Large values of cusp forms on $$\mathrm {GL}_n$$ GL n
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Selecta Mathematica New Series
Large values of cusp forms on GLn Farrell Brumley1 · Nicolas Templier2
© Springer Nature Switzerland AG 2020
Abstract We establish the transition behavior of Jacquet–Whittaker functions on split semisimple Lie groups. As a consequence, we show that for certain finite volume Riemannian manifolds, the local bound for normalized Laplace eigenfunctions does not hold globally. Keywords Sup-norms · Maass forms · Whittaker functions · Oscillatory integrals · Lagrangian mappings · Pearcey function Mathematics Subject Classification Primary: 11F70; Secondary: 58K55
Contents 1 Statement of results . . . . 2 Outline of proofs . . . . . . 3 Notation and preliminaries . 4 Reduction to local estimates 5 Rapid decay estimates . . . 6 Proof of Theorem 1.6 . . . 7 Lagrangian singularities . . 8 Proof of Theorem 1.4 . . . 9 Proof of Theorem 1.8 . . . 10 Proof of Theorem 1.9 . . . References . . . . . . . . . . .
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Nicolas Templier [email protected] Farrell Brumley [email protected]
1
Institut Galilée, Universite Paris 13, 93430 Villetaneuse, France
2
Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853-4201, USA 0123456789().: V,-vol
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Page 2 of 71
F. Brumley, N. Templier
Let M be a complete d-dimensional Riemannian manifold without boundary. A central question in semiclassical analysis is to understand the concentration features of Laplacian L 2 -eigenfunctions f = λ f , in relation with the geometry of M. A touchstone is the well-known bound of Hörmander [6,36] which implies that | f (x)| λ
d−1 4
f 2 ,
(A)
where the multiplicative constant depends continuously on x ∈ M. This bound is local, being based on the principle that if an eigenfunction is large at a point, it remains so in a small neighborhood. When f is bounded globally on M, one may go further and compare the sup-norm f ∞ = supx∈M | f (x)| with the L 2 -norm, as a function of λ. This boundedness is known to hold on any of the following three classes of manifolds: when both the sectional curvature and the injectivity radius of M are bounded [24]; under certain assumptions on the isoperimetr
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