Multiplicity one theorem in the orbit method

Kobayashi, Toshiyuki; Nasrin, Salma

doi:10.1090/trans2/210/12

Cited by 16 publications

(5 citation statements)

References 5 publications

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“…This section introduces briefly this kind of results obtained jointly with Nasrin [43], which is a counterpart of representation-theoretic results (Theorems 34 and 40).…”

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confidence: 93%

Multiplicity-free Representations and Visible Actions on Complex Manifolds

Kobayashi¹

2005

Publ. Res. Inst. Math. Sci.

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115

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“…This section introduces briefly this kind of results obtained jointly with Nasrin [43], which is a counterpart of representation-theoretic results (Theorems 34 and 40).…”

mentioning

confidence: 93%

Multiplicity-free Representations and Visible Actions on Complex Manifolds

Kobayashi¹

2005

Publ. Res. Inst. Math. Sci.

Self Cite

115

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“…This article is an outgrowth of the manuscript [44] which I did not publish, but which has been circulated as a preprint. From then onwards, we have extended the theory, in particular, to the following three directions: 1) the generalization of our main machinery (Theorem 2.2) to the vector bundle case ( [49]), 2) the theory of 'visible actions' on complex manifolds ( [50,51,52]), 3) 'multiplicity-free geometry' for coadjoint orbits ( [53]). …”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs

Kobayashi

Progress in Mathematics

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Summary. The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh-Gordan-Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua-Kostant-Schmid K-type formula, and the canonical representations in the sense of Vershik-GelfandGraev. Our method works in a uniform manner for both finite and infinite dimensional cases, for both discrete and continuous spectra, and for both classical and exceptional cases.

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“…Links between multiplicity-free restrictions and the orbit method were investigated in [2,4,7,25,32]. A key role here is played by the Corwin-Greenleaf multiplicity function n. For a closed subgroup H < G and coadjoint orbits O H ∈ h * /H and O G ∈ g * /G, this function takes the value…”

Section: Multiplicity-free Restrictionsmentioning

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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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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Multiplicity one theorem in the orbit method

Cited by 16 publications

References 5 publications

Multiplicity-free Representations and Visible Actions on Complex Manifolds

Multiplicity-free Representations and Visible Actions on Complex Manifolds

Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs

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

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