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Generalized Hua-operators and parabolic subgroups. The cases of $ {\rm SL}(n,\,{\bf C})$ and $ {\rm SL}(n,\,{\bf R})$

Author: Kenneth D. Johnson
Journal: Trans. Amer. Math. Soc. 281 (1984), 417-429
MSC: Primary 22E46; Secondary 22E30, 43A85
MathSciNet review: 719678
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Abstract: Suppose $ G = {\text{SL}}(n,{\mathbf{C}})$ or $ {\text{SL}}(n,{\mathbf{R}})$ and $ K$ is a maximal compact subgroup of $ G$. If $ P$ is any parabolic subgroup of $ G$, we determine a system of differential equations on $ G/K$ with the property that any function on $ G/K$ satisfies these differential equations if and only if it is the Poisson integral of a hyperfunction on $ G/P$.

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  • [1] Nicole Berline and Michèle Vergne, Équations de Hua et intégrales de Poisson, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 3, A123–A125 (French, with English summary). MR 563956
  • [2] -, Equations du Hua et noyau de Poisson, Lecture Notes in Math., vol. 880, Springer-Verlag, Berlin and New York, 1981, pp. 1-51.
  • [3] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 0219666
  • [4] Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • [5] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I., 1963. MR 0171936
  • [6] Kenneth D. Johnson, Differential equations and the Bergman-Šilov boundary on the Siegel upper half plane, Ark. Mat. 16 (1978), no. 1, 95–108. MR 499140, 10.1007/BF02385985
  • [7] Kenneth D. Johnson and Adam Korányi, The Hua operators on bounded symmetric domains of tube type, Ann. of Math. (2) 111 (1980), no. 3, 589–608. MR 577139, 10.2307/1971111
  • [8] M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Ōshima, and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2) 107 (1978), no. 1, 1–39. MR 485861, 10.2307/1971253
  • [9] Adam Korányi, Poisson integrals and boundary components of symmetric spaces, Invent. Math. 34 (1976), no. 1, 19–35. MR 0425197
  • [10] A. Korányi and P. Malliavin, Poisson formula and compound diffusion associated to an overdetermined elliptic system on the Siegel halfplane of rank two, Acta Math. 134 (1975), no. 3-4, 185–209. MR 0410278
  • [11] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 0311837
  • [12] M. Lasalle, C. R. Acad. Sci. Paris 294 (1982).
  • [13] -, C. R. Acad. Sci. Paris 294 (1982).
  • [14] Hans Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR 0344198

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Keywords: Hua-operators, parabolic subgroups, boundary
Article copyright: © Copyright 1984 American Mathematical Society