Causal compactification and Hardy spaces

Ólafsson, Gestur; Ørsted, Bent

doi:10.1090/s0002-9947-99-02101-7

Cited by 10 publications

(4 citation statements)

References 13 publications

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“…It was shown by Ol'shanskii [80] and Stanton [98] that this "holomorphic" part of the discrete spectrum can be realized as a Hardy space of holomorphic functions on a local tube domain. Those results were generalized to symmetric spaces of Hermitian type (or compactly causal symmetric spaces) in a series of papers [75,76,34,79,2] The last theorem shows in particular that the corresponding highest weight modules are unitary. It was shown by Wallach [103] and Rossi and Vergne [102] that those are not all the unitary highest weight modules.…”

Section: Highest Weight Modulesmentioning

confidence: 99%

Unitary representations and Osterwalder-Schrader duality

Jørgensen¹,

Ólafsson²

2000

The Mathematical Legacy of Harish-Chandra

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The notions of reflection, symmetry, and positivity from quantum field theory are shown to induce a duality operation for a general class of unitary representations of Lie groups. The semisimple Lie groups which have this c-duality are identified and they are placed in the context of Harish-Chandra's legacy for the unitary representations program. Our paper begins with a discussion of path space measures, which is the setting where reflection positivity (Osterwalder-Schrader) was first identified as a useful tool of analysis.Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe. -Jacques HadamardBoth authors are supported in part by the U.S. National Science Foundation. The second author was supported by LEQSF grant (1996-99)-RD-A-12. 1 UNITARY REPRESENTATIONS AND O-S DUALITY 3of P n by passing over to imaginary time. This idea was used in the paper by J. Fröhlich, K. Osterwalder, and E. Seiler in [15], see also [50], to construct quantum field theoretical systems using Euclidean field theory. In this paper we will give some general constructions and ideas related to this problem in the context of the applications (i)-(v) mentioned above, and work out some simple examples.In [95] R. Schrader used this idea to construct, from a complementary series representation of SL(2n, C), a unitary representation of the group SU (n, n) × SU (n, n). In that paper the similarities to the Yang-Baxter relation were also discussed, a theme that we will leave out in this exposition. What was missing in R. Schrader's paper was the identification of the resulting representations and a general procedure how to construct those representations. We will see that we can do this for all simple Lie groups where the associated Riemannian symmetric space G/K is a tube domain, and the that the duality works between complementary series representations and highest weight representations.In general this problem can be formulated in terms of c-duality of Lie groups and the analytic continuation of unitary representations from one real form to another. (See [33] for these terms.) The representations that show up in the case of semisimple groups are generalized principal series representations on the one side and highest weight representations on the other. The symmetric spaces are the causal symmetric spaces, and the duality is between non-compactly causal symmetric spaces and compactly causal symmetric spaces. The latter correspond bijectively to real forms of bounded symmetric domains. Therefore both the geometry and the representations are closely related to the work of Harish-Chandra on bounded symmetric domains and the holomorphic discrete series [21,22,23]. But the ideas are also related to the work of I. Segal and S. Paneitz through the notion of causality and invariant cones, [96,86,87]. A more complete exposure can be found in the joint paper by the coauthors: Unitary Representations of Lie Groups with Reflection Symmetry, [46].There are other interesting and related questions, problems, and directions...

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Section: Highest Weight Modulesmentioning

confidence: 99%

Unitary representations and Osterwalder-Schrader duality

Jørgensen¹,

Ólafsson²

2000

The Mathematical Legacy of Harish-Chandra

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“…Example 7. Cayley-type spaces are considered in [Ólafsson and Ørsted 1999;Faraut and Korányi 1994]. These are…”

Section: More Examplesmentioning

confidence: 99%

Untitled

2016

Pacific J. Math.

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“…Further applications of the original Wolf-Korányi theory include unitary highest weight representations [23,11,2]; Poisson integrals [30,24,33,6]; Hardy spaces on various domains [9,50,3]; parahermitian or Cayley type symmetric spaces [25,26]; Toeplitz operators [58,60].…”

Section: Introductionmentioning

confidence: 99%

Boundary orbit strata and faces of invariant cones and complex Ol’shanskiĭ semigroups

Alldridge¹

2011

Trans. Amer. Math. Soc.

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Abstract. Let D = G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification G C , and there exists a closed complex subsemigroup G ⊂ Γ ⊂ G C , the so-called minimal Ol'shanskiȋ semigroup, characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to Γ • .Parallel to the classical theory of boundary strata for the symmetric domain D, due to Wolf and Korányi, we give a detailed and complete description of the K-orbit type strata of Γ as K-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra.

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Causal compactification and Hardy spaces

Cited by 10 publications

References 13 publications

Unitary representations and Osterwalder-Schrader duality

Unitary representations and Osterwalder-Schrader duality

Untitled

Boundary orbit strata and faces of invariant cones and complex Ol’shanskiĭ semigroups

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