Subconvex bounds for Hecke-Maass forms on compact arithmetic quotients of semisimple Lie groups

TL;DR

This paper extends subconvex bounds for Hecke-Maass forms to compact arithmetic quotients of semisimple Lie groups, covering new classes of automorphic representations and eigenfunctions on curved manifolds, including holomorphic modular forms.

Contribution

It generalizes subconvex bounds to non-trivial $K$-types and compact quotients, providing new bounds for automorphic forms and eigenfunctions on curved spaces.

Findings

Extended subconvex bounds to non-trivial $K$-types

Derived bounds for eigenfunctions on compact manifolds with curvature

Established new bounds for holomorphic modular forms in weight aspect

Abstract

Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $\Gamma\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms on the locally symmetric space $\Gamma \backslash G/K$ to corresponding bounds on the arithmetic quotient $\Gamma \backslash G$ for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial $K$-types, yielding subconvex bounds for new classes of automorphic representations, and constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.