  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.
