Let Γ be a noncommutative free group on finitely many generators. We consider unitary representations of Γ which are weakly contained in the regular representations. Equivalently, these are the "tempered" representations, those whose matrix coefficients are almost in ℓ2. Let Ω be the natural boundary of Γ. A representation which acts in a certain well-defined natural way on some L2-space on Ω is called a boundary representation. All boundary representations are tempered. Conversely, if π is any tempered representation, there is an inclusion of π into some boundary representation. Such an inclusion is a boundary realization of π.
Consideration of examples leads to the duplicity conjecture: a given irreducible tempered representation has at most two inequivalent, irreducible boundary realizations. We give the details of this conjecture.
There are lots of representations of Γ, and one's intuition is that a "generic" representation is irreducibile. However, proving the irreducibility of a specific representation is usually difficult. In many cases, an analysis going by way of boundary realizations works. In certain cases one can prove simultaneously that a representation is irreducible and that it has exactly two inequivalent, irreducible boundary realizations. In other cases one can prove that it has exactly one boundary realization. These theorems are joint work with Waldemar Hebisch.
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In joint work with Gabriella Kuhn we construct a large class of irreducible tempered representations of the free group. Although this class covers almost nothing of the (unmanageably enormous) tempered unitary dual, it does cover most of the provably irreducible tempered representations discussed in the literature. Our methods give a uniform proof of the irreducibility and inequivalence for representations in our class. The duplicity conjecture holds for these representations and the idea of boundary realization is fundamental in the proofs.