabolic Eisenstein series induce usual eigenfunctions used for the spectral resolution of space forms with cusps.