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Some irreducible free group representations in which a linear combination of the generators has an eigenvalue

Part of: Locally compact groups and their algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

William L. Paschke
William L. Paschke
Affiliation:
Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045-2142, USA e-mail: 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.

MSC classification

Secondary: 46L54: Free probability and free operator algebras 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 257 - 286
Copyright
Copyright © Australian Mathematical Society 2002

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