Global Analysis by Hidden Symmetry

Abstract

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 Xℂ is Gℂ′ -spherical. Applications to branching problems of unitary representations, and to spectral analysis on pseudo-Riemannian locally symmetric spaces are also discussed.