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

Subhajit Jana

Published 2020-01-27Version 1

We provide a few natural applications of the analytic newvectors, initiated in arXiv:1911.01880, to some analytic questions in automorphic forms for $\mathrm{PGL}_r(\mathbb{Z})$ with $r\geq 2$, in the archimedean analytic conductor aspect. We prove an orthogonality result of the Hecke eigenvalues, 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. The new ideas of the proofs include use of the 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 in the geometric side of the Kuznetsov trace formula.