Lower bounds for Maass forms on semisimple groups

TL;DR

This paper establishes lower bounds on the growth of sup norms of Laplace eigenfunctions on certain arithmetic manifolds associated with anisotropic semisimple groups, revealing new insights into their spectral behavior.

Contribution

It provides the first lower bounds for the sup norms of eigenfunctions on non-compact arithmetic manifolds linked to non-split, almost simple semisimple groups.

Findings

Existence of eigenfunctions with sup norms growing polynomially with eigenvalue

Lower bounds applicable to a broad class of anisotropic semisimple groups

Advances understanding of quantum unique ergodicity and eigenfunction concentration

Abstract

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_0$. In addition, suppose that $G_{v_0}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated to $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.