A characterization of spherical series representations of the free group

Kuhn, Gabriella; Steger, Tim

doi:10.1090/S0002-9939-1991-1064905-8

A characterization of spherical series representations of the free group

Authors: Gabriella Kuhn and Tim Steger
Journal: Proc. Amer. Math. Soc. 113 (1991), 1085-1096
MSC: Primary 22D10; Secondary 20E05, 43A65
DOI: https://doi.org/10.1090/S0002-9939-1991-1064905-8
MathSciNet review: 1064905
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Abstract: The spherical series representations of a free nonabelian group are here characterized as those irreducible unitary representations possessing a certain type of matrix coefficient. We make some conjectures on the problem of finding a natural boundary realization for a general irreducible representation of the free group, supposed to be weakly contained in the regular representation.

[Angelini] F. Angelini, Rappresentazioni di un gruppo libero associate ad una passegiata a caso, undergraduate thesis submitted at the Università degli Studi di Roma "La Sapienza," 1989.
[Bishop-Steger] C. Bishop and T. Steger, Representation theoretic rigidity in $\operatorname{PSL} (2,\mathbb{R})$ , preprint.
[Cartier 1] Pierre Cartier, Géométrie et analyse sur les arbres, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 407, Springer, Berlin, 1973, pp. 123–140. Lecture Notes in Math., Vol. 317 (French). MR 0425032
[Cartier 2] -, Fonctiones harmoniques sur les arbres, Sympos. Math. 9 (1972), 203-270.
[Cartier 3] P. Cartier, Harmonic analysis on trees, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 419–424. MR 0338272
[Cecchini-Figà-Talamanca] Carlo Cecchini and Alessandro Figà-Talamanca, Projections of uniqueness for 𝐿^{𝑝}(𝐺), Pacific J. Math. 51 (1974), 37–47. MR 0394043
[Culler-Morgan] Marc Culler and John W. Morgan, Group actions on 𝑅-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571–604. MR 907233, https://doi.org/10.1112/plms/s3-55.3.571
[Cowling-Steger] M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math. 420 (1991), 85–98. MR 1124567
[Dunford-Schwartz] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
[Figà-Talamanca-Picardello 1] Alessandro Figà-Talamanca and Massimo A. Picardello, Spherical functions and harmonic analysis on free groups, J. Funct. Anal. 47 (1982), no. 3, 281–304. MR 665019, https://doi.org/10.1016/0022-1236(82)90108-2
[Figà-Talamanca-Picardello 2] Alessandro Figà-Talamanca and Massimo A. Picardello, Restriction of spherical representations of 𝑃𝐺𝐿₂(𝑄_{𝑝}) to a discrete subgroup, Proc. Amer. Math. Soc. 91 (1984), no. 3, 405–408. MR 744639, https://doi.org/10.1090/S0002-9939-1984-0744639-3
[Figà-Talamanca-Picardello 3] Alessandro Figà-Talamanca and Massimo A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker, Inc., New York, 1983. MR 710827
[Figà-Talamanca-Steger 1] Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis on trees, Symposia Mathematica, Vol. XXIX (Cortona, 1984) Sympos. Math., XXIX, Academic Press, New York, 1987, pp. 163–182. MR 951184
[Figà-Talamanca-Steger 2] Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc. 110 (1994), no. 531, xii+68. MR 1219707, https://doi.org/10.1090/memo/0531
[Haagerup] Uffe Haagerup, An example of a nonnuclear 𝐶*-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, https://doi.org/10.1007/BF01410082
[Helgason] SigurÄ‘ur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 0263988, https://doi.org/10.1016/0001-8708(70)90037-X
[Kajiwara] Tsuyoshi Kajiwara, On irreducible decompositions of the regular representation of free groups, Boll. Un. Mat. Ital. A (6) 4 (1985), no. 3, 425–431 (English, with Italian summary). MR 821080
[Kato 1] Shin-ichi Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 929–943 (1982). MR 656065
[Kato 2] Shin-ichi Kato, On eigenspaces of the Hecke algebra with respect to a good maximal compact subgroup of a 𝑝-adic reductive group, Math. Ann. 257 (1981), no. 1, 1–7. MR 630642, https://doi.org/10.1007/BF01450650
[Mantero-Zappa] Anna Maria Mantero and Anna Zappa, The Poisson transform and representations of a free group, J. Funct. Anal. 51 (1983), no. 3, 372–399. MR 703084, https://doi.org/10.1016/0022-1236(83)90019-8
[Ol'shanskii 1] G. I. Ol'shanskii, Representations of groups of automorphisms of trees, Uspekhi Mat. Nauk 30 (1975), 169-170. (Russian)
[Ol'shanskii 2] -, Classification of irreducible representations of groups of automorphisms of Bruhat-Tit trees, Funct. Anal. Appl. 11 (1977), 26-34.
[Pytlik] 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. MR 622348, https://doi.org/10.1515/crll.1981.326.124
[Pytlik-Swarc] T. Pytlik and R. Szwarc, An analytic family of uniformly bounded representations of free groups, Acta Math. 157 (1986), no. 3-4, 287–309. MR 857676, https://doi.org/10.1007/BF02392596
[Rudin] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
[Steger] T. Steger, Finite reducibility of random walk representations of free groups, in preparation.
[Yoshizawa] Hisaaki Yoshizawa, Some remarks on unitary representations of the free group, Osaka Math. J. 3 (1951), 55–63. MR 0041854