Linearization of Poisson actions and singular values of matrix products

Alekseev, Anton; Meinrenken, Eckhard; Woodward, Chris

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.

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

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