  We construct irreducible unitary representations of a finitely generated free
group which are weakly contained in the left regular representation and in
which a given linear combination of the generators has an eigenvalue. When the
eigenvalue is specified, we conjecture that there is only one such
representation. The representation we have found is described explicitly
(modulo inversion of a certain rational map on euclidean space) in terms of a
positive definite function, and also by means of a quasi-invariant probability
measure on the combinatorial boundary of the group.
