An analytic series of irreducible representations of the free group

Szwarc, Ryszard

Szwarc, Ryszard

Annales de l'Institut Fourier, Tome 38 (1988), p. 87-110 / Harvested from Numdam

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Résumé
Abstract

Soit $F_{k}$ un groupe libre avec $k$ générateurs. On construit une série des représentations uniformément bornées $\prod_{z}$ de $F_{k}$ qui opèrent sur un espace de Hilbert commun. Les représentations $\prod_{z}$ sont irréductibles et dépendent analytiquement d’un paramètre complexe $z$ tel que $1 / (2 k - 1) < | z | < 1$ . Pour $z$ réel ou $| z | = 1 / (\sqrt{2 k - 1})$ les $\prod_{z}$ sont unitaires; autrement $\prod_{z}$ ne sont pas unitarisables. Pour $z \neq z^{'}$ les différences $\prod_{z} - \prod_{z^{'}}$ sont des opérateurs compacts.

Let $F_{k}$ be a free group on $k$ generators. We construct the series of uniformly bounded representations $\prod_{z}$ of $F_{k}$ acting on the common Hilbert space, depending analytically on the complex parameter z, $1 / (2 k - 1) < | z | < 1$ , such that each representation $\prod_{z}$ is irreducible. If $z$ is real or $| z | = 1 / (\sqrt{2 k - 1})$ then $\prod_{z}$ is unitary; in other cases $\prod_{z}$ cannot be made unitary. For $z \neq z^{'}$ representations $\prod_{z}$ and $\prod_{z^{'}}$ are congruent modulo compact operators.

Publié le : 1988-01-01

MR 89j:22023 | Zbl 0634.22003

DOI : https://doi.org/10.5802/aif.1124

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     author = {Szwarc, Ryszard},
     title = {An analytic series of irreducible representations of the free group},
     journal = {Annales de l'Institut Fourier},
     volume = {38},
     year = {1988},
     pages = {87-110},
     doi = {10.5802/aif.1124},
     mrnumber = {89j:22023},
     zbl = {0634.22003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1988__38_1_87_0}
}

Szwarc, Ryszard. An analytic series of irreducible representations of the free group. Annales de l'Institut Fourier, Tome 38 (1988) pp. 87-110. doi : 10.5802/aif.1124. http://gdmltest.u-ga.fr/item/AIF_1988__38_1_87_0/

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