Weak Weyl's law for congruence subgroups

Abstract

Let $G$ be a connected and simply connected semisimple algebraic group over $\Bbb Q$ and let $\Gamma\subset G(\Bbb Q)$ be an arithmetic subgroup. Let $K_\infty\subset G(\Bbb R)$ 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 $T\to\infty$.