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The irreducible unitary GL(n-1,\({\mathbb{R}})\)-spherical representations of SL(n,\({\mathbb{R}})\). (English) Zbl 0723.22018
Let \(G=SL(n,{\mathbb{R}})\), \(\sigma\) an involution of G defined by \(\sigma (g)=JgJ\) where \(J=diag(1,1,...,1,-1)\). Let H be the full subgroup of \(\sigma\)-fixed points. It is isomorphic to GL(n-1,\({\mathbb{R}})\). Note H has two connected components. In the paper reviewed the following problem is solved: to describe all unitary irreducible representations (UIRs) of G which are H-spherical (in other words, of class one w.r.t. H), i.e. have a nontrivial H-invariant distribution vector. A list contains 3 series of UIRs (principal, complementary, relative discrete) and the trivial one. All of them are subquotients of the principal (nonunitary) series associated with the symmetric space G/H. The latter is non-Riemannian, non-isotropic, of rank one. The authors use essentially a description of spherical distributions on G/H given in the paper by M. T. Kosters and G. van Dijk [in J. Funct. Anal. 68, No.2, 168-213 (1985; Zbl 0607.43008)].

MSC:
22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
43A90 Harmonic analysis and spherical functions
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References:
[1] J. Bang-Jensen : The Multiplicities of Certain K-types in Irreducible Spherical Representations of Semisimple Lie Groups . Thesis Massachusetts Institute of Technology, 1987.
[2] J.N. Bernstein and S.I. Gelfand : Tensor products of finite and infinite dimensional representations of semisimple Lie algebras . Comp. Math. 41 (1980), p. 245-285. · Zbl 0445.17006
[3] G Van Dijk & M. Poel : The Plancherel formula for the pseudo-Riemannian space SL(n, R)/GL(n-1, R) . Comp. Math. 58 (1986) p. 371-397. · Zbl 0593.43009
[4] J. Dixmier : Enveloping Algebras . North-Holland Publishing Company, Amsterdam/ New York/Oxford, 1977. · Zbl 0339.17007
[5] J. Faraut : Distributions sphériques sur les espaces hyperboliques . J. Math. Pures et Appl. 58 (1979), p. 369-444. · Zbl 0436.43011
[6] M. Flensted-Jensen & T.H. Koornwinder : Positive definite spherical functions on a non-compact rank one symmetric space . In: Lect. Notes in Math. 739, Springer Verlag Berlin etc., 1979, p. 249-282. · Zbl 0433.43014
[7] S. Helgason : A Duality for Symmetric Spaces with Applications to Group Representations I . Adv. Math. 5 (1970), p. 1-154. · Zbl 0209.25403
[8] A.W. Knapp : Representation Theory of Semisimple Groups. An Overview Based on Examples . Princeton University Press, Princeton N.J., 1986. · Zbl 0604.22001
[9] B. Kostant : On the existence and irreducibility of certain series of representations . In: I.M. Gelfand (ed.) Lie groups and their representations , Halsted Press, New-York, 1975, p. 231-329.(See also Bull. A.M.S. 75 (1969), p. 627-642.) · Zbl 0229.22026
[10] M.T. Kosters : Spherical distributions on rank one symmetric spaces . Thesis University of Leiden, 1983.
[11] M.T. Kosters & G. Van Dijk : Spherical Distributions on the Pseudo-Riemannian space SL(n, R)/GL(n - 1, R) . J. Funct. Anal. 68 (1986), p. 168-213. · Zbl 0607.43008
[12] W.A. Kosters : Eigenspaces of the Laplace-Beltrami operator on SL(n, R)/S(GL(1) x GL(n - 1)), I and II . Ind. Math. 47 (1985), p. 99-145. · Zbl 0576.43006
[13] V.F. Molčanov : The Plancherel formula for the pseudo-Riemannian space SL(3, R)/ GL(2, R) . Sibirsk Math. J. 23 (1982), p. 142-151. · Zbl 0515.22012
[14] M. Poel : Harmonic Analysis on SL(n, R)/GL(n - 1, R) . Thesis University of Utrecht, 1986.
[15] E.G.F. Thomas : The theorem of Bochner-Schwartz-Godement for generalized Gelfand pairs . In: K.D. Bierstedt and B. Fuchsteiner (eds.), Functional Analysis: Surveys and recent results III , Elseviers Science Publishers B.V. (North Holland) (1984). · Zbl 0564.43008
[16] V.S. Varadarajan : Harmonic Analysis on Real Reductive Groups . Lecture Notes in Mathematics, No. 576, Springer-Verlag, Berlin/ Heidelberg/New York, 1977. · Zbl 0354.43001
[17] D.A. Vogan : Representations of Real Reductive Lie Groups . Birkhäuser, Boston/ Basel/ Stuttgart, 1981. · Zbl 0469.22012
[18] D.A. Vogan : The unitary dual of GL(n) over an archimedian field . Invent. Math. 83 (1985), p. 449-505. · Zbl 0598.22008
[19] N.R. Wallach : Harmonic Analysis on Homogeneous Spaces . Dekker, New-York, 1977. · Zbl 0265.22022
[20] G. Zuckerman : Tensor products of finite and infinite dimensional representations of semisimple Lie groups . Ann. Math. 106 (1977), p. 295-308. · Zbl 0384.22004
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