Discretely Decomposable Restrictions of Unitary Representations of Reductive Lie Groups — Examples and Conjectures

Table of Contents § 1. Breaking symmetry and hidden symmetry §2.Analytic and algebraic notion of discretely decomposable restrictions §3.Compact-like actions in infinite dimensional groups §4.Criterion for discretely decomposable restrictions §5.Conjectures on discrete branching laws §1.Breaking symmetry and hidden symmetry 1.1.The purpose of this exposition is to give a survey of the recent study of discrete decomposable restrictions of unitary representations of real reductive Lie groups ([24], [30], [31]).We shall also give some perspectives by examples and open problems.There have been recent developments in connection with restrictions of unitary representations such as: i) finding explicit branching laws ([8], [10], [16], [23], [24], [33], [36], [37], [40], [44], [57]), ii) estimate of multiplicities in the branching laws ([30], [35]), iii) topology of modular varieties in locally Riemannian symmetric spaces ([34]), iv) construction of new discrete series representations for non-symmetric homogeneous spaces ([15], [22], [24], [27], [28], [32], [39]), v) finding explicit parameters >. for which Zuckerman-Vogan's derived functor modules Aq (>.) are non-zero with>.singular in some special settings ([22], §4), vi) existence problem of compact Clifford-Klein forms ofnon-Riemannian homogeneous spaces ([21], [25], [26], [42]).