  Hidden symmetry of a G'-space X is defined by an extension of the G'-action
on X to that of a group G containing G' as a subgroup. In this setting, we
study the relationship between the three objects:
  (A) global analysis on X by using representations of G (hidden symmetry);
  (B) global analysis on X by using representations of G';
  (C) branching laws of representations of G when restricted to the subgroup
G'.
  We explain a trick which transfers results for finite-dimensional
representations in the compact setting to those for infinite-dimensional
representations in the noncompact setting when $X_C$ is $G_C$-spherical.
Applications to branching problems of unitary representations, and to spectral
analysis on pseudo-Riemannian locally symmetric spaces are also discussed.
