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

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.**Michael Cowling and Uffe Haagerup,*Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one*, Invent. Math.**96**(1989), no. 3, 507–549. MR**996553**, 10.1007/BF01393695**3.**Alessandro Figà-Talamanca and Claudio Nebbia,*Harmonic analysis and representation theory for groups acting on homogeneous trees*, London Mathematical Society Lecture Note Series, vol. 162, Cambridge University Press, Cambridge, 1991. MR**1152801****4.**Alexander Lubotzky,*Trees and discrete subgroups of Lie groups over local fields*, Bull. Amer. Math. Soc. (N.S.)**20**(1989), no. 1, 27–30. MR**945301**, 10.1090/S0273-0979-1989-15686-3**5.**George W. Mackey,*The theory of unitary group representations*, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955; Chicago Lectures in Mathematics. MR**0396826****6.**Robert P. Martin,*Tensor products for 𝑆𝐿(2,𝑘)*, Trans. Amer. Math. Soc.**239**(1978), 197–211. MR**487045**, 10.1090/S0002-9947-1978-0487045-3**7.**Mizan Rahman and Arun Verma,*Product and addition formulas for the continuous 𝑞-ultraspherical polynomials*, SIAM J. Math. Anal.**17**(1986), no. 6, 1461–1474. MR**860927**, 10.1137/0517104**8.**Joe Repka,*Tensor products of unitary representations of 𝑆𝐿₂(𝑅)*, Amer. J. Math.**100**(1978), no. 4, 747–774. MR**509073**, 10.2307/2373909**9.**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504**

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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