Analyse conforme sur les algèbres de Jordan

Abstract. We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair (V, Q), where V is a complex vector space and Q a homogeneous polynomial of degree 4 on V . The manifold Ξ is an orbit of a covering of Conf(V, Q), the conformal group of the pair (V, Q), in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra g, and furthermore a real form g R . The connected and simply connected Lie group G R with Lie(G R ) = g R acts unitarily on a Hilbert space of holomorphic functions defined on the manifold Ξ.

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“…The involution α defined by α(g) = σ •ḡ • σ −1 is a Cartan involution of K (see[19, Proposition 1.1. ]), and…”

mentioning

confidence: 99%

Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Achab

Faraut

2012

Can. j. math.

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“…[128,129] Signed powers χ m ε , m ∈ C, ε = 0, 1 of the character χ of the structure group can be extended to the characters of the parabolic subgroupP which induce a series of representations π m,ε of the whole conformal group. The latter may be realized on the space The problem of irreducibility of these representations rises naturally.…”

Section: Two Problems Of Representation Theory: Induction and Restricmentioning

confidence: 99%

“…This method was generalized in [128,129]. Starting from a representation of a simple Jordan algebra we defined an analogue of the Weil representation of a conformal Lie algebra g and constructed a family of intertwining operators between this representation and the infinitesimal representations of the maximally degenerate principal series.…”

Section: Parabolic Induction and Weil Representationmentioning

confidence: 99%

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Covariant quantization: spectral analysis versus deformation theory

Pevzner

2008

Jpn. J. Math.

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Formal deformation or rather symbolic calculus? To which extent do these approaches complete each other in the study of symmetry-preserving quantization procedures for homogeneous spaces? The representation theory of underlying Lie groups shows that the answer is much more delicate than initially thought and that it cannot be always reduced to asymptotic expansions with respect to some Planck's constant. The main goal of this survey is to give hints regarding the aims of each approach and, on the domain where these intersect, to compare the answers they lead to.

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“…where (Dg) x ∈ Str(V ) is the differential of the conformal map x → g.x at x and n′ ∈ N (see [21] Prop. 1.4).…”

Section: Berezin Kernels On Makarevich Spacesmentioning

confidence: 99%

Berezin kernels and analysis on Makarevich spaces

Faraut¹,

Pevzner²

2005

Indagationes Mathematicae

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Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels.We consider the conformal group Conf(V ) of a simple real Jordan algebra V . The maximal degenerate representations π s (s ∈ C) we shall study are induced by a character of a maximal parabolic subgroupP of Conf(V ). These representations π s can be realized on a space I s of smooth functions on V . There is an invariant bilinear form B s on the space I s . The problem we consider is to diagonalize this bilinear form B s , with respect to the action of a symmetric subgroup G of the conformal group Conf(V ). This bilinear form can be written as an integral involving the Berezin kernel B ν , an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B ν . From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity :where D(ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I −s to I s . Furthermore we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V ).2000 Mathematics Subject Classification. 43A35, 22E46.

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Analyse conforme sur les algèbres de Jordan

Cited by 7 publications

References 13 publications

Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Covariant quantization: spectral analysis versus deformation theory

Berezin kernels and analysis on Makarevich spaces

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