Concentration properties of theta lifts on orthogonal groups

Abstract

Let $n>m⩾1$ be integers with $n+m⩾4$ even. We prove the existence of Maass forms with large sup norms on anisotropic $\mathrm{O}(n,m)$, by combining a counting argument with a new period relation showing that a certain orthogonal period on $\mathrm{O}(n,m)$ distinguishes theta lifts from ${\mathrm{Sp}}_{2m}$. 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 $\mathrm{O}(n,m)$ and ${\mathrm{Sp}}_{2m}(\mathbb{R})$, 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.