An analytic series of irreducible representations of the free group
Annales de l'Institut Fourier, Volume 38 (1988) no. 1, p. 87-110

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

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

@article{AIF_1988__38_1_87_0,
     author = {Szwarc, Ryszard},
     title = {An analytic series of irreducible representations of the free group},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {38},
     number = {1},
     year = {1988},
     pages = {87-110},
     doi = {10.5802/aif.1124},
     zbl = {0634.22003},
     mrnumber = {89j:22023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1988__38_1_87_0}
}
Szwarc, Ryszard. An analytic series of irreducible representations of the free group. Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 87-110. doi : 10.5802/aif.1124. http://www.numdam.org/item/AIF_1988__38_1_87_0/

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