On the existence and temperedness of cusp forms for \(\text{SL}_3(\mathbb{Z})\) (Q2710297)

The author develops a partial trace formula for the quotient \(X= \text{SL}_3(\mathbb{Z}) \setminus \text{SL}_3(\mathbb{R})/ \text{SO}_3(\mathbb{R})\) whose proof circumvents some of the technical difficulties that come up in proving the full Selberg trace formula. As is well known it is very difficult to separate the contribution of the discrete spectrum to the eigenvalue asymptotics from the contribution of the continuous one if the space under consideration is noncompact. An interesting success of the present paper is a technical device for separating the two contributions, and this is used to establish the Weyl asymptotic law for the discrete spectrum of the Laplacian (in accordance with a well-known conjecture of Phillips and Sarnak). The nonconstant eigenfunctions of the Laplacian on \(L^2(X)\) are cusp forms. The existence of odd cusp forms is rather easy to see and the only known even cusp forms on \(X\) are Gelbart-Jacquet lifts on Maass forms on \(\text{SL}_2(\mathbb{Z}) \setminus \mathbb{H}\). NEWLINENEWLINENEWLINEOne of the main results of the paper under review states that those cusp forms on \(X\) that are not Gelbart-Jacquet lifts of Maass forms comprise asymptotically 100\% of the spectrum. The same holds for the cusp forms that are not self-dual in the sense that \(\phi(g)= \phi((g^t)^{-1})\) for all \(g\in \text{SL}_3(\mathbb{R})\). In addition it is shown that the tempered cusp forms also comprise asymptotically 100\% of the spectrum. NEWLINENEWLINENEWLINEThe main technical tools are truncation and the Maass-Selberg relations. To explain the main ideas free from technicalities, the author proves the Weyl law for the eigenvalues of the Laplacian on \(\text{SL}_2(\mathbb{Z}) \setminus \mathbb{H}\) by means of his approach by a partial trace formula.