A program for branching problems in the representation theory of real reductive groups

Abstract

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G ′ (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are a priori known to be "nice" in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.

An extension problem related to the fractional Branson–Gover operators

“…For 1 2 < s < n 2 − p one of these representations is the corresponding complementary series representation of O(1, n) onḢ s− 1 2 ,p (R n−1 ), and the restriction mapḢ s,p (R n ) →Ḣ s− 1 2 ,p (R n−1 ) projects onto this component. This observation makes it possible to use the machinery of symmetry breaking operators whose study was recently initiated by Kobayashi [6] (see also [4,7,8,9]). In this language the differential operator ∆ a,p (a = 2(1 − s)) corresponds to the action of the Casimir element of O(1, n) inḢ s,p (R n ) and the fractional Branson-Gover operators are the standard Knapp-Stein intertwining operators between principal series representations of the group O(1, n).…”

“…The irreducible representations of G ′ which occur inside π ∞ λ,p | G ′ are described in terms of so-called symmetry breaking operators (see e.g. Kobayashi [6]). In our setting, a continuous linear operator T :…”

Conformal Symmetry Breaking on Differential Forms and Some Applications

“…Finding a formula of m(Π, π) is a substitute of the branching law Π| G ′ when Π is not a unitary representation. The author proposed in [19] a program for branching problems in the following three stages: Loosely speaking, Stage B concerns a decomposition of representations, whereas Stage C asks for a decomposition of vectors.…”

Analytic Continuation of Equivariant Distributions

“…One can view both [MOO16, MO, Moe] and the ∆H ⊂ G × H-case of Corollary C and Proposition D above as part of the general program of constructing symmetry breaking operators, see [Kob15] and references therein. Some of the operators are constructed in that project through their kernel distribution (as in the present paper), while some others are given by explicit differential operators.…”