The Selberg trace formula for $SL(3,\mathbf Z)$

A. B. Venkov

Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part II, Tome 63 (1976), pp. 8-66

Résumé

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space $S$ and a discrete group $\Gamma$ for which the fundamental region $\Gamma\setminus S$ goes to infinity nontrivially appears. For $S$ we use the space $SL(3,\mathbf R)/SO(3)$ and for $\Gamma$ we use $SL(3,\mathbf Z)$. The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator $K$ which appears in the right side of the trace formula of the expression $\int h(\lambda)d\nu^c(\lambda)$, where $\nu^c(\lambda)$ is the continuous part of the spectral measure of the quasiregular representation on the space $L_2(\Gamma\setminus S)$.