Analytic properties of spherical cusp forms on GL(n)

Let $\varphi$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of ${\mathrm{P}\mathrm{G}\mathrm{L}}_n$ over $\mathbb{Q}$ with Laplace eigenvalue ${\lambda }_{\varphi }$. We establish explicit estimates for various quantities related to $\varphi$ that are uniform in ${\lambda }_{\varphi }$. This includes uniforms bounds for spherical Whittaker functions on ${\mathrm{G}\mathrm{L}}_n(\mathbb{R})$, uniform bounds for the global sup-norm of $\varphi$, and uniform bounds for the “essential support” of $\varphi$, i.e. the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools.