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Integral formulas for the minimal representation of \(O(p,2\)). (English) Zbl 1100.22007

Summary: The minimal representation \(\pi\) of \(O(p,q)\) (\(p+q\) even) is realized on the Hilbert space of square integrable functions on the conical subvariety of \(\mathbb R^{p+q-2}\). This model presents a close resemblance of the Schrödinger model of the Segal–Shale–Weil representation of the metaplectic group. We shall give explicit integral formulas for the “inversion” together with the analytic continuation to a certain semigroup of \(O(p+2,\mathbb C\)) of the minimal representation of \(O(p,2)\) by using Bessel functions.

MSC:

22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations
20M20 Semigroups of transformations, relations, partitions, etc.
43A80 Analysis on other specific Lie groups
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[1] Binegar, B. and Zierau, R.: Unitarization of a singular representation of SO(p,q), Comm. Math. Phys. 138 (1991), 245-258. · Zbl 0748.22009 · doi:10.1007/BF02099491
[2] Dvorsky, A. and Sahi, S.: Explicit Hilbert spaces for certain unipotent representations II, Invent. Math. 138 (1999), 203-224. · Zbl 0937.22006 · doi:10.1007/s002220050347
[3] Folland, B.: Harmonic Analysis in Phase Space, Ann. of Math. Studies 122, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001
[4] Howe, R.: The oscillator semigroup, In: Proc. Sympos. Pure Math. 48, Amer. Math. Soc., 1988, pp. 61-132. · Zbl 0687.47034
[5] Huang, J.-S. and Zhu, C.-B.: On certain small representations of indefinite orthogonal groups, Represent. Theory 1 (1997), 190-206. · Zbl 0887.22016 · doi:10.1090/S1088-4165-97-00031-9
[6] Kobayashi, T.: Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds, In: Proceedings of the 22nd Winter School ?Geometry and Physics? (Srni, 2002), Rend. Circ. Mat. Palermo (2) Suppl. 71, 2003, pp. 15-40.
[7] Kobayashi, T. and Mano, G.: The inversion formula and holomorphic extension of the minimal representation of O(p,2), in preparation. · Zbl 1230.22007
[8] Kobayashi, T. and Ørsted, B.: Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry, Adv. Math. 180 (2003), 486-512. · Zbl 1046.22004 · doi:10.1016/S0001-8708(03)00012-4
[9] Kobayashi, T. and Ørsted, B.: Analysis on the minimal representation of O(p,q) III. Ultrahyperbolic equations on Rp?1,q?1, Adv. Math. 180 (2003), 551-595. · Zbl 1039.22005 · doi:10.1016/S0001-8708(03)00014-8
[10] Kostant, B.: 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, Progress in Math. 92, Birkhäuser, Boston, 1990, pp. 85-124. · Zbl 0739.22012
[11] Sahi, S.: Explicit Hilbert spaces for certain unipotent representations, Invent. Math. 110 (1992), 409-418. · Zbl 0779.22006 · doi:10.1007/BF01231340
[12] Torasso, P.: Méthode des orbites de Kirillov?Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle, Duke Math. J. 90 (1997), 261-377. · Zbl 0941.22017 · doi:10.1215/S0012-7094-97-09009-8
[13] Vogan, D. Jr.: Singular unitary representations, In: Lecture Notes in Math. 880, Springer, 1980, pp. 506-535.
[14] Watson, G. N.: A Treatise on the Theory of Bessel Functions, Cambridge, 1922. · JFM 48.0412.02
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