Concentration properties of theta lifts on orthogonal groups

TL;DR

This paper establishes the existence of Maass forms with large sup norms on orthogonal groups by using a new period relation and counting argument, extending previous methods to higher ranks and improving known bounds.

Contribution

It introduces a novel period relation distinguishing theta lifts on orthogonal groups, generalizing Rudnick and Sarnak's method to higher ranks and providing sharper lower bounds.

Findings

Constructs Maass forms with large sup norms on ${ m O}(n,m)$.

Provides lower bounds expressed via Plancherel measure ratios.

Improves growth exponent results for hyperbolic spaces, aligning with the purity conjecture.

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.