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Analysis on the minimal representation of \(\mathrm O(p,q)\). II: Branching laws. (English) Zbl 1049.22006
[For Part I, see ibid. 180, No. 2, 486–512 (2003; Zbl 1046.22004).]
Let \(G\) be a Lie group and \(G'\) its subgroup, \(\widehat G\) the unitary dual of \(G\). If \(\pi\in\widehat G\), the restriction \(\pi_{\widehat G'}\) is generally not irreducible and we have an irreducible decomposition \[ \pi_{G'}\cong\int_{\widehat G'}m_{\pi}(\tau)\tau \,d\mu(\tau). \] This is the so-called branching law. The present paper is the second in a series of papers devoted to the analysis of the minimal representation \(\widetilde\omega^{p,q}\) of the indefinite orthogonal group \(G = O(p,q)\), i.e., from the point of view of conformal geometry, the authors find the restriction of \(\widetilde\omega^{p,q}\) with respect to the symmetric pair \((G,G') = (O(p,q),O(p'q')\times O(p'',q''))\): A canonical conclusion obtained by them is the branching law for \(O(pq)\downarrow O(p,q')\times O(q'')\): If \(q''\geq 1\) and \(q'+ q'' = q\), the twisted pull-back \(\widetilde\Phi{*_1}\) of the local conformal map between spheres and hyperboloids gives an explicit irreducible decomposition of the unitary representation \(\tilde\omega^{p,q}\) when restricted to \(O(p,q'')\times O(q'')\) \[ \widetilde\Phi*_1: \widetilde\omega^{p,q}|_{O(p,q')\times O(q'')} \to \sum_{l=0}^{\infty} \pi^{p,q'}_{+,l+q''/2-1}\otimes \mathcal H^l (R^{q''}), \] where the \(\mathcal H^l(R^q)\) are the usual spherical harmonics for the compact orthogonal group \(O(q)\) and the \(\pi^{p,q}_{+,\lambda}\) are the representations of the non-compact orthogonal group \(O(p,q)\), which can be regarded as discrete series representations on hyperboloids, or as cohomological induced representations from characters of certain \(\theta\)-stable parabolic subalgebras. The authors give also the Parseval-Plancherel formula for the above restriction: let \(\widetilde\bigtriangleup_M\) be the Yamabe operator on \(M = S^{p-1}\times S^{q-1}\) and \(F\in \operatorname{Ker} \widetilde\bigtriangleup_M\) and develop \(F\) as \(f =\sum_{l=0}^{\infty} F_l^{(1)}F_l^{(2)}\) according to the above decomposition, then \[ \| F\|^2_{\widetilde\omega^{p,q}}= \sum_{l=0}^{\infty}\| F^{(1)}\|^2_{\pi^{p,q'}_{+,l+q''/2-1}} \| F^{(2)}\|^2_{L^2 (S^{q''}-1)}. \] Finally the authors use certain Sobolev estimates to construct infinitely many discrete spectra when both factors in \(G'\) are non-compact, they also conjecture the form of the fall discrete spectrum.
Reviewer: Fuliu Zhu (Hubei)

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)
Full Text: DOI arXiv
[1] Binegar, B.; Zierau, R., Unitarization of a singular representation of SO(p,q), Comm. math. phys., 138, 245-258, (1991) · Zbl 0748.22009
[2] A. Erdélyi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
[3] A. Erdélyi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954.
[4] Faraut, J., Distributions sphériques sur LES espaces hyperboliques, J. math. pures appl., 58, 369-444, (1979) · Zbl 0436.43011
[5] Howe, R.; Tan, E., Homogeneous functions on light cones, Bull. amer. math. soc., 28, 1-74, (1993) · Zbl 0794.22012
[6] T. Kobayashi, Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds \(U(p,q;F)/U(p−m,q;F)\), Mem. Amer. Math. Soc., Vol. 462, Amer. Math. Soc., Providence, RI, 1992. · Zbl 0752.22007
[7] 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
[8] 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, 1997 pp. 9-17.
[9] Kobayashi, T., Discrete decomposability of the restriction of \(Aq(λ)\) with respect to reductive subgroups II—micro-local analysis and asymptotic K-support, Ann. math., 147, 709-729, (1998) · Zbl 0910.22016
[10] 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
[11] 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
[12] T. Kobayashi, Multiplicity-free theorem in branching problems of unitary highest weight modules, preprint.
[13] 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, Vol. 26, 2000, pp. 98-126.
[14] T. Kobayashi, Branching laws of O(p,q) associated to minimal elliptic orbits, in preparation.
[15] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p,q)—I, Realization via conformal geometry, to appear in Adv. in Math. · Zbl 1046.22004
[16] T.H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions: Group theoretical aspects and applications, Math. Appl. (1984) 1-85.
[17] B. Kostant, The vanishing scalar curvature and the minimal unitary representation of SO(4,4) in: A. Connes et al. (Eds.), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Vol. 92, Birkhäuser, Boston, 1990, pp. 85-124. · Zbl 0739.22012
[18] Schlichtkrull, H., Eigenspaces of the Laplacian on hyperbolic spacescomposition series and integral transforms, J. funct. anal., 70, 194-219, (1987) · Zbl 0617.43005
[19] D. Vogan Jr., Representations of Real Reductive Lie Groups, Progress in Math. Vol. 15, Birkhäuser, Basel, 1981. · Zbl 0469.22012
[20] Vogan, D., Unitarizability of certain series of representations, Ann. math., 120, 141-187, (1984) · Zbl 0561.22010
[21] D. Vogan Jr., Unitary Representations of Reductive Lie Groups, Ann. Math. Stud. Vol. 118, Princeton University Press, Princeton, 1987. · Zbl 0626.22011
[22] Vogan, D., Irreducibility of discrete series representations for semisimple symmetric spaces, Adv. stud. pure math., 14, 191-221, (1988)
[23] H. Wong, Dolbeault cohomologies and Zuckerman modules associated with finite rank representations, Ph. D. Dissertation, Harvard University, 1992.
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