Applications of analytic newvectors for $\mathrm{GL}(n)$

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 Lindel\"of 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.