On irreducible representations of the free group (Q6614516)

Using the results by \textit{C. Pensavalle} and \textit{T. Steger} [Pac. J. Math. 173, No. 1, 181--202 (1996; Zbl 0848.22012)], the author studies tensor products of irreducible unitary representations of a finitely generated free group \(G\) that are weakly contained in the regular representation. In particular, it is proved that, for such a representation \(\eta\) of \(G\) on an infinite-dimensional separable Hilbert space \(M\) (the set of these representations is denoted by \(\operatorname{Rep}(G,M)\) and is naturally equipped with the structure of a complete separable metric space), the set of such representations \(\pi\in\operatorname{Rep}(G,M)\) for which the tensor product \(\eta\otimes\pi\) is irreducible is of second category in \(\operatorname{Rep}(G,M)\). Using anisotropic principal series representations, the author proves that there exists an irreducible boundary realization (see [\textit{W. Hebisch} et al., Trans. Am. Math. Soc. 375, No. 3, 1825--1860 (2022; Zbl 1531.22003)]) on an infinite-dimensional Hilbert space (which is called the fiber of the boundary realization of the representation). Some applications are indicated.