Uniform bounds for maximal flat periods on $SL_n(\mathbb{R})$

TL;DR

This paper establishes upper bounds in the spectral aspect for Maass forms on compact locally symmetric spaces associated with SL(n,R), focusing on maximal flat submanifolds and using Euclidean approximation techniques.

Contribution

It provides the first uniform bounds for maximal flat periods on SL(n,R) spaces, extending previous results to higher rank symmetric spaces.

Findings

Derived explicit upper bounds for Maass forms on flat submanifolds

Applied Euclidean approximation to analyze spectral integrals

Extended flat period bounds to higher rank symmetric spaces

Abstract

Let $X$ be a compact locally symmetric space associated to $SL_n(\mathbb{R})$ and $Y \subset X$ a maximal flat submanifold, not necessarily closed. Using a Euclidean approximation, we give an upper bound in the spectral aspect for Maass forms integrated against a smooth cutoff function on $Y$ .