We relate the analytic conductor of a generic irreducible representation of GL n (R) to the invariance properties of vectors in that representation. The relationship is an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory of Casselman and Jacquet-Piatetski-Shapiro-Shalika. We illustrate how this relationship may be applied in trace formulas to majorize sums over automorphic forms on PGL n (Z)\PGL n (R) ordered by analytic conductor.

We provide a few natural applications of the analytic newvectors, initiated in Jana and Nelson (Analytic newvectors for $$\text {GL}_n(\mathbb {R})$$ GL n ( R ) , arXiv:1911.01880 [math.NT], 2019), to some analytic questions in automorphic forms for $$\mathrm {PGL}_n(\mathbb {Z})$$ PGL n ( Z ) with $$n\ge 2$$ n ≥ 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.

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We prove an asymptotic expansion of the second moment of the central values of the $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L-functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over the family of $\pi $ with analytic conductors bounded by a quantity that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.

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