Global analysis by hidden symmetry

TL;DR

This paper explores the concept of hidden symmetry in G'-spaces, analyzing how it influences global analysis, representation theory, and branching laws, with applications to unitary representations and spectral analysis on symmetric spaces.

Contribution

It introduces a method to transfer finite-dimensional representation results to infinite-dimensional cases in noncompact settings with spherical varieties.

Findings

Transfer of results from compact to noncompact settings.

Application to branching laws of unitary representations.

Spectral analysis on pseudo-Riemannian symmetric spaces.

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