Spherical functions and uniformly bounded representations of free groups

Tadeusz Pytlik

Studia Mathematica (1991)

  • Volume: 100, Issue: 3, page 237-250
  • ISSN: 0039-3223

Abstract

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We give a construction of an analytic series of uniformly bounded representations of a free group G, through the action of G on its Poisson boundary. These representations are irreducible and give as their coefficients all the spherical functions on G which tend to zero at infinity. The principal and the complementary series of unitary representations are included. We also prove that this construction and the other known constructions lead to equivalent representations.

How to cite

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Pytlik, Tadeusz. "Spherical functions and uniformly bounded representations of free groups." Studia Mathematica 100.3 (1991): 237-250. <http://eudml.org/doc/215886>.

@article{Pytlik1991,
abstract = {We give a construction of an analytic series of uniformly bounded representations of a free group G, through the action of G on its Poisson boundary. These representations are irreducible and give as their coefficients all the spherical functions on G which tend to zero at infinity. The principal and the complementary series of unitary representations are included. We also prove that this construction and the other known constructions lead to equivalent representations.},
author = {Pytlik, Tadeusz},
journal = {Studia Mathematica},
keywords = {free group; uniformly bounded representation; spherical functions; analytic series of representations; unitary representations},
language = {eng},
number = {3},
pages = {237-250},
title = {Spherical functions and uniformly bounded representations of free groups},
url = {http://eudml.org/doc/215886},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Pytlik, Tadeusz
TI - Spherical functions and uniformly bounded representations of free groups
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 3
SP - 237
EP - 250
AB - We give a construction of an analytic series of uniformly bounded representations of a free group G, through the action of G on its Poisson boundary. These representations are irreducible and give as their coefficients all the spherical functions on G which tend to zero at infinity. The principal and the complementary series of unitary representations are included. We also prove that this construction and the other known constructions lead to equivalent representations.
LA - eng
KW - free group; uniformly bounded representation; spherical functions; analytic series of representations; unitary representations
UR - http://eudml.org/doc/215886
ER -

References

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  1. [1] P. Cartier, Géométrie et analyse sur les arbres, Séminaire Bourbaki 24 (1971/72), Exposé 407. Zbl0267.14010
  2. [2] P. Cartier, Harmonic analysis on trees, in: Proc. Sympos. Pure Math. 26 (1972), 419-424. 
  3. [3] A. Figà-Talamanca and A. M. Picardello, Spherical functions and harmonic analysis on free groups, J. Funct. Anal. 47 (1982), 281-304. Zbl0489.43008
  4. [4] A. Figà-Talamanca and A. M. Picardello, Harmonic Analysis on Free Groups, M. Dekker, New York 1983. Zbl0536.43001
  5. [5] A. M. Mantero, T. Pytlik, R. Szwarc and A. Zappa, Equivalence of two series of spherical representations of a free group, Ann. Mat. Pura Appl., to appear. Zbl0801.22010
  6. [6] A. M. Mantero and A. Zappa, The Poisson transform and representations of a free group, J. Funct. Anal. 51 (1983), 372-399. Zbl0532.43006
  7. [7] A. M. Mantero and A. Zappa, Irreducibility of the analytic continuation of the principal series of a free group, J. Austral. Math. Soc. 43 (1987), 199-210. 
  8. [8] M. Pimsner, Cocycles on trees, J. Operator Theory 17 (1987), 121-128. Zbl0645.46056
  9. [9] T. Pytlik, Radial functions on free groups and a decomposition of the regular representation into irreducible components, J. Reine Angew. Math. 326 (1981), 124-135. Zbl0464.22004
  10. [10] T. Pytlik, Spherical functions and uniformly bounded representations of free groups, preprint 60 (1986), Math. Inst. Univ. Wrocław. Zbl0754.22002
  11. [11] T. Pytlik and R. Szwarc, An analytic family of uniformly bounded representations of free groups, Acta Math. 157 (1986), 287-309. Zbl0681.43011
  12. [12] S. Sawyer, Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete 12 (1978), 279-292. Zbl0362.60075
  13. [13] R. Szwarc, An analytic series of irreducible representations of the free group, Ann. Inst. Fourier (Grenoble) 38 (1) (1988), 87-110. Zbl0634.22003
  14. [14] A. Valette, Cocycles d'arbres et représentations uniformément bornées, C. R. Acad. Sci. Paris 310 (1990), 703-708. Zbl0828.22007

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