Conjectures on reductive homogeneous spaces

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 $G\mathrm{/}H$; (2) discontinuous group $\mathrm{\Gamma }$ for $G\mathrm{/}H$ beyond the Riemannian setting; (3) analysis on pseudo-Riemannian locally homogeneous spaces.