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A characterization of the Cauchy-type distributions on boundaries of semisimple groups. (English) Zbl 0738.60006
The author proves the following theorem: Let \(G\) be a connected semisimple Lie group and \(P\) a parabolic subgroup of \(G\). Let \(K\) be a maximal compact subgroup of \(G\) and let \(m\) be the \(K\)-invariant probability measure on \(G/P\). Moreover let \(H\) be a subgroup of \(P\) such that \(HK=G\). Then for a probability measure \(\mu\) on \(G/P\) the following assertions are equivalent: (i) \(\{g\mu:g\in G\}=\{h\mu:h\in H\}\), i.e. the sets of \(G\)-types and of \(H\)-types of \(\mu\) coincide; (ii) \(\mu=hm\) for some \(h\in H\), i.e. \(\mu\) is of the same \(H\)-type as \(m\). In particular, if \(G=SL(n,\mathbb{R})\) and \(P=\{g\in G:gv\in\mathbb{R} v\}\) for some \(v\in\mathbb{R}^ n\), \(v\neq 0\), this theorem yields a characterization of the Cauchy distributions on \(\mathbb{R}^ n\) due to F. B. Knight and P. A. Meyer [Z. Wahrscheinlichkeitstheorie Verw. Geb. 34, 129-134 (1976; Zbl 0353.60020)]. Moreover, the theorem strengthens results by J.-L. Dunau and H. Senateur [J. Multivariate Anal. 22, 74-78 (1987; Zbl 0655.62007)] concerning measures on boundaries of semisimple Lie groups.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
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[1] Borel, A., and Tits, J. (1965). Groupes r?ductifs,Publ. I.H.E.S. 27, 55-150.
[2] Dani, S. G. (1982). On ergodic quasi-invariant measures of group automorphisms.Israel J. Math. 43, 62-74. · Zbl 0553.28015 · doi:10.1007/BF02761685
[3] Dunau, J.-L., and Senateur, H. (1989). Characterization of the type of some generalizations of the Cauchy distributions. In H. Heyer (Ed.),Probability Measures in Groups, IX (Oberwolfach 1988), Lecture Notes in Mathematics, Vol. 1379, pp. 64-74, Springer-Verlag, Berlin, Heidelberg, New York.
[4] Helgason, S. (1978).Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, San Francisco, London. · Zbl 0451.53038
[5] Hochschild, G. (1981).Basic Theory of Algebraic Groups and Lie Algebras. Springer-Verlag, Berlin, Heidelberg, New York. · Zbl 0589.20025
[6] Knight, F. B. (1976). A characterization of the Cauchy type.Proc. Amer. Math. Soc. 55, 130-135. · Zbl 0341.60009 · doi:10.1090/S0002-9939-1976-0394803-6
[7] Knight, F. B., and Meyer, P. A. (1976). Une characterization de la loi de Cauchy.Z. Wahrsch. verw. Gebiete 34, 129-134. · Zbl 0353.60020 · doi:10.1007/BF00535680
[8] Montgomery, D., and Zippin, L. (1955).Topological Transformation Groups. Interscience Publishers, New York and London. · Zbl 0068.01904
[9] Warner, G. (1972).Harmonic Analysis on Semisimple Lie Groups I. Springer-Verlag, Berlin, Heidelberg, New York. · Zbl 0265.22020
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