A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I

Authors:
Donald I. Cartwright, Gabriella Kuhn and Paolo M. Soardi

Journal:
Trans. Amer. Math. Soc. **353** (2001), 349-364

MSC (2000):
Primary 20E08, 20C15; Secondary 22E40

DOI:
https://doi.org/10.1090/S0002-9947-00-02584-8

Published electronically:
September 18, 2000

MathSciNet review:
1707193

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Abstract | References | Similar Articles | Additional Information

Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.

**1.**D.I. Cartwright, G. Kuhn and P.M. Soardi,*A product formula for spherical representations of a group of automorphisms of a homogeneous tree, II*, To appear, Trans. Amer. Math. Soc.**2.**M. Cowling and U. Haagerup,*Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one*, Invent. Math.**96**, 1989, 507-549. MR**90h:22008****3.**A. Figà-Talamanca and C. Nebbia,*Harmonic analysis and representation theory for groups acting on homogeneous trees*, London Mathematical Society Lecture Note Series**162**, Cambridge University Press, Cambridge 1991. MR**93f:22004****4.**A. Lubotzky,*Trees and discrete subgroups of Lie groups over local fields*, Bull. Amer. Math. Soc.**20**(1989), 27-30. MR**89g:22016****5.**G. W. Mackey,*The theory of unitary group representations*, Chicago Lectures in Mathematics, The University of Chicago Press, 1976. MR**53:686****6.**R. P. Martin,*Tensor products for*, Trans. Amer. Math. Soc.**239**(1978), 197-211. MR**80i:22033****7.**M. Rahman and A. Verma,*Product and addition formulas for the continuous -ultraspherical polynomials*, SIAM J. Math. Anal.**17**(1986), 1461-1474. MR**87k:33007****8.**J. Repka,*Tensor products of unitary representations of*, Amer. J. Math.**100**(1978), 747-774. MR**80g:22014****9.**J.-P. Serre,*Trees*, Springer-Verlag, Berlin, Heidelberg, New York, 1980. MR**82c:20083**

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Additional Information

**Donald I. Cartwright**

Affiliation:
School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia

Email:
donaldc@maths.usyd.edu.au

**Gabriella Kuhn**

Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Viale Sarca 202, Edificio U7, 20126 Milano, Italy

Email:
kuhn@matapp.unimib.it

**Paolo M. Soardi**

Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Viale Sarca 202, Edificio U7, 20126 Milano, Italy

Email:
soardi@matapp.unimib.it

DOI:
https://doi.org/10.1090/S0002-9947-00-02584-8

Keywords:
Spherical representation,
homogeneous tree

Received by editor(s):
January 22, 1996

Received by editor(s) in revised form:
April 23, 1999

Published electronically:
September 18, 2000

Article copyright:
© Copyright 2000
American Mathematical Society