Analytic Newvectors for GL(n,R) and Applications

Abstract

We introduce an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory formulated by Casselman and Jacquet--Piatetski-Shapiro--Shalika. We relate the analytic conductor of a generic irreducible representation of GL(n,R) to the invariance properties of some special vectors in that representation, which we name analytic newvectors. We also provide a few natural applications of analytic newvectors to some analytic questions concerning automorphic forms for GL(n,Z) in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate for the non-tempered forms, an equidistribution result for the Satake parameters with respect to the Sato--Tate measure, as well as a second moment estimate for the central L-values as strong as Lindeloef on average. We also verify the random matrix prediction concerning the distribution of the low-lying zeros of the Langlands L-functions in the analytic conductor aspect.