Abstract
Let φ be an L2-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of PGLn over ℚ with Laplace eigenvalue λφ. We establish explicit estimates for various quantities related to φ that are uniform in λφ. This includes uniform bounds for spherical Whittaker functions on GLn(ℝ), uniform bounds for the global sup-norm of φ, and uniform bounds for the “essential support” of φ, i.e., the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools.
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