Branching laws for square integrable representations

Duflo, Michel; Vargas, Jorge

doi:10.3792/pjaa.86.49

Cited by 28 publications

(59 citation statements)

References 15 publications

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“…This can be computed via Kawasaki's index theorem, formula (7) in [16]. If (η + ρ K )/i is a singular value of Φ, then Paradan and Vergne [40] showed how to still define this index in a meaningful way, essentially by replacing (η + ρ K )/i by a nearby regular value; see Subsection 2.1.…”

Section: Background and Motivationmentioning

confidence: 99%

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A geometric formula for multiplicities of 𝐾-types of tempered representations

Hochs

Song

Shi

2019

Trans. Amer. Math. Soc.

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Let G be a connected, linear, real reductive Lie group with compact centre. Let K < G be compact. Under a condition on K, which holds in particular if K is maximal compact, we give a geometric expression for the multiplicities of the K-types of any tempered representation (in fact, any standard representation) π of G. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π| K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU(p, 1), SO 0 (p, 1) and SO 0 (2, 2) restrict multiplicity-freely to maximal compact subgroups. Contents3 Ingredients of the proof Hecht-Schmid [11] and later also in [6]) is an explicit combinatorial expression for the multiplicities of the K-types of π. For general tempered representations, there exist algorithms to compute these multiplicities. See for example the ATLAS software package 1 and its documentation [1]. This involves representations of disconnected subgroups of G, which cannot be classified via Lie algebra methods. That is one of the reasons why it is a challenge to deduce general properties of multiplicities of K-types of tempered representations from such algorithms. Another reason is the cancellation of terms, that already occurs in Blattner's formula. That can make it hard, for example, to determine which multiplicities are zero.Paradan [35] gave a geometric expression for the multiplicities of the K-types of discrete series representations π. This was based on a version of the quantisation commutes with reduction principle for a certain class of noncompact Spin c -manifolds, and a geometric realisation of π| K based in turn on Blattner's formula and index theory of transversally elliptic operators. The main result in this paper, Theorem 2.7, is a generalisation of Paradan's result to arbitrary tempered representations. (In fact, it applies more generally to standard representations.) This generalisation is now possible, because of a quantisation commutes with reduction result for general noncompact Spin c -manifolds proved recently by the first two authors of this paper [14]. Theorem 2.7 can in fact be generalised to more general compact subgroups K < G; see Corollary 2.8. For tempered representations with regular parameters, the multiplicity formula was proved by Duflo and Vergne [8], via very different methods. Our result has applications to multiplicityfree restrictions of admissible representations.

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Section: Background and Motivationmentioning

confidence: 99%

“…Duflo and Vargas showed that in the case of a discrete series representation π, properness of Φ with K replaced by a possibly noncompact, closed, reductive subgroup H < G is equivalent to the restriction of π to H being admissible (i.e. decomposing into irreducibles with finite multiplicities); see Proposition 4 in [7].…”

Section: Background and Motivationmentioning

confidence: 99%

A geometric formula for multiplicities of 𝐾-types of tempered representations

Hochs

Song

Shi

2019

Trans. Amer. Math. Soc.

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“…• (discretely decomposable case) branching problems may be purely algebraic and combinatorial ( [12,13,15,26,28,29,32,49,50,59]); • (continuous spectrum) branching problems may have analytic features [8,52,57,63]. (For example, some special cases of branching laws of unitary representations are equivalent to a Plancherel-type theorem for homogeneous spaces.…”

Section: A2mentioning

confidence: 99%

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

Kobayashi

2015

Progress in Mathematics

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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.

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“…where the multiplicities m O O 1 are finite. We will use the following geometrical characterization of the G-admissibility obtained by Duflo and Vargas [5,6]. The main result of this article is the following…”

Section: Introductionmentioning

confidence: 99%