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

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.