Analysis on the minimal representation of \(O(p,q)\). I: Realization via conformal geometry. (English) Zbl 1046.22004

For a semisimple Lie group \(G\) an interesting unitary irreducible representation (called the minimal representation) is the one corresponding via geometric quantization to the minimal nilpotent coadjoint orbit. This representation coincides with the minimal representation studied by Kostant [B. Kostant, The vanishing scalar curvature and the minimal representation of \(SO(4,4)\). Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 85–124 (1990; Zbl 0739.22012)] and Binegar-Zierau [B. Binegar and R. Zierau, Commun. Math. Phys. 138, No. 2, 245–258 (1991; Zbl 0748.22009)].
The present paper deals with the study of the minimal unitary representation of the semisimple Lie group \(G = O (p,q)\) and plays a basic role for subsequent papers by the authors on this topic.
The authors work on a single unitary representation and analyze it by examining its restrictions to natural subgroups of the semisimple Lie group. They study the symmetries of the representation space by breaking the large symmetry originally present in the Lie group \(O (p, q)\) by passing to a subgroup. In this way they consider the restriction from the conformal group \(O (p, q)\) to the subgroup of isometries, where different geometries (which are all locally conformal equivalent) are associated to different choices of the subgroup. Changing the subgroup, one obtains different models of the representation and one can compute the spectrum of the subgroup. Applying methods from conformal geometry of pseudo-Riemannian manifolds (like the functorial properties of the Yamabe operator) they obtain different models of the unitary representation and new proofs of unitarity and irreducibility.
Reviewer: Anna Fino (Torino)


22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E46 Semisimple Lie groups and their representations
53A30 Conformal differential geometry (MSC2010)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI arXiv


[1] Beckner, W., Geometric inequalities in Fourier analysis, (), 36-68 · Zbl 0888.42006
[2] B. Binegar, R. Zierau, Unitarization of a singular representation of SO(p,q), Comm. Math. Phys. 138 (1991) 245-258. · Zbl 0748.22009
[3] Brylinski, R.; Kostant, B., Minimal representations of E6,E7 and E8 and the generalized capelli identity, Proc. nat. acad. sci. U.S.A., 91, 2469-2472, (1994) · Zbl 0812.22009
[4] Brylinski, R.; Kostant, B., Differential operators on conical Lagrangian manifolds, Lie theory and geometry, progress math., 123, 65-96, (1994) · Zbl 0878.58033
[5] Brylinski, R.; Kostant, B., Lagrangian models of minimal representations of E6,E7 and E8, Functional analysis on the eve of the 21st century, 1, progress math., 13, 13-63, (1995)
[6] A. Erdélyi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
[7] A. Erdélyi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954.
[8] Gelfand, I.M.; Shilov, G.E., Generalized functions, I, (1964), Academic Press New York · Zbl 0115.33101
[9] Helgason, S., Differential geometry, Lie groups and symmetric spaces, Pure appl. math., 80, (1978) · Zbl 0451.53038
[10] Howe, R., Transcending classical invariant theory, J. amer. math. soc., 2, 535-552, (1989) · Zbl 0716.22006
[11] Howe, R.; Tan, E., Homogeneous functions on light cones, Bull. amer. math. soc., 28, 1-74, (1993) · Zbl 0794.22012
[12] Huang, J.-S.; Zhu, C.-B., On certain small representations of indefinite orthogonal groups, Representation theory, 1, 190-206, (1997) · Zbl 0887.22016
[13] Kobayashi, S., Transformation groups in differential geometry, classics math, (1995), Springer Berlin
[14] Kobayashi, T., Singular unitary representations and discrete series for indefinite Stiefel manifolds \(U(p,q;F)/U(p−m,q;F)\), Mem. amer. math. soc., 462, (1992) · Zbl 0752.22007
[15] Kobayashi, T., Discrete decomposability of the restriction of \(Aq(λ)\) with respect to reductive subgroups and its applications, Invent. math., 117, 181-205, (1994) · Zbl 0826.22015
[16] T. Kobayashi, Multiplicity free branching laws for unitary highest weight modules, in: K. Mimachi (Ed.), Proceedings of the Symposium on Representation Theory held at Saga, Kyushu 1997, pp. 9-17.
[17] Kobayashi, T., Discrete decomposability of the restriction of \(Aq(λ)\) with respect to reductive subgroups II—micro-local analysis and asymptotic K-support, Ann. of math., 147, 709-729, (1998) · Zbl 0910.22016
[18] Kobayashi, T., Discrete decomposability of the restriction of \(Aq(λ)\) with respect to reductive subgroups III—restriction of harish-chandra modules and associated varieties, Invent. math., 131, 229-256, (1998) · Zbl 0907.22016
[19] T. Kobayashi, Discretely decomposable restrictions of unitary representations of reductive Lie groups, in: T. Kobayashi, et al. (Eds.), Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, Vol. 26, Kinokuniya, Tokyo, 2000, pp. 98-126.
[20] T. Kobayashi, Multiplicity-free theorem in branching problems of unitary highest weight modules, preprint.
[21] T. Kobayashi, Branching laws of O(p,q) associated to minimal elliptic orbits, in preparation.
[22] Kobayashi, T.; Ørsted, B., Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits, C. R. acad. sci. Paris, 326, 925-930, (1998) · Zbl 0910.22010
[23] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p,q)—II. Branching laws, to appear in Adv. Math. · Zbl 1049.22006
[24] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p,q)—III. Ultrahyperbolic equations on \(R\^{}\{p−1,q−1\}\), to appear in Adv. Math. · Zbl 1039.22005
[25] B. Kostant, The vanishing scalar curvature and the minimal unitary representation of SO(4,4), in: Connes et al. (Eds.), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Vol. 92, Birkhäuser, Boston, 1990, pp. 85-124. · Zbl 0739.22012
[26] Lee, J.M.; Parker, T.H., The Yamabe problem, Bull. amer. math soc., 17, 37-91, (1987) · Zbl 0633.53062
[27] Ørsted, B., A note on the conformal quasi-invariance of the Laplacian on a pseudo-Riemannian manifold, Lett. math. phys., 1, 183, (1977) · Zbl 0338.53046
[28] Ørsted, B., Conformally invariant differential equations and projective geometry, J. funct. anal., 44, 1-23, (1981) · Zbl 0507.58048
[29] Sabourin, H., Une représentation unipotente associée à l’orbite minimalele cas de SO(4,3), J. funct. anal., 137, 394-465, (1996) · Zbl 0849.22016
[30] Schlichtkrull, H., Eigenspaces of the Laplacian on hyperbolic spacescomposition series and integral transforms, J. funct. anal., 70, 194-219, (1987) · Zbl 0617.43005
[31] W. Schmid, Boundary value problems for group invariant differential equations, Asterisque, hors s’erie. Élie Cartan et les Mathématiques d’aujourd’hui, 1985, pp. 311-321.
[32] Torasso, P., Méthode des orbites de Kirillov-Duflo et representations minimales des groupes simples sur un corps local de caractéristique nulle, Duke math J., 90, 261-377, (1997) · Zbl 0941.22017
[33] D. Vogan, Jr., Singular unitary representations, Noncommutative harmonic analysis and Lie groups, Lecture Notes in Mathematics, Vol. 880, Springer, Berlin, 1980, pp. 506-535.
[34] D. Vogan Jr., Representations of Real Reductive Lie Groups, Progress Math., Vol. 15, Birkhäuser, Basel, 1981. · Zbl 0469.22012
[35] Vogan, D., Unitarizability of certain series of representations, Ann. of math., 120, 141-187, (1984) · Zbl 0561.22010
[36] D. Vogan Jr., Unitary Representations of Reductive Lie Groups, Ann. Math. Stud., Vol. 118, Princeton University Press, Princeton, 1987. · Zbl 0626.22011
[37] Vogan, D., Irreducibility of discrete series representations for semisimple symmetric spaces, Adv. stud. pure math., 14, 191-221, (1988)
[38] H. Wong, Dolbeault cohomologies and Zuckerman modules associated with finite rank representations, Ph.D. Dissertation, Harvard University, 1992.
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.