J. Austral. Math. Soc.
72 (2002), 257-286
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Some irreducible free group representations in which a linear combination of the generators has an eigenvalue
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William L. Paschke
Department of Mathematics
University of Kansas
405 Snow Hall
Lawrence, KS 66045-2142
USA
paschke@math.ukans.edu
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Abstract
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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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