Weak Weyl's law for congruence subgroups

Abstract

Let $G$ be a connected and simply connected semisimple algebraic group over $\mathbb{Q}$ and let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup. Let $K_{\infty }\subset G(\mathbb{R})$ be a maximal compact subgroup and let $d$ be the dimension of the symmetric space $G(\mathbb{R})/K_{\infty }$. Let $\sigma$ be an irreducible unitary representation of $K_{\infty }$. We prove that for every $\Gamma$ there exists a normal subgroup ${\Gamma }_1\subset \Gamma$ of finite index such that the quotient of the counting function of the ${\Gamma }_1$-cuspidal spectrum of weight $\sigma$ and $T^{d/2}$ has a positive lower bound as $T\to \infty$.

13 pages