J. Austral. Math. Soc.  72 (2002), 257-286
Some irreducible free group representations in which a linear combination of the generators has an eigenvalue

William L. Paschke
  Department of Mathematics
  University of Kansas
  405 Snow Hall
  Lawrence, KS 66045-2142

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.
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