For the scalar holomorphic discrete series representations of and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside . We construct a Cayley transform between the Ol’shanskiĭ semigroup having as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for . This allows calculating the composition series in terms of harmonic analysis on . In particular we show that the Ol’shanskiĭ Hardy space for is different from the classical Hardy space for ; this provides a counterexample to a statement in a paper by Gindikin.
Nous étudions la série discrète holomorphe de et son prolongement analytique par restriction à la forme réelle non-compacte . Nous construisons pour cela une transformation de Cayley entre le sous semi-groupe de Ol’shanskiĭ de ayant comme frontière de Šilov et un ouvert dense du domaine tube de . Nous déterminons ainsi la décomposition de la restriction à d’une représentation de la série discrète holomorphe. En particulier nous montrons que la représentation de dans l’espace de Hardy construit par Gel’fand et Gindikin n’est pas obtenue par restriction d’une représentation de la série discrète holomorphe de . Ceci infirme un résultat de Gindikin.
@article{AIF_1996__46_3_689_0, author = {Koufany, Khalid and {\O}rsted, Bent}, title = {Function spaces on the {Ol\'shanski\u{i}semigroup} and the {Gel'fand-Gindikin} program}, journal = {Annales de l'Institut Fourier}, pages = {689--722}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {46}, number = {3}, year = {1996}, doi = {10.5802/aif.1528}, zbl = {0853.22010}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1528/} }
TY - JOUR AU - Koufany, Khalid AU - Ørsted, Bent TI - Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program JO - Annales de l'Institut Fourier PY - 1996 SP - 689 EP - 722 VL - 46 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1528/ DO - 10.5802/aif.1528 LA - en ID - AIF_1996__46_3_689_0 ER -
%0 Journal Article %A Koufany, Khalid %A Ørsted, Bent %T Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program %J Annales de l'Institut Fourier %D 1996 %P 689-722 %V 46 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1528/ %R 10.5802/aif.1528 %G en %F AIF_1996__46_3_689_0
Koufany, Khalid; Ørsted, Bent. Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program. Annales de l'Institut Fourier, Volume 46 (1996) no. 3, pp. 689-722. doi : 10.5802/aif.1528. https://aif.centre-mersenne.org/articles/10.5802/aif.1528/
[1]Hardy spaces in non-commutative harmonic analysis, Summer School, Tuczno, 1994.
,[2]Analysis on Symmetric Cones, Oxford Mathematical Monographs, Clarendon Press, 1994. | MR | Zbl
and ,[3]Complex manifolds whose skeletons are semi-simple real Lie groups, and analytic discrete series of representations, Funct. Anal. Appl., 11 (1977), 258-265. | Zbl
and ,[4]Generalized conformal structures on classical real Lie groups and related problems on the theory of representations, C. R. Acad. Sci. Paris, 315 (1992), 675-679. | MR | Zbl
,[5]Fourier transform and Hardy spaces of ∂-cohomology in tube domains, C. R. Acad. Sci. Paris, 315 (1992), 1139-1143. | MR | Zbl
,[6]The characters of some representations of Harish-Chandra, Math. Ann., 210 (1976), 213-226. | MR | Zbl
,[7]A note on Howe's oscillator semigroup, Annal. Inst. Fourier, 39-3 (1989), 663-688. | Numdam | MR | Zbl
,[8]Lie Semigroups and theirs Applications, Lecture Notes in Math. Springer-Verlag, 1993, t. 1552. | MR | Zbl
and ,[9]Analytic continuations of representations, the solvable case, Jap. Journal of Math., 18 (1992), 213-290. | Zbl
and ,[10]The Oscillator Semigroup, Proc. Symp. Pure Math., 48 (1988), 61-132. | MR | Zbl
,[11]Wave and Dirac operators, and representations of the conformal group, J. Funct. Anal., 24 (1977), 52-106. | MR | Zbl
and ,[12]Restrictions and expansions of holomorphic representations, J. Funct. Anal., 34 (1979), 29-53. | MR | Zbl
and ,[13]K-types and singular spectrum, Lecture Note in Math., Springer-Verlag t. 728, 1978.
and ,[14]Espace de Hardy sur le semi-groupe métaplectique, C. R. Acad. Sci. Paris, 322, Série I (1996), 113-116. | MR | Zbl
et ,[15]Hardy spaces on two-sheeted covering semigroups, in preparation, 1996.
and ,[16]Bounded symmetric domains and Jordan pairs, Lecture notes, University of California, Irvine, 1977. | Zbl
,[17]Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. Appl., 15 (1982), 275-285. | Zbl
,[18]Complex Lie semigroups, Hardy spaces and the Gel'fand Gindikin program, Topics in group theory and homological algebra, Yaroslavl University Press, 1982, p. 85-98 (Russian); English translation in Differential Geomerty and its Applications, 1 (1991), 235-246. | Zbl
,[19]Composition series for analytic continuations of holomorphic discrete series representations of SU(n,n), Trans. Amer. Math. Soc., 260 (1980), 563-573. | MR | Zbl
,[20]The conformal invariance of Huygen's principle, J. Diff. Geom., 16 (1981), 1-9. | MR | Zbl
,[21] Quantization of wave equations and hermitian structures in partial differential varieties, Proc. Nat. Acad. Sci. USA, 77 (1980), 6943-6947. | MR | Zbl
and ,[22]Analytic continuation of the holomorphic discrete series, Acta Math., 136 (1976), 1-59. | MR | Zbl
and ,[23]Degenerate series representations of the universal covering group of SU(2,2), J. Funct. Anal., 33 (1978), 95-118. | MR | Zbl
,[24]Analytic extension of the holomorphic discrete series, Amer. J. of Math., 108 (1986), 1411-1424. | MR | Zbl
,[25]The classical groups, Princeton Univ. Press, Princeton, N.J., 1946.
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