Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $G{L}_m(\mathbb{Z})$

Let $f$ be a full-level cusp form for $G{L}_m(\mathbb{Z})$ with Fourier coefficients $A_f(n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_{m-1})$. In this paper an asymptotic expansion of Voronoi’s summation formula for $f$ is established. As applications of this formula, a smoothly weighted average of $A_f(n,1,\mathrm{.}\mathrm{.}\mathrm{.},1)$ against $e(\alpha \mathrm{\mid }n{\mathrm{\mid }}^{\beta })$ is proved to be rapidly decayed when $0<\beta <1\mathrm{/}m$. When $\beta =1\mathrm{/}m$ and $\alpha$ equals or approaches $\pm m{q}^{1\mathrm{/}m}$ for a positive integer $q$, this smooth average has a main term of the size of $\mathrm{\mid }{A}_f(1,\mathrm{.}\mathrm{.}\mathrm{.},1,q)+A_f(1,\mathrm{.}\mathrm{.}\mathrm{.},1,-q)\mathrm{\mid }{X}^{1\mathrm{/}(2m)+1\mathrm{/}2}$, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients $A_f(n,1,\mathrm{.}\mathrm{.}\mathrm{.},1)$. Similar estimate is also proved for a sharp-cut sum.