Probability Theory
Realization of Virasoro unitarizing measures on the set of Jordan curves
[Réalisation de mesures unitarisantes sur l'ensemble des courbes de Jordan]
Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 429-434.

Deux fonctions univalentes sont équivalentes, fg, si elles ont même dérivée Schwarzienne. La relation d'équivalence ∼ étant définie à une transformation homographique près, on obtient un isomorphisme entre la variété 𝒥 des courbes de Jordan et la variété quotient 𝒮 ˜. Cela permet de déduire des champs de vecteurs sur 𝒮 ˜ et sur 𝒥. On explicite l'action de ces champs de vecteurs sur les polynômes de Neretin. On étudie l'existence de mesures unitarisantes sur le quotient de l'ensemble des fonctions univalentes par cette relation d'équivalence et pour une telle mesure, on établit des relations d'orthogonalité entre les polynômes de Neretin. Ce travail est une réalisation concrète du quotient Diff (S 1 )/ SL (2,R) de Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626 sur l'espace complexe C produit d'une infinité dénombrable de C.

Two univalent functions are equivalent, fg, if they have the same Schwarzian derivative. The equivalence relation ∼ being defined up to an homographic transformation, it gives an isomorphism between the manifold 𝒥 of Jordan curves and the quotient manifold 𝒮 ˜. It permits to obtain vector fields on 𝒮 ˜ and on 𝒥. The action of these vector fields on the Neretin polynomials is explicited. The existence of a unitarizing measure on the quotient manifold 𝒮 ˜ is discussed and for such a measure, orthogonality relations for the Neretin polynomials are obtained. This work is a concrete realization on the complex space C of the abstract quotient Diff (S 1 )/ SL (2,R) considered in Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626.

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DOI : 10.1016/S1631-073X(03)00085-2
Airault, Helene 1, 2 ; Bogachev, Vladimir 3

1 INSSET, Université de Picardie, 48, rue Raspail, 02100 Saint-Quentin (Aisne), France
2 Laboratoire CNRS UMR 6140, LAMFA, 33, rue Saint-Leu, 80039 Amiens, France
3 Dept. Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia
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Airault, Helene; Bogachev, Vladimir. Realization of Virasoro unitarizing measures on the set of Jordan curves. Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 429-434. doi : 10.1016/S1631-073X(03)00085-2. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00085-2/

[1] Airault, H. Mesure unitarisante; algèbre de Heisenberg, algèbre de Virasoro, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 787-792

[2] Airault, H.; Malliavin, P. Unitarizing probability measures for representations of Virasoro algebra, J. Math. Pures Appl., Volume 80 (2001) no. 6, pp. 627-667

[3] Airault, H.; Malliavin, P.; Thalmaier, A. Support of Virasoro unitarizing measures, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 621-626

[4] Airault, H.; Ren, J. An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. (2002)

[5] Bogachev, V. Cours à l'Université de Pise. Differentiable measures and the Malliavin calculus (Mai 1995), Scuola Normale Superiore, Pisa, J. Math. Sci., Volume 87 (1997) no. 4, pp. 3577-3731

[6] Kirillov, A.A. Geometric approach to discrete series of unireps for Virasoro, J. Math. Pures Appl., Volume 77 (1998), pp. 735-746

[7] Lehto, O. Univalent Functions and Teichmüller Spaces, Graduate Texts in Math., 109, Springer-Verlag, 1987

[8] Neretin, Yu.A. Holomorphic extensions of representations of the group of diffeomorphisms of the circle, Math. USSR-Sb., Volume 67 (1990) no. 1, pp. 75-96

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