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The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels

Authors: G. Ólafsson and B. Ørsted
Journal: Trans. Amer. Math. Soc. 326 (1991), 385-405
MSC: Primary 22E46; Secondary 22E30, 43A85
MathSciNet review: 1002923
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Abstract: Consider a semisimple connected Lie group $ G$ with an affine symmetric space $ X$. We study abstractly the intertwining operators from the discrete series of $ X$ into representations with reproducing kernel and, in particular, into the discrete series of $ G$; each such is given by a convolution with an analytic function. For $ X$ of Hermitian type, we consider the holomorphic discrete series of $ X$ and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question.

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  • [1] E. P. van den Ban, Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces, Proc. Koninklijke Nederl. Akad. Wetensch. A 90 (1987).
  • [2] F. Bien, Multiplicity one for exceptional symmetric spaces, Preprint, May 1987; Spherical $ \mathcal{D}$-modules and representations of reductive Lie groups, Ph.D. Thesis, M.I.T., June 1986.
  • [3] Mogens Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (1980), no. 2, 253–311. MR 569073, 10.2307/1971201
  • [4] -, $ K$-finite joint eigenfunctions of $ U{(\mathfrak{g})^K}$ on a non-Riemannian semisimple symmetric space $ G/H$, Lecture Notes in Math., vol. 880, Springer-Verlag, Berlin, 1981.
  • [5] -, Analysis on non-Riemannian symmetric spaces, CBMS Regional Conf. Ser. in Math., no. 61, Amer. Math. Soc., Providence, R.I., 1987.
  • [6] E. Gutkin, Invariant elliptic operators and unitary representations, Ph.D. Thesis, Brandeis Univ., 1978; Preprint, 1979.
  • [7] Henryk Hecht and Wilfried Schmid, On integrable representations of a semisimple Lie group, Math. Ann. 220 (1976), no. 2, 147–149. MR 0399358
  • [8] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • [9] Ryoshi Hotta, On a realization of the discrete series for semisimple Lie groups, J. Math. Soc. Japan 23 (1971), 384–407. MR 0306405
  • [10] Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239
  • [11] Shuichi Matsumoto, Discrete series for an affine symmetric space, Hiroshima Math. J. 11 (1981), no. 1, 53–79. MR 606834
  • [12] Toshio Ōshima and Toshihiko Matsuki, A description of discrete series for semisimple symmetric spaces, Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 331–390. MR 810636
  • [13] Gestur Ólafsson, Die Langlands-Parameter für die Flensted-Jensensche fundamentale Reihe, Math. Scand. 55 (1984), no. 2, 229–244 (German). MR 787199
  • [14] G. Ólafsson and B. Ørsted, The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal. 81 (1988), no. 1, 126–159. MR 967894, 10.1016/0022-1236(88)90115-2
  • [15] -, Imbedding of the discrete series of $ G$ into $ {{\mathbf{L}}^2}(X)$, Mathematica Gottingensis, no. 5, 1988.
  • [16] M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1976), no. 1-2, 1–59. MR 0480883
  • [17] H. Schlichtkrull, On some series of representations related to symmetric spaces, Mém. Soc. Math. France (N.S.) 15 (1984), 277–289. Harmonic analysis on Lie groups and symmetric spaces (Kleebach, 1983). MR 789088
  • [18] Henrik Schlichtkrull, The Langlands parameters of Flensted-Jensen’s discrete series for semisimple symmetric spaces, J. Funct. Anal. 50 (1983), no. 2, 133–150. MR 693225, 10.1016/0022-1236(83)90064-2
  • [19] Henrik Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, vol. 49, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 757178
  • [20] P. C. Trombi and V. S. Varadarajan, Asymptotic behaviour of eigen functions on a semisimple Lie group: the discrete spectrum, Acta. Math. 129 (1972), no. 3-4, 237–280. MR 0393349
  • [21] V. S. Varadarajan, Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer-Verlag, Berlin-New York, 1977. MR 0473111
  • [22] Nolan R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. MR 531967, 10.1090/S0002-9947-1979-0531967-2
  • [23] Nolan R. Wallach and Joseph A. Wolf, Completeness of Poincaré series for automorphic forms associated to the integrable discrete series, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 265–281. MR 733818
  • [24] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. MR 0498999

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