Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program

Koufany, Khalid ; Ørsted, Bent

Annales de l'Institut Fourier, Tome 46 (1996), p. 689-722 / Harvested from Numdam

Résumé

Nous étudions la série discrète holomorphe de ${\rm SU}(2,2)$ et son prolongement analytique par restriction à la forme réelle non-compacte ${\rm S}\bigl ({\rm U}(1,1)\times {\rm U}(1,1)\bigr )$. Nous construisons pour cela une transformation de Cayley entre le sous semi-groupe de Ol’shanskiĭ de ${\rm GL}(2,{\Bbb C})$ ayant ${\rm U}(1,1)$ comme frontière de Šilov et un ouvert dense du domaine tube de ${\rm SU}(2,2)$. Nous déterminons ainsi la décomposition de la restriction à ${\rm S}\bigl ({\rm U}(1,1)\times {\rm U}(1,1)\bigr )$ d’une représentation de la série discrète holomorphe. En particulier nous montrons que la représentation de ${\rm S}\bigl ({\rm U}(1,1)\times {\rm U}(1,1)\bigr )$ 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 ${\rm SU}(2,2)$. Ceci infirme un résultat de Gindikin.

Abstract

For the scalar holomorphic discrete series representations of ${\rm SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside ${\rm SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having ${\rm U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for ${\rm SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on ${\rm U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for ${\rm U}(1,1)$ is different from the classical Hardy space for ${\rm U}(2)$; this provides a counterexample to a statement in a paper by Gindikin.

Détail

Publié le : 1996-01-01

Zbl 0853.22010 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1528

Bibliographie

[1]J. Faraut, Hardy spaces in non-commutative harmonic analysis, Summer School, Tuczno, 1994.

[2]J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Clarendon Press, 1994. | MR 98g:17031 | Zbl 0841.43002

[3]I.M. Gel'Fand and S.G. Gindikin, Complex manifolds whose skeletons are semi-simple real Lie groups, and analytic discrete series of representations, Funct. Anal. Appl., 11 (1977), 258-265. | Zbl 0449.22018

[4]S.G. Gindikin, 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 93h:32045 | Zbl 0788.22008

[5]S.G. Gindikin, Fourier transform and Hardy spaces of ∂-cohomology in tube domains, C. R. Acad. Sci. Paris, 315 (1992), 1139-1143. | MR 93m:32006 | Zbl 0771.32001

[6]H. Hecht, The characters of some representations of Harish-Chandra, Math. Ann., 210 (1976), 213-226. | MR 55 #573 | Zbl 0301.22009

[7]J. Hilgert, A note on Howe's oscillator semigroup, Annal. Inst. Fourier, 39-3 (1989), 663-688. | Numdam | MR 91b:22008 | Zbl 0674.47029

[8]J. Hilgert and K.-H. Neeb, Lie Semigroups and theirs Applications, Lecture Notes in Math. Springer-Verlag, 1993, t. 1552. | MR 96j:22002 | Zbl 0807.22001

[9]J. Hilgert and H. Ólafsson, Analytic continuations of representations, the solvable case, Jap. Journal of Math., 18 (1992), 213-290. | Zbl 0788.22014

[10]R. Howe, The Oscillator Semigroup, Proc. Symp. Pure Math., 48 (1988), 61-132. | MR 90f:22014 | Zbl 0687.47034

[11]H.P. Jakobsen and M. Vergne, Wave and Dirac operators, and representations of the conformal group, J. Funct. Anal., 24 (1977), 52-106. | MR 55 #12876 | Zbl 0361.22012

[12]H.P. Jacobsen and M. Vergne, Restrictions and expansions of holomorphic representations, J. Funct. Anal., 34 (1979), 29-53. | MR 80m:22020 | Zbl 0433.22011

[13]M. Kashiwara and M. Vergne, K-types and singular spectrum, Lecture Note in Math., Springer-Verlag t. 728, 1978.

[14]K. Koufany et B. Ørsted, Espace de Hardy sur le semi-groupe métaplectique, C. R. Acad. Sci. Paris, 322, Série I (1996), 113-116. | MR 96m:22017 | Zbl 0880.22003

[15]K. Koufany and B. Ørsted, Hardy spaces on two-sheeted covering semigroups, in preparation, 1996.

[16]O. Loos, Bounded symmetric domains and Jordan pairs, Lecture notes, University of California, Irvine, 1977. | Zbl 0228.32012

[17]G.I. Ol'Shanskiĭ, Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. Appl., 15 (1982), 275-285. | Zbl 0503.22011

[18]G.I. Ol'Shanskiĭ, 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 0789.22011

[19]B. Ørsted, Composition series for analytic continuations of holomorphic discrete series representations of SU(n,n), Trans. Amer. Math. Soc., 260 (1980), 563-573. | MR 81g:22020 | Zbl 0439.22017

[20]B. Ørsted, The conformal invariance of Huygen's principle, J. Diff. Geom., 16 (1981), 1-9. | MR 83c:58080 | Zbl 0447.35059

[21] S. Paneitz and I. Segal, Quantization of wave equations and hermitian structures in partial differential varieties, Proc. Nat. Acad. Sci. USA, 77 (1980), 6943-6947. | MR 82g:81040 | Zbl 0469.35065

[22]H. Rossi and M. Vergne, Analytic continuation of the holomorphic discrete series, Acta Math., 136 (1976), 1-59. | MR 58 #1032 | Zbl 0356.32020

[23]B. Speh, Degenerate series representations of the universal covering group of SU(2,2), J. Funct. Anal., 33 (1978), 95-118. | MR 80j:22017 | Zbl 0415.22012

[24]R.J. Stanton, Analytic extension of the holomorphic discrete series, Amer. J. of Math., 108 (1986), 1411-1424. | MR 88b:22013 | Zbl 0626.43008

[25]H. Weyl, The classical groups, Princeton Univ. Press, Princeton, N.J., 1946.