Function spaces on the Ol’shanskiĭ semigroup and the Gel’fand-Gindikin program. (English) Zbl 0853.22010
Summary: For the scalar holomorphic discrete series representations of \(\text{SU}(2,2)\) and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside \(\text{SU}(2,2)\). We construct a Cayley transform between the Ol’shanskiĭ semigroup having \(\text{U}(1,1)\) as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for \(\text{SU}(2,2)\). This allows calculating the composition series in terms of harmonic analysis on \(\text{U}(1,1)\). In particular we show that the Ol’shanskiĭ Hardy space for \(\text{U}(1,1)\) is different from the classical Hardy space for \(\text{U}(2)\); this provides a counterexample to a statement in a paper by Gindikin.
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 22E30 | Analysis on real and complex Lie groups |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 43A17 | Analysis on ordered groups, \(H^p\)-theory |
| 43A85 | Harmonic analysis on homogeneous spaces |
Keywords:
Cauchy-Szegö kernel; Cayley transform; composition series; Hardy space; holomorphic discrete series; Ol’shanskiĭ semigroupReferences:
| [1] | [1], Hardy spaces in non-commutative harmonic analysis, Summer School, Tuczno, 1994. · Zbl 1254.43004 |
| [2] | [2] and , Analysis on Symmetric Cones, Oxford Mathematical Monographs, Clarendon Press, 1994. · Zbl 0841.43002 |
| [3] | [3] and , 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] | [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. · Zbl 0788.22008 |
| [5] | [5], Fourier transform and Hardy spaces of ∂-cohomology in tube domains, C. R. Acad. Sci. Paris, 315 (1992), 1139-1143. · Zbl 0771.32001 |
| [6] | [6], The characters of some representations of Harish-Chandra, Math. Ann., 210 (1976), 213-226. · Zbl 0301.22009 |
| [7] | [7], A note on Howe’s oscillator semigroup, Annal. Inst. Fourier, 39-3 (1989), 663-688. · Zbl 0674.47029 |
| [8] | [8] and , Lie Semigroups and theirs Applications, Lecture Notes in Math. Springer-Verlag, 1993, t. 1552. · Zbl 0807.22001 |
| [9] | [9] and , Analytic continuations of representations, the solvable case, Jap. Journal of Math., 18 (1992), 213-290. · Zbl 0788.22014 |
| [10] | [10], The Oscillator Semigroup, Proc. Symp. Pure Math., 48 (1988), 61-132. · Zbl 0687.47034 |
| [11] | [11] and , Wave and Dirac operators, and representations of the conformal group, J. Funct. Anal., 24 (1977), 52-106. · Zbl 0361.22012 |
| [12] | [12] and , Restrictions and expansions of holomorphic representations, J. Funct. Anal., 34 (1979), 29-53. · Zbl 0433.22011 |
| [13] | [13] and , K-types and singular spectrum, Lecture Note in Math., Springer-Verlag t. 728, 1978. · Zbl 0404.22012 |
| [14] | [14] et , Espace de Hardy sur le semi-groupe métaplectique, C. R. Acad. Sci. Paris, 322, Série I (1996), 113-116. · Zbl 0880.22003 |
| [15] | [15] and , Hardy spaces on two-sheeted covering semigroups, in preparation, 1996. |
| [16] | [16], Bounded symmetric domains and Jordan pairs, Lecture notes, University of California, Irvine, 1977. · Zbl 0228.32012 |
| [17] | [17], Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. Appl., 15 (1982), 275-285. · Zbl 0503.22011 |
| [18] | [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 0789.22011 |
| [19] | [19], Composition series for analytic continuations of holomorphic discrete series representations of SU(n,n), Trans. Amer. Math. Soc., 260 (1980), 563-573. · Zbl 0439.22017 |
| [20] | [20], The conformal invariance of Huygen’s principle, J. Diff. Geom., 16 (1981), 1-9. · Zbl 0447.35059 |
| [21] | [21] and , Quantization of wave equations and hermitian structures in partial differential varieties, Proc. Nat. Acad. Sci. USA, 77 (1980), 6943-6947. · Zbl 0469.35065 |
| [22] | [22] and , Analytic continuation of the holomorphic discrete series, Acta Math., 136 (1976), 1-59. · Zbl 0356.32020 |
| [23] | [23], Degenerate series representations of the universal covering group of SU(2,2), J. Funct. Anal., 33 (1978), 95-118. · Zbl 0415.22012 |
| [24] | [24], Analytic extension of the holomorphic discrete series, Amer. J. of Math., 108 (1986), 1411-1424. · Zbl 0626.43008 |
| [25] | [V] , Caractérisation des intégrales orbitales sur un groupe réductif p-adique · Zbl 1024.20502 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
