Semi-groupe de Lie associé à un cône symétrique

Koufany, Khalid

Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 1-29.

Soit $V$ une algèbre de Jordan simple euclidienne de dimension finie et $\Omega$ le cône symétrique associé. Nous étudions dans cet article le semi-groupe $\Gamma$, naturellement associé à $V$, formé des automorphismes holomorphes du domaine tube $T_{\Omega }:=V+i\Omega$ qui appliquent le cône $\Omega$ dans lui-même.

To any formally real Jordan algebra one may attach a symmetric cone. We study the sub-semigroup of elements of the conformal group which map the cone into itself.

[1] Ph. Bougerol, Kalman filtring with random coefficients and contractions, SIAM. J. Control and Optimization, 31 (1993), 942-959. | Zbl

[2] H. Braun & M. Koecher, Jordan Algebren, Springer, 1975. | Zbl

[3] N. Dörr, On Ol'shanskiĩ semigroup, Math. Ann., 288 (1990), 21-33. | Zbl

[4] J. Faraut, Algèbres de Jordan et cônes symétriques, Écoles d'été du CIMPA, Université de Poitiers, 1989.

[5] J. Faraut, Espaces symétriques ordonnés et algèbres de Volterra, J. of Math. Soc. Japan, 43, 1 (1991), 133-146. | MR | Zbl

[6] J. Faraut, Fonctions sphériques sur un espace symétrique ordonné de type Cayley, Preprint, avril 1994.

[7] J. Faraut & A. Koranyi, Analysis on symmetric cones, À paraître, Oxford University Press. | Zbl

[8] J. Faraut, J. Hilgert & G. Ólafsson, Spherical functions on ordered symmetric spaces, Preprint, July 1993. | Numdam | Zbl

[9] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press. New York-London, 1978. | Zbl

[10] J. Hilgert, K.H. Hofmann & J.D. Lawson, Lie groups, convex cones, and semigroups Oxford University Press, 1989. | MR | Zbl

[11] J. Hilgert, G. Ólafsson & B. Øersted, Hardy spaces on affine symmetric spaces, J. reine angew. Math., 415 (1991), 189-218. | MR | Zbl

[12] M. Koecher, Die Geodätischen von Positivitätsbereichen, Math. Ann., 135 (1958), 192-202. | MR | Zbl

[13] M. Koecher, Jordan algebras and their applications, Lectures notes, Univ. of Minnesota, Minneapolis, 1962. | Zbl

[14] M. Koecher, An elementary approach to bounded symmetric domains, Lectures notes, Rice University, 1969. | MR | Zbl

[15] A. Koranyi, Complex analysis and symmetric domains, École d'été du CIMPA, Université de Poitiers, 1988.

[16] K. Koufany, Semi-groupe de Lie associé à une algèbre de Jordan euclidienne, Doctorat de l'Université de Nancy I, octobre 1993.

[17] K. Koufany, Réalisation des espaces symétriques de type Cayley, C.R. Acad. Sci. Paris, 318, Série I (1994), 425-428. | MR | Zbl

[18] J.D. Lawson, Polar and Ol'shanskiĩ decompositions, Seminaire Sophus Lie, 1, 2 (1991), 163-173. | MR | Zbl

[19] O. Loos, Jordan pairs, Lectures Notes in Math. 460, Springer, 1974.

[20] O. Loos, Bounded symmetric domains and Jordan pairs, Lectures notes, University of California, Irvine, 1977. | Zbl

[21] K.-H. Neeb, Holomorphic representation theory I, Preprint, 1993.

[22] K.-H. Neeb, Holomorphic representation theory II, Preprint, 1993.

[23] K.-H. Neeb, Holomorphic representation theory III, Preprint, 1993.

[24] G. Ólafsson, Causal symmetric spaces, Math. Gotting., 15 1990.

[25] G.I. Ol'Shanskiĩ, Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. Appl., 15 (1982), 275-285. | Zbl

[26] G.I. Ol'Shanskiĩ, Convex cones in symmetric Lie algebras, Lie semigroups and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Soviet Math. Dokl., 26 (1982), 97-101. | Zbl

[27] G.I. Ol'Shanskiĩ, Complex Lie semigroups, Hardy spaces and the Gelfand Gindikin program, Topics in group theory and homological algebra, Varoslavl University Press, 1982, p. 85-98 (Russian); English translation in Differential Geometry and its Applications, 1 (1991), 235-246.

[28] I. Satake, Algebraic structures of symmetric domains, Iwanami-Shoten and Princeton Univ. Press, 1980. | MR | Zbl

[29] E.B. Vinberg, Homogeneous cones, Soviet Math. Dokl., 1 (1960), 787-790. | MR | Zbl

[30] M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergod. Theo. and Dyn. Sys., 5 (1985), 145-161. | MR | Zbl