Existence and Weyl’s law for spherical cusp forms

Abstract

Let G be a split adjoint semisimple group over $${\user2{\mathbb{Q}}} $$ and $$ K _\infty \subset \mathbf{G} \user2{\mathbb{(R)}} $$ a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of $$ {\mathbf{G}}({\user2{\mathbb{R}}})/K_{\infty } $$ . This proves a conjecture of Sarnak for $$ \user2{\mathbb{Q}} $$ -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula.