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arXiv · 2309.06433

Concentration properties of theta lifts on orthogonal groups

Farrell Brumley·Simon Marshall
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Abstract

Let $n>m\geqslant 1$ be integers with $n+m\geqslant 4$ even. We prove the existence of Maass forms with large sup norms on anisotropic ${\rm O}(n,m)$, by combining a counting argument with a new period relation showing that a certain orthogonal period on ${\rm O}(n,m)$ distinguishes theta lifts from ${\rm Sp}_{2m}$. This generalizes a method of Rudnick and Sarnak in the rank one case, when $m = 1$. Our lower bound is naturally expressed as a ratio of the Plancherel measures for the groups ${\rm O}(n,m)$ and ${\rm Sp}_{2m}(\mathbb{R})$, up to logarithmic factors, and strengthens the lower bounds of our previous paper for such groups. In the case of odd-dimensional hyperbolic spaces, the growth exponent we obtain improves on a result of Donnelly, and is optimal under the purity conjecture of Sarnak.

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Farrell Brumley, Simon Marshall. 2023-09-12. Concentration properties of theta lifts on orthogonal groups. https://arxiv.org/abs/2309.06433

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