Analytic properties of spherical cusp forms on GL(n)

Abstract

Let $\varphi$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of ${\mathrm{PGL}}_n$ over $\mathbb{Q}$ with Laplace eigenvalue ${\lambda }_{\varphi }$. 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 ${\mathrm{GL}}_n(\mathbb{R})$, 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.

18 pages, LaTeX2e, submitted; v2: revised version fixing mainly (45) and its consequences; v3: revised version incorporating suggestions by the referee, e.g. Theorem 2 is now more general than before; v4: final version to appear in Journal d'Analyse Mathématique; v5: small corrections added in proof

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• Hybrid sup-norm bounds for Maass newforms of powerful level
• The sup-norm problem for GL(2) over number fields
• Sup-norm bounds for Eisenstein series
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Cited by in corpus (2)

• Archimedean Newform Theory for ${\mathrm{GL}}_n$
• Bounds for standard $L$-functions