Let \(\Gamma\) be a discrete subgroup of a Lie group \(G\) for \(\text{GL}_2\). Denote by \(L_{\psi}\) the \(L\)-function attached to \(\Gamma\)-automorphic function \(\psi\), and define \[ {\mathcal M}(L_{\psi}, g)=\int_{-\infty}^\infty| L_{\psi}({1\over 2}+it)| ^2g(t)\,dt, \] where \(g(t)\) is an even, entire, real on \(\mathbb R\), and of rapid decay in any fixed horizontal strip. The author continues his and joint with R. W. Bruggeman numerous investigations on the mean value \({\mathcal M}(L_{\psi}, g)\). More precisely, he considers the spectral decomposition of \({\mathcal M}(L_{\psi}, g)\) for some functions \(L_{\psi}\) in terms of notions from representation theory. Let, as usual, \(\zeta(s)\) denote the Riemann zeta-function, \(\zeta_K(s)\) be the Dedekind zeta-function of the Gaussian number field \(K=\mathbb Q(i)\), and let \(\psi\) be the Eisenstein series.
In the paper, first a new proof of the spectral decomposition for \({\mathcal M}(\zeta^2, g)\) given in [J. Reine Angew. Math. 579, 75–114 (2005; Zbl 1064.11059)] is presented. In this case, \(G=\text{PSL}_2(\mathbb R)\) and \(\Gamma=\text{PSL}_2(\mathbb Z)\). Further, this extended to the complex case \(G=\text{PSL}_2(\mathbb C)\), \(\Gamma=\text{PSL}_2(\mathbb Z[-1])\), and the spectral decomposition for \({\mathcal M}(\zeta_K^2, g)\) is obtained. For this, the author proves two new lemmas on the operator \[ {\mathcal K}\phi(u)=| u| ^{1-\nu}({u /| u| })^p{\mathcal A}_u\phi(1), \] where \({\mathcal A}_u\) is the Jacquet operator. The latter results allows to simplify the proof of the spectral formula for \({\mathcal M}(\zeta_K^2, g)\) given in [Funct. Approximatio, Comment. Math. 31, 23–92 (2003; Zbl 1068.11057)].
For the entire collection see [Zbl 1104.11004].