We provide a few natural applications of the analytic newvectors, initiated in Jana and Nelson (Analytic newvectors for \(\text {GL}_n(\mathbb {R})\), arXiv:1911.01880 [math.NT], 2019), to some analytic questions in automorphic forms for \(\mathrm {PGL}_n(\mathbb {Z})\) with \(n\ge 2\), in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato–Tate measure, and a second moment estimate of the central L-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribution of the low-lying zeros of automorphic L-function in the analytic conductor aspect. The new ideas of the proofs include the use of analytic newvectors to construct an approximate projector on the automorphic spectrum with bounded conductors and a soft local (both at finite and infinite places) analysis of the geometric side of the Kuznetsov trace formula.

我们提供了一些在 Jana 和 Nelson 中发起的分析新向量的自然应用（Analytic newvectors for \(\text {GL}_n(\mathbb {R})\) , arXiv:1911.01880 [math.NT], 2019），在阿基米德解析导体方面，对于\(\mathrm {PGL}_n(\mathbb {Z})\)和\(n\ge 2\) 的一些自守形式的分析问题。我们证明了傅立叶系数的正交性结果、非回火形式的密度估计、Satake 参数相对于 Sato-Tate 测度的等分布结果，以及中心L值的二阶矩估计强为林德洛夫平均。我们还证明了关于自守L的低位零点分布的随机矩阵预测-在分析导体方面的功能。证明的新思想包括使用解析新向量在具有有界导体的自守谱上构建近似投影仪，以及对库兹涅佐夫迹公式的几何边的软局部（在有限和无限处）分析。