We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on $SL\left(n,\mathbb{Z}\right)\setminus SL\left(n,ℝ\right)∕SO\left(n\right)$. As part of the proof, we establish a uniform upper-bound for spherical functions on semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square $L$-functions, we deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan, including an improvement on the previous literature in the case $n=2$.

中文翻译：

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SL(n, ℝ)∕SO(n) 上 Hecke-Maass 形式族的 Sato-Tate 均衡分布

我们建立了 Hecke-Maass 尖点形式族的 Hecke 特征值的 Sato-Tate 等分布 $SL\left(n,\mathbb{Z}\right)\setminus SL\left(n,ℝ\right)∕所以\left(n\right)$. 作为证明的一部分，我们在具有独立兴趣的半单李群上建立球函数的统一上界。对于主、对称正方形和外正方形中的每一个$升$-functions，我们推导出低位零的有限支持的水平分布。我们还推导出了对拉马努金的平均估计，包括对先前文献的改进$n=2$.