Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-12T02:38:42.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Irreducibility of the analytic continuation of the principal series of a free group

Part of: Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Anna Maria Mantero  and
Anna Zappa
Anna Maria Mantero
Affiliation:
Istituto di Matematica Universita di GenovaVia L. B. Alberti 16132 Genova, Italy
Anna Zappa
Affiliation:
Istituto di Matematica Universita di GenovaVia L. B. Alberti 16132 Genova, Italy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper it is proved that the principal series of representations of Γ = Z2*…*Z2 may be analytically continued to give uniformly bounded representations on the same Hilbert space, and that these representations are irreducible. Further, the reducibility of the restrictions to Γ ⊂ SL(2, Qp) of the irreducible unitary representations of SL(2, Qp) is examined.

MSC classification

Secondary: 43A65: Representations of groups, semigroups, etc. 22E35: Analysis on $p$-adic Lie groups 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 2 , October 1987 , pp. 199 - 210
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bamazi, L., Représentations sphériques uniformement bornées des groupes de Lorentz; Analyse Harmonique sur les Groupes de Lie II (Lecture Notes in Math. 739, Springer-Verlag, Berlin, Heidelberg, New York, 1979).Google Scholar
[2]Choucroun, F., ‘Groups opérant simplement transitivement sur un arbre homogène et prolongements dans PGL(2, k)’, C. R. Acad. Sc. Paris 298 (1984), 313–315.Google Scholar
[3]Cowling, M. G., ‘Unitary and uniformly bounded representations of some simple Lie groups, Harmonic Analysis and Group Representations’, Atti del corso C.I.M.E., Cortona 1980, Liguori, Ed., (1984), 49–128.Google Scholar
[4]Dunford, N. and Schwartz, J. T., Linear operators, Part I (Interscience Publ., New York, 1958, 1963).Google Scholar
[5]Figà-Talamanca, A. and Picardello, A. M., ‘Spherical functions and harmonic analysis on free groups’, J. Functional Anal. 47 (1982), 281–304.CrossRefGoogle Scholar
[6]Figà-Talamanca, A. and Picardello, A. M., Harmonic analysis on free groups, (Lecture Notes in Pure and Applied Math., Vol. 87, Marcel Dekker, New York, 1983).Google Scholar
[7]Figà-Talamanca, A. and Picardello, A. M., ‘Restriction of spherical representations of PGL(2, Qp) to a discrete subgroup’, Proc. Amer. Math. Soc. 91 (1984), 405–408.Google Scholar
[8]Gel'fand, I., Graev, M. and Pyatetskii-Shapiro, I., Representation theory and automorphic functions (W. B. Saunders Co., Philadelphia, 1969).Google Scholar
[9]Kunze, R. A. and Stein, E. M., ‘Uniformly bounded representations and harmonic analysis on the 2×2 real unimodular groups,’ Amer. J. Math. 82 (1960), 1–62.CrossRefGoogle Scholar
[10]Kunze, R. A. and Stein, E. M., ‘Uniformly bounded representations and harmonic analysis on the n×n complex unimodular groups,’ Amer. J. Math. 83 (1961), 723–786.CrossRefGoogle Scholar
[11]Kunze, R. A. and Stein, E. M., ‘Uniformly bounded representations III: Intertwining operators for the principal series on semisimple groups,’ Amer. J. Math. 89 (1976), 385–442.CrossRefGoogle Scholar
[12]Lipsman, R. L., ‘Uniformly bounded representations of SL(2, C),’ Amer. J. Math. 91 (1969), 47–66.CrossRefGoogle Scholar
[13]Lipsman, R. L., ‘Harmonic analysis on SL(n, C),’ J. Functional Anal. 3, (1969), 126–155.CrossRefGoogle Scholar
[14]Lipsman, R. L., ‘Uniformly bounded representations on the Lorentz group,’ Amer. J. Math. 91 (1969), 938–962.CrossRefGoogle Scholar
[15]Lipsman, R. L., ‘An explicit realisation of Kostant's principal series with applications to uniformly bounded representations’, preprint.Google Scholar
[16]Lohoué, N., ‘Sur les représentations uniformement bornées et le théorème de convolution de Kunze-Stein,’ Osaka J. Math. 18 (1981), 465–480.Google Scholar
[17]Mantero, A. M. and Zappa, A., ‘The Poisson transform and representations of a free group,’ J. Functional Anal. 51 (1983), 372–399.CrossRefGoogle Scholar
[18]Mantero, A. M. and Zappa, A., ‘Uniformly bounded representations and Lp-convolution operators in a free group’, Harmonic analysis, Proc. Conf. Cortona 1982, pp. 333–343 (Lecture Notes in Math. 992, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983).CrossRefGoogle Scholar
[19]Mantero, A. M. and Zappa, A., ‘Special series of unitary representations of groups acting on homogeneous tree,’ J. Austral. Math. Soc. 40 (1986), 83–88.CrossRefGoogle Scholar
[20]Pytlik, T. and Szwarc, R., ‘An analytic family of uniformly bounded representations of free groups,’ preprint.Google Scholar
[21]Sally, P. J. Jr, ‘Analytic continuation of the irreducible unitary representations of the universal covering group of SL(2, R),’ Mem. Amer. Math. Soc. 69 (1967).Google Scholar
[22]Sally, P. J. Jr, ‘Unitary and uniformly bounded representations of the two by two unimodular group over local fields,’ Amer. J. Math. 90 (1968), 406–443.CrossRefGoogle Scholar
[23]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
[24]Serre, J. P., ‘Arbres, amalgames, SL2,’ Asterisque 46 (Soc. Math. de France, Paris 1977).Google Scholar
[25]Steger, T. (Ph.D. dissertation, Washington University, St. Lous, 1985).Google Scholar
[26]Wilson, E. N., ‘Uniformly bounded representations for the Lorentz groups,’ Trans. Amer. Math. Soc. 166 (1972), 431–438.CrossRefGoogle Scholar