Integral formula of the unitary inversion operator for the minimal representation of \(\text{O}(p,q)\). (English) Zbl 1230.22007

Summary: The indefinite orthogonal group \(G = \text{O}(p,q)\) has a distinguished infinite dimensional unitary representation \(\pi\), called the minimal representation for \(p+q\) even and greater than 6. The Schrödinger model realizes \(\pi\) on a very simple Hilbert space, namely, \(L^2(C)\) consisting of square integrable functions on a Lagrangian submanifold \(C\) of the minimal nilpotent coadjoint orbit, whereas the \(G\)-action on \(L^2(C)\) has not been well-understood. This paper gives an explicit formula of the unitary operator \(\pi(w_0)\) on \(L^2(C)\) for the ‘conformal inversion’ \(w_0\) as an integro-differential operator, whose kernel function is given by a Bessel distribution. Our main theorem generalizes the classic Schrödinger model on \(L^2(\mathbb R^n)\) of the Weil representation, and leads us to an explicit formula of the action of the whole group \(\text{O}(p,q)\) on \(L^2(C)\). As its corollaries, we also find a representation theoretic proof of the inversion formula and the Plancherel formula for Meijer’s \(G\)-transforms.


22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations
43A80 Analysis on other specific Lie groups
Full Text: DOI arXiv Euclid


[1] B. Binegar and R. Zierau, Unitarization of a singular representation of \({\mathrm SO}(p,q)\), Comm. Math. Phys. 138 (1991), no. 2, 245-258. · Zbl 0748.22009
[2] R. Brylinski and B. Kostant, Differential operators on conical Lagrangian manifolds, in Lie theory and geometry , 65-96, Progr. Math., 123, Birkhäuser, Boston, Boston, MA, 1994. · Zbl 0878.58033
[3] G. B. Folland, Harmonic analysis in phase space , Ann. of Math. Stud., 122, Princeton Univ. Press, Princeton, NJ, 1989. · Zbl 0682.43001
[4] C. Fox, The \(G\) and \(H\) functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429.
[5] R. Howe, The oscillator semigroup, in The mathematical heritage of Hermann Weyl ( Durham, NC , 1987), 61-132, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988. · Zbl 0687.47034
[6] R. E. Howe and E.-C. Tan, Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 1, 1-74. · Zbl 0794.22012
[7] C.-B. Zhu and J.-S. Huang, On certain small representations of indefinite orthogonal groups, Represent. Theory 1 (1997), 190-206. · Zbl 0887.22016
[8] T. Kobayashi, Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds, Rend. Circ. Mat. Palermo (2) Suppl. No. 71 (2003), 15-40. · Zbl 1074.53031
[9] T. Kobayashi and G. Mano, Integral formulas for the minimal representation of \(O(p,2)\), Acta Appl. Math. 86 (2005), no. 1-2, 103-113. · Zbl 1100.22007
[10] T. Kobayashi and G. Mano, The inversion formula and holomorphic extension of the minimal representation of the conformal group, in Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory: In Honour of Roger E. Howe (J.-S. Li, E.-C. Tan, N. Wallach, and C.-B. Zhu, eds.), Singapore University Press and World Scientific Publishing. (to appear). · Zbl 1390.22010
[11] T. Kobayashi and G. Mano, The Schrödinger model for the minimal representation of the indefinite orthogonal group \(O(p,q)\). (in preparation). · Zbl 1225.22001
[12] T. Kobayashi and B. Ørsted, Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 8, 925-930. · Zbl 0910.22010
[13] T. Kobayashi and B. Ørsted, Analysis on the minimal representation of \(O(p,q)\), I, II, III. Adv. Math. 180 (2003), 486-512, 513-550, 551-595. · Zbl 1046.22004
[14] B. Kostant, The vanishing of scalar curvature and the minimal representation of \(SO(4,4)\), in Operator algebras, unitary representations, enveloping algebras, and invariant theory ( Paris , 1989), 85-124, Progr. Math., 92, Birkhäuser, Boston, Boston, MA, 1990. · Zbl 0739.22012
[15] G. Mano, Radon transform of functions supported on a homogeneous cone, Ph.D. thesis, Research Institute for Mathematical Sciences, Kyoto University, 2007.
[16] E. C. Titchmarsh, Introduction to the theory of Fourier integrals , Oxford Univ. Press, Oxford, 1937; Third edition, Chelsea, New York, 1986. · Zbl 0017.40404
[17] P. Torasso, Méthode des orbites de Kirillov–Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle, Duke Math. J. 90 (1997), no. 2, 261-377. · Zbl 0941.22017
[18] D. A. Vogan, Jr., Singular unitary representations, in Noncommutative harmonic analysis and Lie groups ( Marseille , 1980), 506-535, Lecture Notes in Math., 880, Springer, Berlin. · Zbl 0464.22007
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.