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.