Abstract
Let V be a Euclidean Jordan algebra, Гthe associated symmetric cone and G be the identity component of the linear automorphism group of Г.In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain Г +iV.
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References
Clerc, J.-L. Laplace transform and unitary highest weight modules,J. Lie Theory,5(2), 225–240, (1995).
Faraut, J. and Koranyi, A.Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994.
Gindikin, S. Analysis on homogeneous domains,Russ. Math. Surveys,19, 1–89, (1964).
Gindikin, S. Invariant generalized functions in homogeneous domains,J. Funct. Anal. Appl.,9, 50–52, (1975).
Hilgert, J. and Neeb, K.-H.Unitary highest weight representations on tube domains, preprint. Mittag-Leffler Institut, 1996.
Hilgert, J., Neeb, K.-H., and Ørsted, B. Conal Heisenberg algebras and associated Hilbert spaces,J. reine angew. Math.,474, 67–122, (1996).
Hilgert, J., Neeb, K.-H., and Ørsted, B.Unitary representations defined by operator valued Riesz kernels, in preparation.
Helgason, S.Groups and Geometric Analysis, Academic Press, London, 1984.
Hewitt, E. and Ross, K.A.Abstract Harmonic Analysis II, Die Grundlehren der mathematischen Wissenschaften,152, Springer-Verlag, Berlin, Heidelberg, 1970.
Hilgert, J.and Ólafsson,G.Causal Symmetric Spaces, Geometry and Harmonic Analysis, Academic Press, Orlando, 1996.
Hörmander, L.The Analysis of Linear Partial Differential Operators I, Springer, Berlin, 1983.
Neeb, K.-H. A convexity theorem for semisimple symmetric spaces,Pacific Journal of Math.,162(2), 305–349, (1994).
Neeb, K.-H. Operator valued positive definite kernels on tubes,Monatshefte für Math.,126, 125–160, (1998).
Riesz, M. L’intégrale de Riemann-Liouville et le problème de Cauchy,Acta Math.,81, 1–223, (1949).
Rossi, H. and Vergne, M. Analytic continuation of the holomorphic discrete series of a semisimple Lie group,Acta Math.,136, 1–59, (1976).
Satake, I.Algebraic Structures of Symmetric Domains, Publications of theMath. Soc. of Japan,14, Princeton University Press, 1980.
Schwartz, L.Théorie des distributions, Hermann Paris, 1966.
Shintani, T. On Dirichlet series whose coefficients are class numbers of integral binary cubic forms,J. Math. Soc. Japan,24, 132–188, (1972).
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Hilgert, J., Neeb, KH. Vector valued Riesz distributions on Euclidian Jordan algebras. J Geom Anal 11, 43–75 (2001). https://doi.org/10.1007/BF02921953
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DOI: https://doi.org/10.1007/BF02921953