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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

##### MSC:
 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
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##### References:
 [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 Unpublished manuscript (1960).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
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