Asymptotic Voronoi summation and Resonance Sums for Automorphic Representations on GL(n)

Let [Formula: see text] be a Galois extension of [Formula: see text] of degree [Formula: see text], and let [Formula: see text] be an automorphic cuspidal representation of [Formula: see text]. Let [Formula: see text] denote the coefficients of the [Formula: see text]-function [Formula: see text] over [Formula: see text]. If the automorphic induction of [Formula: see text] down to [Formula: see text] is known, in this paper a Voronoi summation formula for the coefficients [Formula: see text] is derived using the Meier G-function. As an application, under the assumption that the imaginary part of the sum of the Archimedean parameters of the [Formula: see text]-function over [Formula: see text] vanishes, using well-known results on base change from [1], an asymptotic expansion for the resonance sum of [Formula: see text] against [Formula: see text] is computed when [Formula: see text] is cyclic of prime degree [Formula: see text] and [Formula: see text] is Galois invariant. Consistent with previous cases as in [8] and [31], we show when [Formula: see text] the main term has size [Formula: see text] when the phase [Formula: see text] is close or equal to [Formula: see text] where [Formula: see text] is a positive integer and [Formula: see text] denotes the conductor of [Formula: see text]. When [Formula: see text] we show that the sum decays rapidly, and this exhibits the oscillatory nature of the sequence [Formula: see text] coming from an automorphic representation over a number field.