zbMATH — the first resource for mathematics

Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula. (English) Zbl 0645.43009
Author’s introduction: Let G be a connected real semisimple Lie group with finite centre, and let \(\tau\) be an involutive automorphism of G. Put \(G^{\tau}=\{x\in G:\tau (x)=x\}\), and let H be a closed subgroup of G with \((G^{\tau})_ e\subset H\subset G^{\tau}\); here \((G^{\tau})_ e\) denotes the identity component of \(G^{\tau}.\)
In this paper we investigate some properties of the algebra D(X) of invariant differential operators on the semisimple symmetric space \(X=G/H\). Our main results are that the action of D(X) diagonalizes over the discrete part of L 2(X) (Theorem 1.5), and that the irreducible constituents of an abstract Plancherel formula for X occur with finite multiplicities (Theorem 3.1). Both results are proved by using techniques of Harish-Chandra adapted to the situation at hand.
Reviewer: M.Flensted-Jensen

43A85 Harmonic analysis on homogeneous spaces
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
53C35 Differential geometry of symmetric spaces
PDF BibTeX Cite
Full Text: DOI
[1] Casselman, W. andMiliĉić, D., Asymptotic behavior of matrix coefficients of admissible representations.Duke Math. J.49 (1982), 869–930. · Zbl 0524.22014
[2] Van Dijk, G., On generalized Gelfand paris,Proc. Japan Acad., Ser. A60 (1984), 30–34. · Zbl 0555.43010
[3] Faraut, J., Distributions sphériques sur les espaces hyperboliques,J. Math. Pures Appl.,58 (1979), 369–444. · Zbl 0436.43011
[4] Flensted-Jensen, M., Spherical functions on a real semisimple Lie group. A method of reduction to the complex case,J. Funct. Anal. 30 (1978), 106–146. · Zbl 0419.22019
[5] Flensted-Jensen, M., Discrete series for semisimple symmetric spaces,Ann. Math. 111 (1980), 253–311. · Zbl 0462.22006
[6] Harish-Chandra, Representations of a semisimple Lie group on a Banach space I,Trans. Am. Math. Soc. 75 (1953), 185–243. · Zbl 0051.34002
[7] Harish-Chandra, Differential equations and semisimple Lie groups. Unpublished manuscript (1960). In:Collected Papers Vol. III, Springer-Verlag, New York 1984. · Zbl 0080.10202
[8] Harish-Chandra, Discrete series for semisimple Lie groups II,Acta Math. 116 (1966), 1–111. · Zbl 0199.20102
[9] Helgason, S., Differential operators on homogeneous spaces,Acta Math. 102 (1959), 239–299. · Zbl 0146.43601
[10] Helgason, S., Fundamental solutions of invariant differential operators on symmetric spaces,Am. J. Math. 86 (1964), 565–601. · Zbl 0178.17001
[11] Kengmana, T., Characters of the discrete series for pseudo-Riemannian symmetric spaces. In:Representation Theory of Reductive Groups, Proc. of the univ. of Utah Conf. 1982 (P. Trombi, ed.), Birkhaüser, Boston-Basel 1983.
[12] Mackey, G. W.,The theory of unitary group representations, The Univ. of Chicago Press, Chicago 1976. · Zbl 0344.22002
[13] Nelson, E., Analytic vectors,Ann. Math. 70 (1959), 572–615. · Zbl 0091.10704
[14] Oshima, T. andSekigughi, J., Eigenspaces of invariant differential operators on an affine symmetric space,Invent. Math. 57 (1980), 1–81. · Zbl 0434.58020
[15] Oshima, T. andMatsuki, T., A description of discrete series for semisimple symmetric spaces,Adv. Stud. Pure Math. 4 (1984), 331–390.
[16] Penney, R., Abstract Plancherel theorems and a Frobenius reciprocity theorem,J. Funct. Anal. 18 (1975), 177–190. · Zbl 0305.22016
[17] Rossmann, W., Analysis on real hyperbolic spaces,J. Funct. Anal. 30 (1978), 448–477. · Zbl 0395.22014
[18] Rossmann, W., The structure of semisimple symmetric spaces,Can. J. Math. 31 (1979), 157–180. · Zbl 0393.53032
[19] Varadarajan, V. S.,Harmonic Analysis on Real Reductive Groups, Lecture Notes in Math., 5/6, Springer-Verlag, Berlin-Heidelberg 1977. · Zbl 0354.43001
[20] Warner, G.,Harmonic Analysis on Semi-Simple Lie Groups I, Springer-Verlag, Berlin-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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.