Analytic properties of spherical cusp forms on GL(n)

TL;DR

This paper provides explicit, uniform estimates for spherical cusp forms on GL(n), including bounds on Whittaker functions, the global sup-norm, and decay regions, using combined analytic and arithmetic methods.

Contribution

It introduces new uniform bounds for spherical cusp forms on GL(n), advancing understanding of their analytic and decay properties in relation to Laplace eigenvalues.

Findings

Uniform bounds for spherical Whittaker functions on GL(n, R)

Global sup-norm bounds for cusp forms

Exponential decay regions outside essential support

Abstract

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