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

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 z≠z ′ les différences ∏ z -∏ z ′ sont des opérateurs compacts.

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 z≠z ′ representations ∏ z and ∏ z ′ are congruent modulo compact operators.

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     author = {Szwarc, Ryszard},
     title = {An analytic series of irreducible representations of the free group},
     journal = {Annales de l'Institut Fourier},
     pages = {87--110},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {1},
     year = {1988},
     doi = {10.5802/aif.1124},
     zbl = {0634.22003},
     mrnumber = {89j:22023},
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Szwarc, Ryszard. An analytic series of irreducible representations of the free group. Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 87-110. doi : 10.5802/aif.1124. https://aif.centre-mersenne.org/articles/10.5802/aif.1124/

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