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
\[ C_\Gamma(\sigma)C_S(\Gamma)\leq \liminf_{\lambda\to\infty} \frac{N_{\text{cusp}}^\Gamma (\lambda,\sigma)} {\lambda^{d/2}}. \]


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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