The \(L^2\) restriction norm of a \(\mathrm{GL}_3\) Maass form
From the introduction: ``In this paper, we study a novel restriction problem for a \(\mathrm{GL}_3\) Maass form restricted to a codimension two submanifold (essentially \(\mathrm{GL}_2 \times \mathbb{R}^+\)). Such a restricted function has nice invariance properties; it is invariant by \(\mathrm{SL}_2(\mathbb{Z})\) on the left and by \(\mathrm{O}_2(\mathbb{R})\) on the right, and it is natural to understand how it fits into the \(\mathrm{GL}_2\) picture. For instance, one can ask what is the inner product of this restricted function with a given \(\mathrm{SL}_2(\mathbb{Z})\) Maass form (or, more generally, we ask for the spectral decomposition). In fact, the Rankin-Selberg \(L\)-function for \(\mathrm{GL}_3 \times \mathrm{GL}_2\) is constructed along these lines. There are many examples of such period integrals giving values of \(L\)-functions''.NEWLINENEWLINE``Our main result is the following.NEWLINENEWLINETheorem. Let \(F\) be a Hecke-Maass form of type \((\nu_1, \nu_2)\) for \(\mathrm{SL}_3(\mathbb{Z})\) that is in the tempered spectrum of \(\Delta\) (meaning \(\mathrm{Re}(\nu_1) = \mathrm{Re}(\nu_2) = 1/3\) or alternately the Langlands parameters \(i\alpha\), \(i\beta\), \(i\gamma\)'' \(\ldots\) ``are purely imaginary), with Laplace eigenvalue \(\lambda_F (\Delta) = 1 + \frac{1}{2} ( \alpha^2 +\beta^2 + \gamma^2)\), and with \(L^2\) norm equal to 1. Then we have NEWLINE\[NEWLINE N(F) := \int_0^\infty \int_{\mathrm{SL}_2(\mathbb{Z}) \backslash \mathcal{H}^2} \left| F \begin{pmatrix} z_2y_1 & \\ & 1 \end{pmatrix} \right|^2 \frac{dx_2 \, dy_2}{y_2^2} \frac{dy_1}{y_1} \ll_\varepsilon \lambda_F(\Delta)^\varepsilon \, \left| A_F(1,1)\right|^2, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE z_2 = \begin{pmatrix} 1& x_2 \\ & 1 \end{pmatrix} \begin{pmatrix} y_2 & \\ & 1 \end{pmatrix} y_2^{-\frac{1}{2}}, NEWLINE\]NEWLINE \(A_F (1,1)\) is the first Fourier coefficient of \(F\), and the implied constant depends only on \(\varepsilon > 0\).'' NEWLINENEWLINE\vdotsNEWLINENEWLINE `` Corollary. Let notation be as in the Theorem. If \(F\) is self-dual, then NEWLINE\[NEWLINE N(F) \ll_\varepsilon \lambda_F(\Delta)^\varepsilon. NEWLINE\]NEWLINE It would be interesting to find an asymptotic for \(N(F)\); it is not clear if our techniques could be modified to lead to such an asymptotic. This would be a delicate problem.''
scientific article; zbMATH DE number 6051022
scientific article