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arXiv · 1712.05065

Analytic properties of spherical cusp forms on GL(n)

Valentin Blomer·Gergely Harcos·Péter Maga
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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.

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Valentin Blomer, Gergely Harcos, Péter Maga. 2017-12-14. Analytic properties of spherical cusp forms on GL(n). https://doi.org/10.1007/s11854-020-0094-7

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