×

Fourier and Poisson transformation associated to a semisimple symmetric space. (English) Zbl 0665.43004

Let G/H be a semisimple symmetric space: G is a semisimple Lie group and H is the fixed point group of an involution \(\tau\) of G. Let \(\theta\) be a Cartan involution commuting with \(\tau\) and K be the fixed point group of \(\theta\). One considers a parabolic subgroup P with a decomposition \(P=MAN\) such that MA is \(\tau\) and \(\theta\) stable, and \(M/M\cap H\) is compact. By a result of Matsuki there is a finite number of open orbits \({\mathcal O}_ 1,...,{\mathcal O}_ r\) of P in G/H. Using the finite- dimensional H-K-spherical representations of G, the author gives a system of equations for the set \(G/H-{\mathcal O}\), with \({\mathcal O}=\cup {\mathcal O}_ j.\)
For a character \(\nu\) of A, and an irreducible finite-dimensional \(M\cap H\)-spherical representation \(\sigma\) of M one defines the representation \(\sigma_{\nu}\) of P, \(\sigma_{\nu}(man)=a^{\nu}\sigma (m)\), and the induced representation \(\pi_{\sigma,\nu}\) of G, acting in a space \(I(\sigma,\nu)\) of functions on G with values in the space of \(\sigma\). The Poisson kernel \(p^ j_{\sigma,\nu}(x)\) is an analytic function on \({\mathcal O}_ j\), extended by 0 outside of \({\mathcal O}_ j\). For \(\nu\) in a certain tube, \(p^ j_{\sigma,\nu}(x)\) is locally integrable and defines a distribution \(u^ j_{\sigma,\nu}\) belonging to the space \(I(\sigma,-\nu)^ H_{-\infty}\). With respect to suitable coordinates \(p^ j_{\sigma,\nu}\) is a product of powers of polynomials. By the Bernstein theorem it follows that \(\nu \mapsto u^ j_{\sigma,\nu}\) extends to a meromorphic function with values in \(I(\sigma,\nu)_{- \infty}.\)
The Poisson transformation \({\mathcal P}^ j_{\sigma,\nu}\) is the map from \(I(\sigma,\nu)_{\infty}\) into \({\mathcal C}^{\infty}(G)\) defined by \({\mathcal P}^ j_{\sigma,\nu}(f)(x)=<\pi^{-\infty}_{\sigma,- \nu}(x)u^ j_{\sigma,\nu},f>\), and the Fourier transformation, essentially its dual, is the map from \({\mathcal C}_ c^{\infty}(G/H)\) into \(I({\bar\sigma},\nu)_{\infty}\) given by \({\mathcal F}^ j_{\sigma,\nu}(\phi)(x)= \int_{G/H}\phi(y)p^ j_{\sigma,\nu}(y^{-1}x)dy\). A Fatou type theorem is proved for the Poisson transformation which involves intertwining operators.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
32D15 Continuation of analytic objects in several complex variables
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Ban, E.P. van den: A convexity theorem for semisimple symmetric spaces. Pac. J. Math.124, 21-55 (1986) · Zbl 0599.22014
[2] Ban, E.P. van den: Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces. Centre for Math. and Computer Science. Amsterdam, 1984 (Preprint) · Zbl 0629.43008
[3] Ban, E.P. van den: The principal series for a reductive symmetric spaces I,H-fixed distribution vectors. Utrecht, 1986 (Preprint) · Zbl 0714.22009
[4] Bernstein, I.N.: The analytic continuation of generalized functions with respect to a parameter. Funct. Anal. Applic6, 4 (1972)
[5] Bernstein, I.N., Gelfand, S.I.: Meromorphic property of the functionsP ?. Funct. Anal. Applic.3, 1 (1969) · Zbl 0249.34035 · doi:10.1007/BF01078269
[6] Björk, J.-E.: Rings of differential operators. North-Holland Publishing Company: Amsterdam Oxford New York 1979
[7] Dijk, G. van, Poel, M.: The plancherel formula for the pseudo-Riemannian spaceSL(n,?)/GL(n?1,?). Compos. Math.58, 371-397 (1986) · Zbl 0593.43009
[8] Faraut, J.: Distributions sphériques sur les espaces hyperboliques. J. Math. Pures Appl. (9)58, 369-444 (1979) · Zbl 0436.43011
[9] Flensted-Jensen, M.: Harmonic analysis on semisimple symmetric spaces. A method of duality. In: Herb, R., Johnson, R., Lipsmann, R., Rosenberg, J. (eds.) Lie Groups Representations III. Proceedings, University of Maryland, 1982-1983, Lect. Notes Math. vol. 1077, 1984
[10] Gaal, S.: Linear analysis and representation theory. Springer: Berlin Heidelberg New York, 1973 · Zbl 0275.43008
[11] Harish-Chandra: Spherical functions on a semi-simple Lie Groups I Am. J. Math.80, 241-310 (1958) · Zbl 0093.12801 · doi:10.2307/2372786
[12] Helgason, S.: A duality for symmetric spaces with application to group prepresentations. Adv. Math.5, 1-164 (1970) · Zbl 0209.25403 · doi:10.1016/0001-8708(70)90037-X
[13] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. Academic Press: New York London, 1978 · Zbl 0451.53038
[14] Helgason, S.: Groups and geometric analysis, integral geometry, invariant differential operators and spherical functions. Academic Press: New York London, 1984 · Zbl 0543.58001
[15] Holdgrün, H.S.: Topologische Gruppen. Lecture Notes, Göttingen, 1983
[16] Hoogenboom, B.: Intertwining Functions on compact Lie Groups. Thesis, University of Leiden, 1983 · Zbl 0553.43005
[17] Knapp, A.W., Stein, E.M.: Intertwining operators for semisimple groups, II. Invent Math.60, 9-84 (1980) · Zbl 0454.22010 · doi:10.1007/BF01389898
[18] Kosters, M.T.: Sperical distributions on rank one symmetric spaces. Thesis, University of Leiden, 1984
[19] Kosters, W.A.: Harmonic analysis on symmetric spaces. Thesis, University of Leiden, 1984
[20] Kunze, R., Stein, E.M.: Uniformly bounded representations III. Am. J. Math.89, 385-442 (1967) · Zbl 0195.14202 · doi:10.2307/2373128
[21] Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Jpn31, 331-357 (1979) · Zbl 0396.53025 · doi:10.2969/jmsj/03120331
[22] Olafsson, G.: Integral formulas and induced representations associated to an affine symmetric space. Mathematica Gottingensis, No. 33, Göttingen 1984
[23] Oshima, T.: Fourier analysis on semisimple symmetric spaces. In: (ed.) Carmona, H., Vergne, M. Non commutative harmonic analysis and Lie groups. Proceedings Marseille Luminy 1980 Lect. Notes Math.880, 357-369 (1981)
[24] Oshima, T., Sekiguchi, J.: Eigenspaces of invariant differential operators on an affine symmetric space. Invent. Math.57, 1-81 (1980) · Zbl 0434.58020 · doi:10.1007/BF01389818
[25] Schlichtkrull, H.: Hyperfunctions and harmonic analysis on symmetric spaces. Birkhäuser: Boston Basel Stuttgart, 1984 · Zbl 0555.43002
[26] Wallach, N.: Harmonic analysis on homogeneous spaces. Marcel Dekker, Inc., New York, 1973 · Zbl 0265.22022
[27] Warner, G.: Harmonic analysis on semi-simple Lie groups I. Springer: Berlin Heidelberg New York, 1972 · Zbl 0265.22020
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.