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Discrete decomposability of restriction of \(A_{\mathfrak q}(\lambda)\) with respect to reductive subgroups and its applications. (English) Zbl 0826.22015

Sei \(G\) eine lokalkompakte Gruppe vom Typ 1 und \((\pi, V)\) eine unitäre Darstellung von \(G\) auf dem Hilbertraum \(V\). \((\pi,V)\) heißt diskret zerlegbar, falls sie unitär äquivalent zu \(\sum^\oplus_m (\tau) (\tau, H_\tau)\), \((\tau, H_\tau)\) irreduz., ist (mit \(m(\tau) := \dim_\mathbb{C} \text{Hom}_G (H_\tau, V)\)). Man nennt \((\pi, V)\) \(G\)- zulässig, falls die Multiplizität \(m(\tau) < \infty\). Ein erstes Ergebnis der vorliegenden Arbeit ist, daß die \(K\)-Zulässigkeit von \((\pi|_K, V)\) für eine Untergruppe \(K\) die \(G\)-Zulässigkeit von \((\pi, V)\) impliziert. Hauptresultat ist eine hinreichende Bedingung für die \(G'\)-Zulässigkeit der Zuckermanschen induzierten Darstellung \(\overline {A_q (\lambda)}|_{G'}\) für den Fall symmetrischer Paare \((G,G')\) und holomorpher Einbettungen. Als Anwendung ergeben sich neue Erkenntnisse über diskrete Serien für gewisse pseudoriemannsche (nicht-symmetrische) sphärische homogene Räume.

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22D10 Unitary representations of locally compact groups
22D30 Induced representations for locally compact groups
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[1] [A1] Adams, J.: Discrete spectrum of the dual reductive pair (O(p, q), Sp(2m)). Invent. Math.74, 449-475 (1984) · Zbl 0561.22011
[2] [A2] Adams, J.: Unitary highest weight modules. Adv. Math.63, 113-137 (1987) · Zbl 0623.22014
[3] [Bo] Borel, A.: Some remarks about Lie groups transitive on spheres and tori. Bull.Am. Math. Soc.55, 580-587 (1949) · Zbl 0034.01603
[4] [BoW] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Princeton: Princeton University Press 1980 · Zbl 0443.22010
[5] [Br] Brion, M.: Classification des espaces homogenes spheriques. Compos. Math.63-2, 189-208 (1987)
[6] [C] Chang, J.T.: Remarks on localization and standard modules: The duality theorem on a generalized flag variety. Proc. Am. Math. Soc.117, 585-591 (1993) · Zbl 0833.22020
[7] [EPWW] Enright, T.J., Parthasarathy, R., Wallach, N.R., Wolf, J.A.: Unitary derived functor module with small spectrum. Acta. Math.154, 105-136 (1985) · Zbl 0568.22007
[8] [FJ] Flensted-Jensen, M.: Analysis on Non-Riemannian Symmetric Spaces. (Conf. Board, vol. 61) Providence, RI: Am. Math. Soc. 1986 · Zbl 0589.43008
[9] [GG] Gelfand, I.M., Graev, M.I.: Geometry of homogeneus spaces, representations of groups in homogeneous spaces, and related questions of integral geometry. Transl., II. Ser., Am. Math. Soc.37, 351-429 (1964) · Zbl 0136.43404
[10] [GGP] Gelfand, I.M., Graev, M.I., Piatecki-?apiro, I.: Representation theory and automorphic functions, Phiadelphia: Saunders 1969
[11] [HMSW] Hecht, H., Mili?ic, D., Schmid, W., Wolf, J.A.: Localization and standard modules for real semisimple Lie groups. Invent. Math.90, 297-332 (1987) · Zbl 0699.22022
[12] [HS] Hecht, H., Schmid, W.: A proof of Blattner’s conjecture. Invent. Math.31, 129-154 (1976) · Zbl 0319.22012
[13] [He] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. (Pure Appl. Math., vol. 80) New York London: Academic Press, 1978 · Zbl 0451.53038
[14] [Ho] Howe, R.: ?-series and invariant theory. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations and L-functions. (Proc. Symp. Pure Math., vol. 33, pp. 275-285) Providence, RI: Am. Math. Soc. 1979
[15] [HT] Howe, R., Tan, E.: Homogeneous functions on light cones: The infinitesimal structures of some degenerate principal series representations. Bull. Am. Math. Soc.28, 1-74 (1993) · Zbl 0794.22012
[16] [J] Jakobsen, H.P.: Tensor products, reproducing kernels, and power series. J. Funct. Anal.31, 293-305 (1979) · Zbl 0403.22011
[17] [JV] Jakobsen, H.P., Vergne, M.: Restrictions and expansions of holomorphic representations. J. Funct. Anal.34, 29-53 (1979) · Zbl 0433.22011
[18] [KV] Kashiwara, M., Vergne, M.: On the Segal-Shale-Weil representations and harmonic polynomials. Invent. Math.44, 1-47 (1978) · Zbl 0375.22009
[19] [Ko1] Kobayashi, T.: Proper action on a homogeneous space of reductive type. Math. An.,285, 249-263 (1989) · Zbl 0662.22008
[20] [Ko2] Kobayashi, T.: Unitary representations realized in L2-sections of vector bundles over semisimple symmetric spaces. In: Proceedings at the 27-th. Symp. of Functional Analysis and Real Analysis, pp. 39-54 (in Japanese). Math. Soc. Japan 1989
[21] [Ko3] Kobayashi, T.: Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds \(U(p,q; \mathbb{F})/U(p - m,q; \mathbb{F})\) . Mem. Am. Math. Soc.462 (1992)
[22] [Ko4] Kobayashi, T.: Discrete decomposability of the restriction ofA g(?) with respect to reductive subgroups. II. Classification for classical symmetric pairs. (in preparation)
[23] [Kr1] Krämer, M.: Multilicity free subgroups of compact connected Lie groups. Arch. Math.27, 28-35 (1976) · Zbl 0322.22011
[24] [Kr2] Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math.38-2, 129-153 (1979) · Zbl 0402.22006
[25] [L] Lipsman, R.: Restrictions of principal series to a real form. Pac. J. Math.89, 367-390 (1980) · Zbl 0453.22009
[26] [M] Martens, S.: The characters of the holomorphic discrete series. Proc. Natl. Acad. Sci. USA72, 3275-3276 (1975) · Zbl 0308.22013
[27] [MO1] Matsuki, T., Oshima, T.: A description of discrete series for semisimple symmetric spaces. Adv. Stud. Pure Math.4, 331-390 (1984) · Zbl 0577.22012
[28] [MO2] Matsuki, T., Oshima, T.: Embeddings of discrete series into principal series In: Dulfo, M. et al. (eds.) The orbit method in representation theory. (Prog. Math., vol. 80, pp. 147-175. Boston Buset Stuttgart: Birkhäuser 1990 · Zbl 0746.22011
[29] [OO] Olafsson, G., Ørsted, B.: The holomorphic discrete series of an affine symmetric space. I. J. Funct. Anal.81, 126-159 (1988) · Zbl 0678.22008
[30] [Os] Oshima, T.: Asymptotic behavior of spherical functions on semisimple symmetric spaces. Adv. Stud. Pure Math.14, 561-601 (1988)
[31] [R] Repka, J.: Tensor products of holomorphic discrete series representations. Can. J. Math.31, 836-844 (1979) · Zbl 0407.22013
[32] [S1] Schlichtkrull, H.: A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group. Invent. Math.68, 497-516 (1982) · Zbl 0501.22019
[33] [S2] Schlichtkrull, H.: Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms. J. Funct. Anal.70, 194-219 (1987) · Zbl 0617.43005
[34] [V1] Vogan, D.: Representations of real reductive Lie groups, Boston Basel Stuttgart: Birkhäuser 1981 · Zbl 0469.22012
[35] [V2] Vogan, D.: Unitary representations of reductive Lie groups. Princeton, NJ: Princeton University Press 1987 · Zbl 0626.22011
[36] [V3] Vogan, D.: Irreducibility of discrete series representations for semisimple symmetric spaces. Adv. Stud. Pure Math.14, 191-221 (1988)
[37] [VZ] Vogan, D., Zuckerman, G.J.: Unitary representations with non-zero cohomology. Compos. Math.53, 51-90 (1984) · Zbl 0692.22008
[38] [Wa1] Wallach, N.: Real reductive groups. I. (Pure Appl. Math., vol. 132), Boston: Academic Press 1988
[39] [War] Warner, G.: Harmonic analysis on semisimple Lie groups. I. Berlin Heidelberg, New York: Springer 1972
[40] [Wi] Williams, F.: Tensor products of principal series representations. (Lect. Notes Math., vol. 358) Berlin Heidelberg New York: Springer 1973
[41] [Y] Yamashita, H.: Criteria for the finiteness of restriction of U(g) to subalgebras and applications to Harish-Chandra modules. Proc. Japan Acad.68, 316-321 (1992) · Zbl 0773.17006
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