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Special series of unitary representations of groups acting on homogeneous trees

Published online by Cambridge University Press:  09 April 2009

Anna Maria Mantero
Affiliation:
Istituto di Matematica, Via L. B. Alberti, 4 16132 Genova, Italy
Anna Zappa
Affiliation:
Istituto di Matematica, Via L. B. Alberti, 4 16132 Genova, Italy
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Abstract

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Let G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let be the space of functions on the Poission boundary ω, of zero mean on ω. When p is a prime. G is a discrete subgroup of PGL2(Qp) of finite covolume. The representations of the special series of PGL2(Qp), Which are irreducible and unitary in an appropriate completion of , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on and topologically irreducible on endowed with the week topology.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Betori, W. and Pagliacci, M., ‘Harmonic analysis for groups acting on trees’, Boll. Un. Mat. Ital. 63-B (1984), 333349.Google Scholar
[2]Cecchini, C. and Figà-Talamanca, A., ‘Projections of uniqueness for Lp(G)’, Pacific J. Math. 51 (1974), 3747.CrossRefGoogle Scholar
[3]Figà-Talamanca, A. and Picardello, M. A., ‘Spherical functions and harmonic analysis on free groups’, J. Fund. Anal. 47 (1982), 281304.CrossRefGoogle Scholar
[4]Figà-Talamanca, A. and Picardello, M. A., Harmonic analysis on free groups (Lecture Notes in Pure and Applied Math., Vol. 87, Marcel Dekker, New York, 1983).Google Scholar
[5]Figà-Talamanca, A. and Picardello, M. A., ‘Restriction of spherical representations of PGL 2 (Qp) to a discrete subgroup’, Proc. Amer. Math. Soc. 91 (1984), 405408.Google Scholar
[6]Ol'shanskii, G. I., ‘Classification of irreducible representations of groups of automorphisms of Bruhat-Tits trees’, Functional Anal. Appl. 11 (1977), 2634.CrossRefGoogle Scholar
[7]Serre, J. P., ‘Arbres, amalgames, SL 2’, Astérisque 46 (1977).Google Scholar
[8]Silberger, A. J., PGL2 over the p-adics: its representations, spherical functions, and Fourier analysis (Lecture Notes in Math. 166, Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar