Large values of Maass forms on hyperbolic Grassmannians in the volume aspect

TL;DR

This paper proves the existence of exceptional Maass forms on certain hyperbolic Grassmannian quotients in the volume aspect, extending previous methods to higher rank symmetric spaces.

Contribution

It generalizes the existence results of exceptional Maass forms to higher rank hyperbolic Grassmannians using advanced counting and period relation techniques.

Findings

Existence of exceptional Maass forms in volume aspect for compact quotients.

Lower bounds expressed as volume ratios with logarithmic factors.

Extension of previous methods to higher rank symmetric spaces.

Abstract

Let $n > m \geq 1$ be integers such that $n+ m \geq 4$ is even. We prove the existence, in the volume aspect, of exceptional Maass forms on compact quotients of the hyperbolic Grassmannian of signature $(n,m)$. The method builds upon the work of Rudnick and Sarnak, extended by Donnelly and then generalized by Brumley and Marshall to higher rank manifolds. It combines a counting argument with a period relation, showing that a certain period distinguishes theta lifts from an auxiliary group. The congruence structure is defined with respect to this period and the auxiliary group is either $U(m,m)$ or $Sp_{2m}(\mathbb{R})$, making $(U(n,m),U(m,m))$ or $(O(n,m),Sp_{2m}(\mathbb{R}))$ a type 1 dual reductive pair. The lower bound is naturally expressed, up to a logarithmic factor, as the ratio of the volumes, with the principal congruence structure on the auxiliary group.