Symmetric pairs with finite-multiplicity property for branching laws of admissible representations

Abstract

Abstract: We accomplish the classification of the reductive symmetric pairsðG;HÞ forwhich the dimension of the space Hom H ð j H ; Þ of H-intertwining operators is finite for anyirreducible smooth representation of G and for any irreducible smooth representation of H.Key words: Branching law; restriction of representation; reductive group; real sphericalvariety; symmetric pair.1. Finite-multiplicity in induction andrestriction. One of the basic problems in repre-sentation theory is to understand how a givenrepresentation is decomposed into irreducible rep-resentations. Given a pair of groups G H,thereare two important settings for this problem:I) (Induction) For a simple H-module , un-derstand Ind GH ð Þ as a G-module.II) (Restriction) For a simple G-module ,understand j H as an H-module.We shall highlight the case where G is a realreductive linear Lie group.Concerning Induction Problem (I), a specialcase is the unitary induction Ind GH ð Þ from thetrivial one-dimensional representation ¼ 1 of H,which is unitarily equivalent to the regular repre-sentation of G on L