Conjectures on reductive homogeneous spaces

TL;DR

This paper explores open problems in the analysis of symmetries on reductive homogeneous spaces, focusing on non-commutative harmonic analysis, discontinuous groups, and pseudo-Riemannian spaces, extending classical frameworks.

Contribution

It proposes conjectures and addresses open questions related to discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces beyond traditional settings.

Findings

Formulated conjectures on discrete series for non-symmetric spaces

Identified challenges in discontinuous groups beyond Riemannian cases

Outlined analysis approaches for pseudo-Riemannian locally homogeneous spaces

Abstract

We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete series for non-symmetric homogeneous spaces $G/H$; (2) discontinuous group $\Gamma$ for $G/H$ beyond the Riemannian setting; (3) analysis on pseudo-Riemannian locally homogeneous spaces.