The trace class conjecture in the theory of automorphic forms. II

Abstract.

Let G be the group of real points of a reductive algebraic $\Bbb {Q}$-group satisfying the assumptions made in [H, Chapter I] and let $\Gamma$ be an arithmetic subgroup of G. Let $R_{\Gamma}$ be the right regular representation of G on $L^2(\Gamma \backslash G)$ and denote by $R^d_\Gamma$ the restriction of $R_\Gamma$ to the discrete subspace. In this paper we prove that for every integrable rapidly decreasing function f on G, the operator $R^d_\Gamma (f)$ is of the trace class.