## Weak Weyl’s law for congruence subgroups.(English)Zbl 1162.11344

Summary: Let $$G$$ be a connected and simply connected semisimple algebraic group over $$\mathbb Q$$. Let $$\Gamma\subset G(\mathbb Q)$$ be an arithmetic subgroup, $$K_\infty\subset G(\mathbb R)$$ a maximal compact subgroup and $$\sigma$$ an irreducible unitary representation of $$K_\infty$$. Denote by $$N_{\text{cusp}}^\Gamma (\lambda,\sigma)$$ the counting function of the eigenvalues of the Casimir operator acting in the space of $$\Gamma$$-cusp forms of weight $$\sigma$$. Let $$C_\Gamma$$ be Weyl’s constant and set $$C_\Gamma(\sigma)= \dim(\sigma)C_\Gamma$$. Let $$d=\dim G(R)/K_\infty$$. A conjecture of Sarnak states that Weyl’s law holds for $$\Gamma$$, i.e.,
$\lim_{\lambda\to\infty} \frac{N_{\text{cusp}}^\Gamma (\lambda,\sigma)} {\lambda^{d/2}}= C_\Gamma(\sigma).$
We prove the following weaker result: Let $$S$$ be a finite set of primesq containing at least two finite primes. Then there exists a constant $$C_S(\Gamma)\leq 1$$ which is nonzero for $$\Gamma$$ a deep enough congruence subgroup, such that
$C_\Gamma(\sigma)C_S(\Gamma)\leq \liminf_{\lambda\to\infty} \frac{N_{\text{cusp}}^\Gamma (\lambda,\sigma)} {\lambda^{d/2}}.$

### MSC:

 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11F75 Cohomology of arithmetic groups 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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