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The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras

Published online by Cambridge University Press:  20 November 2018

Salem Ben Saïd
Salem Ben Saïd*
Affiliation:
Université Henri Poincaré – Nancy I, Institut Elie Cartan, Département de Mathématiques, B.P. 239, 54506 Vandoeuvre-Lès-Nancy Cedex, France e-mail: [email protected]
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Abstract

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This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

Keywords

11M4117C2011S90Zeta distributionsfunctional equationsBernstein polynomialsnon-Euclidean Jordan algebras
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 1 , 01 February 2006 , pp. 3 - 22
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Atiyah, M. F., Resolution of singularities and division of distributions. Comm. Pure Appl. Math. 23(1970), 145–150.Google Scholar
[2] Angeli, Y., Analyse harmonique sur les cones satellites. Thèse, Université de Nancy, 2001.Google Scholar
[3] Barchini, L., Sepanski, M., and Zierau, R., Positivity of zeta distributions and small unitary representations. Preprint (2002).Google Scholar
[4] Bernstein, I. N., Analytic continuation of generalized functions with respect to a parameter. Functional Anal. Appl. 6(1972), 273–285.Google Scholar
[5] Bernstein, I. N. and Gel’fand, S. I., Meromorphic property of the function Pλ. Functional Anal. Appl. 3(1969), 68–69.Google Scholar
[6] Bopp, N. and Rubenthaler, H., Fonction zêta associée à la série principal sphérique de certains espaces symétriques. Ann. Sci. Ec. Norm. Sup. (4) 26(1993), 701–745.Google Scholar
[7] Bopp, N. and Rubenthaler, H., Une fonction zêta associée à certaines familles d’espaces symétrique réels. C. R. Acad. Sci. Paris Sér. I Math. 325(1997), 355–360.Google Scholar
[8] Clerc, J.-L., Zeta distributions associated to a representation of a Jordan algebra. Math. Z. 239(2002), 263–276.Google Scholar
[9] Dvorsky, A. and Sahi, S., Explicit Hilbert spaces for certain unipotent representations. III. J. Funct. Anal. 201(2003), 430–456.Google Scholar
[10] Faraut, J. and Korányi, A., Analysis on Symmetric Cones. Oxford Mathematical Monographs, Oxford University Press, Oxford, 1994.Google Scholar
[11] Godment, R. and Jacquet, H., Zeta functions of simple algebras. Lecture Notes in Mathematics 260, Springer-Verlag, Berlin, 1972.Google Scholar
[12] Gindikin, S. G., Invariant generalized functions in homogeneous domains. Functional. Anal. Appl. 9(1975), 50–52.Google Scholar
[13] Muller, I., Décomposition orbitale des espaces préhomogènes réguliers de type parabolique commutatif et application. C. R. Acad. Sci. Paris Sér I Math. 303(1986), 495–498.Google Scholar
[14] Pevsner, M., Analyse conforme sur les algebres de Jordan. Thèse, Université Paris VI 1998.Google Scholar
[15] Satake, I., Algebraic Structures of Symmetric Domains. Iwanami-Shoten, Tokyo; Princeton University Press, Princeton, NJ, 1980.Google Scholar
[16] Satake, I. and Faraut, J., The functional equation of zeta distributions associated with formally real Jordan algebras. Tohoku Math. J. 88(1984), 469–482.Google Scholar
[17] Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces. Ann. of Math. 100(1974), 131–170.Google Scholar
[18] Sato, M. and Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nogoya Math. J. 65(1977), 1–155.Google Scholar
[19] Shintani, T., On the Dirichlet series whose coefficients are class numbers of integral cubic forms. J. Math. Soc. Japan 24(1972), 132–188.Google Scholar
[20] Tate, J., Fourier analysis in number fields and Hecke's zeta function. In: Algebraic Number Theory, Thompson, Washington, D.C., 1967, pp. 305–347.Google Scholar