no. 6
Linearization of Poisson actions and singular values of matrix products
[Linéarisation des actions de Poisson et valeurs singulières de produits de matrices]
Alekseev, Anton1 ; Meinrenken, Eckhard2 ; Woodward, Chris3
1 Université de Genève, Section de Mathématiques, 2-4 rue du Lièvre, Case Postale 240, 1211 Genève 24 (Suisse)
2 University of Toronto, Department of Mathematics, 100 St George Street, Toronto, Ont. (Canada)
3 Rutgers University, Mathematics, Hill Center,110 Frelinghuysen road, Piscataway NJ 08854-8019 (USA )
Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1691-1717.

Nous démontrons que le foncteur de linéarisation de la catégorie des K-actions hamiltoniennes vers celle des K-actions à valeurs dans un groupe de Lie (définie par J.- H. Lu) préserve l’opération de produit à isomorphismes symplectiques près. Ceci donne une nouvelle démonstration de la conjecture de Thompson sur les valeurs singulières des produits de matrices, et l’extension de cette conjecture au cas des matrices réelles. Nous donnons une formule pour les volumes de ces espaces, et obtenons la version hyperbolique de l’isomorphisme de Duflo.

We prove that the linearization functor from the category of Hamiltonian K-actions with group-valued moment maps in the sense of Lu, to the category of ordinary Hamiltonian K- actions, preserves products up to symplectic isomorphism. As an application, we give a new proof of the Thompson conjecture on singular values of matrix products and extend this result to the case of real matrices. We give a formula for the Liouville volume of these spaces and obtain from it a hyperbolic version of the Duflo isomorphism.

DOI : 10.5802/aif.1871
Classification : 53D20, 15A18
Keywords: moment maps, Poisson-Lie groups, singular values
Mot clés : applications du moment, groupes de Lie-Poisson, valeurs singulières

Alekseev, Anton 1 ; Meinrenken, Eckhard 2 ; Woodward, Chris 3

1 Université de Genève, Section de Mathématiques, 2-4 rue du Lièvre, Case Postale 240, 1211 Genève 24 (Suisse)
2 University of Toronto, Department of Mathematics, 100 St George Street, Toronto, Ont. (Canada)
3 Rutgers University, Mathematics, Hill Center,110 Frelinghuysen road, Piscataway NJ 08854-8019 (USA ) 
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Alekseev, Anton; Meinrenken, Eckhard; Woodward, Chris. Linearization of Poisson actions and singular values of matrix products. Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1691-1717. doi : 10.5802/aif.1871. https://aif.centre-mersenne.org/articles/10.5802/aif.1871/

[1] A. Alekseev On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom., Volume 45 (1997) no. 2, pp. 241-256 | MR | Zbl

[2] A. Alekseev; A. Malkin; E. Meinrenken Lie group valued moment maps, J. Differential Geom., Volume 48 (1998) no. 3, pp. 445-495 | MR | Zbl

[3] A. Berenstein; R. Sjamaar Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion, J. Amer. Math. Soc., Volume 13 (2000), pp. 433-466 | DOI | MR | Zbl

[4] P. Boalch Stokes matrices and Poisson Lie groups (2000) (Preprint, SISSA)

[5] V. G. Drinfeld Quantum groups, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Volume Vol. 1, 2 (1987), pp. 798-820

[6] J. J. Duistermaat Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution, Trans. Amer. Math. Soc., Volume 275 (1983), pp. 412-429 | MR | Zbl

[7] S. Evens; J.-H. Lu Poisson harmonic forms, Kostant harmonic forms, and the S 1 -equivariant cohomology of K/T, Adv. Math., Volume 142 (1999) no. 2, pp. 171-220 | MR | Zbl

[8] H. Flaschka; T. Ratiu A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. Ecole Norm. Sup., Volume 29 (1996) no. 6, pp. 787-809 | EuDML | Numdam | MR | Zbl

[9] W. Fulton Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki (Astérisque (1998)), Volume n°252, exp. 845 (1997/98), pp. 255-269 | EuDML | Numdam | Zbl

[10] S. Helgason Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-Londonc, 1962 | MR | Zbl

[11] S. Helgason Geometric analysis on symmetric spaces, American Mathematical Society, Providence, RI, 1994 | MR | Zbl

[12] J. Hilgert; K.H. Neeb Poisson Lie groups and non-linear convexity theorems, Math. Nachr., Volume 191 (1998), pp. 153-187 | DOI | MR | Zbl

[13] M. Kapovich; B. Leeb; J. Millson Polygons in symmetric spaces and euclidean buildings (Preprint, in preparation)

[14] A. Klyachko Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), Volume 4 (1998) no. 3, pp. 419-445 | DOI | MR | Zbl

[15] A. Klyachko; 1-3 Random walks on symmetric spaces and inequalities for matrix spectra, Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Sprectral Problem (Coimbra, 1999) (Linear Algebra Appl.), Volume 319 (2000), pp. 37-59 | Zbl

[16] S. Levendorski; Y. Soibelman Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys., Volume 139 (1991) no. 1, pp. 141-170 | DOI | MR | Zbl

[17] J.-H. Lu Momentum mappings and reduction of Poisson actions, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) (1991), pp. 209-226 | Zbl

[18] J.-H. Lu Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat Poisson structure on G/B, Transform. Groups, Volume 4 (1999) no. 4, pp. 355-374 | DOI | MR | Zbl

[19] J.-H. Lu; T. Ratiu On the nonlinear convexity theorem of Kostant, J. Amer. Math. Soc., Volume 4 (1991) no. 2, pp. 349-363 | DOI | MR | Zbl

[20] J.-H. Lu; A. Weinstein Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom., Volume 31 (1990) no. 2, pp. 501-526 | MR | Zbl

[21] L. O' Shea; R. Sjamaar Moment maps and Riemannian symmetric pairs, Math. Ann., Volume 317 (2000) no. 3, pp. 415-457 | DOI | MR | Zbl

[22] F. Rouvière Espaces symétriques et méthode de Kashiwara-Vergne, Ann. Sci. École Norm. Sup. (4), Volume 19 (1986) no. 4, pp. 553-581 | Numdam | MR | Zbl

[23] C. Torossian L'homomorphisme de Harish-Chandra pour les paires symétriques orthogonales et parties radiales des opérateurs différentiels invariants sur les espaces symétriques, Bull. Soc. Math. France, Volume 126 (1998) no. 3, pp. 295-354 | Numdam | MR | Zbl

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