论文标题
$ \ mathrm {gl}（n）$的分析新向量的应用
Applications of analytic newvectors for $\mathrm{GL}(n)$
论文作者
论文摘要
我们提供了分析新向量的一些自然应用，以\ cite {jn} arxiv：1911.01880发起，以$ \ mathrm {pgl} _n（\ mathbb {z}）$ N（\ Mathbb {Z}）$ n \ n \ ge 2 $ n \ ge 2 $ $ n agraumen the Archimean condactorconorcontocor in Archimean condactorconor condactorconor the Archimean condactorconor condactorconor contractorconticalticalconticor condactorconor confactorconticalcosor的某些分析形式的某些分析问题。我们证明了傅立叶系数的正交性结果，非速度形式的密度估计，是Satake参数相对于SATO-TATE度量的等分分配结果，以及对平均Lindelöf的中央$ L $ l $ l $ VALUES的第二刻估计值。我们还证明了关于分析导体方面的自动形态$ l $ function的低洼零零的分布的随机矩阵预测。证明的新思想包括使用分析新向量在具有有限导体的自形频谱上构建近似投影仪，以及对Kuznetsov痕量公式的几何侧的柔软局部局部分析（在有限和无限位置）分析。
We provide a few natural applications of the analytic newvectors, initiated in \cite{JN} arXiv:1911.01880, 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.
