The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels
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 by G. Ólafsson and B. Ørsted PDF
 Trans. Amer. Math. Soc. 326 (1991), 385405 Request permission
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.References

E. P. van den Ban, Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces, Proc. Koninklijke Nederl. Akad. Wetensch. A 90 (1987).
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.
 Mogens FlenstedJensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (1980), no. 2, 253–311. MR 569073, DOI 10.2307/1971201 —, $K$finite joint eigenfunctions of $U{(\mathfrak {g})^K}$ on a nonRiemannian semisimple symmetric space $G/H$, Lecture Notes in Math., vol. 880, SpringerVerlag, Berlin, 1981. —, Analysis on nonRiemannian symmetric spaces, CBMS Regional Conf. Ser. in Math., no. 61, Amer. Math. Soc., Providence, R.I., 1987. E. Gutkin, Invariant elliptic operators and unitary representations, Ph.D. Thesis, Brandeis Univ., 1978; Preprint, 1979.
 Henryk Hecht and Wilfried Schmid, On integrable representations of a semisimple Lie group, Math. Ann. 220 (1976), no. 2, 147–149. MR 399358, DOI 10.1007/BF01351699
 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1978. MR 514561
 Ryoshi Hotta, On a realization of the discrete series for semisimple Lie groups, J. Math. Soc. Japan 23 (1971), 384–407. MR 306405, DOI 10.2969/jmsj/02320384
 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, DOI 10.1515/9781400883974
 Shuichi Matsumoto, Discrete series for an affine symmetric space, Hiroshima Math. J. 11 (1981), no. 1, 53–79. MR 606834
 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, NorthHolland, Amsterdam, 1984, pp. 331–390. MR 810636, DOI 10.2969/aspm/00410331
 Gestur Ólafsson, Die LanglandsParameter für die FlenstedJensensche fundamentale Reihe, Math. Scand. 55 (1984), no. 2, 229–244 (German). MR 787199, DOI 10.7146/math.scand.a12078
 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, DOI 10.1016/00221236(88)901152 —, Imbedding of the discrete series of $G$ into ${{\mathbf {L}}^2}(X)$, Mathematica Gottingensis, no. 5, 1988.
 M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math. 136 (1976), no. 12, 1–59. MR 480883, DOI 10.1007/BF02392042
 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
 Henrik Schlichtkrull, The Langlands parameters of FlenstedJensen’s discrete series for semisimple symmetric spaces, J. Functional Analysis 50 (1983), no. 2, 133–150. MR 693225, DOI 10.1016/00221236(83)900642
 Henrik Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, vol. 49, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 757178, DOI 10.1007/9781461252986
 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. 34, 237–280. MR 393349, DOI 10.1007/BF02392217
 V. S. Varadarajan, Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, SpringerVerlag, BerlinNew York, 1977. MR 0473111, DOI 10.1007/BFb0097814
 Nolan R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. MR 531967, DOI 10.1090/S00029947197905319672
 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
 Garth Warner, Harmonic analysis on semisimple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, SpringerVerlag, New YorkHeidelberg, 1972. MR 0498999
Additional Information
 © Copyright 1991 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 326 (1991), 385405
 MSC: Primary 22E46; Secondary 22E30, 43A85
 DOI: https://doi.org/10.1090/S00029947199110029230
 MathSciNet review: 1002923