An analytic series of irreducible representations of the free group
Let \({\mathbb{F}}_ k\) be a free group on k generators. We construct the series of uniformly bounded representations \(\prod _ z\) of \({\mathbb{F}}_ k\) acting on the common Hilbert space, depending analytically on the complex parameter z, \(1/(2k-1)<| z| <1\), such that each representation \(\prod _ z\) is irreducible. If z is real or \(| z| =1/(\sqrt{2k-1})\) 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.
scientific article; zbMATH DE number 4031978
scientific article