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Minimal representations via Bessel operators. (English) Zbl 1294.22011

A unified construction of \(L^2\)-models for a family of “smallest” irreducible unitary representations including the minimal ones is presented. The construction provides a unified way to realize the irreducible unitary representations of the Lie groups as Schrödinger models in \(L^2\)-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two. Specifically, the authors use the Bessel operators which are naturally defined in terms of the Jordan structure. The definition of these operators is given and it is proven that they restrict to differential operators on the orbits of the structure group. In addition, it is shown that they are symmetric operators with respect to the \(L^2\) inner product corresponding to certain equivariant measures on the orbits. The Jordan theoretic orbits are related with the minimal nilpotent orbits of the complexified groups. The construction of the Lie algebra representation on a Lagrangian submanifold is illustrated by the examples of the Segal-Shale-Weil representation and the minimal representation of \(O(p,q)\). In particular, the construction applied to Jordan algebras of split rank one provides the entire complementary series representations of the \(SO(n,1)_0\). The explicit \(K\)-finite \(L^2\)-functions for every \(K\)-type are found by means of the “special functions” associated to certain order four differential operators.

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
20G05 Representation theory for linear algebraic groups
17C30 Associated groups, automorphisms of Jordan algebras
33E30 Other functions coming from differential, difference and integral equations
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[1] L. Barchini, M. Sepanski and R. Zierau, Positivity of zeta distributions and small unitary representations, In: The Ubiquitous Heat Kernel, Boulder, 2003, (eds. J. Jorgenson and L. Walling), Contemp. Math., 398 , Amer. Math. Soc., Providence, RI, 2006, pp. 1-46. · Zbl 1096.22008 · doi:10.1090/conm/398/07482
[2] V. Bargmann and I. T. Todorov, Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of \({\mathrm SO}(n)\), J. Mathematical Phys., 18 (1977), 1141-1148. · Zbl 0364.46016 · doi:10.1063/1.523383
[3] Y. Berest and G. Wilson, Differential isomorphism and equivalence of algebraic varieties, In: Topology, Geometry and Quantum Field Theory, Oxford, 2002, (ed. U. Tillmann), London Math. Soc. Lecture Note Ser., 308 , Cambridge University Press, Cambridge, 2004, pp.,98-126. · Zbl 1166.16307 · doi:10.1017/CBO9780511526398.007
[4] W. Bertram, The Geometry of Jordan and Lie Structures, Lecture Notes in Math., 1754 , Springer-Verlag, Berlin, 2000. · Zbl 1014.17024
[5] R. Brylinski and B. Kostant, Minimal representations, geometric quantization, and unitarity, Proc. Nat. Acad. Sci. U.S.A., 91 (1994), 6026-6029. · Zbl 0803.58023 · doi:10.1073/pnas.91.13.6026
[6] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Math. Ser., Van Nostrand Reinhold Co., New York, 1993. · Zbl 0972.17008
[7] H. Dib, Fonctions de Bessel sur une algèbre de Jordan, J. Math. Pures Appl. (9), 69 (1990), 403-448. · Zbl 0648.33006
[8] A. Dvorsky and S. Sahi, Explicit Hilbert spaces for certain unipotent representations. II, Invent. Math., 138 (1999), 203-224. · Zbl 0937.22006 · doi:10.1007/s002220050347
[9] A. Dvorsky and S. Sahi, Explicit Hilbert spaces for certain unipotent representations. III, J. Funct. Anal., 201 (2003), 430-456. · Zbl 1029.22021 · doi:10.1016/S0022-1236(03)00069-7
[10] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1994. · Zbl 0841.43002
[11] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud., 122 , Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001
[12] W. T. Gan and G. Savin, Uniqueness of Joseph ideal, Math. Res. Lett., 11 (2004), 589-597. · Zbl 1112.17014 · doi:10.4310/MRL.2004.v11.n5.a4
[13] I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol.,I: Properties and Operations, Translated by Eugene Saletan, Academic Press, New York, 1964. · Zbl 0115.33101
[14] S. Gindikin and S. Kaneyuki, On the automorphism group of the generalized conformal structure of a symmetric \(R\)-space, Differential Geom. Appl., 8 (1998), 21-33. · Zbl 0914.53029 · doi:10.1016/S0926-2245(97)00015-6
[15] K.-H. Helwig, Halbeinfache reelle Jordan-Algebren, Math. Z., 109 (1969), 1-28. · Zbl 0174.32301 · doi:10.1007/BF01135571
[16] J. Hilgert, T. Kobayashi, G. Mano and J. Möllers, Special functions associated with a certain fourth-order differential equation, Ramanujan J., 26 (2011), 1-34. · Zbl 1232.33014 · doi:10.1007/s11139-011-9315-0
[17] J. Hilgert, T. Kobayashi, G. Mano and J. Möllers, Orthogonal polynomials associated to a certain fourth order differential equation, Ramanujan J., 26 (2011), 295-310. · Zbl 1236.33017 · doi:10.1007/s11139-011-9338-6
[18] R. Howe, The oscillator semigroup, In: The Mathematical Heritage of Hermann Weyl, Durham, NC, 1987, (ed. R. O. Wells, Jr.), Proc. Sympos. Pure Math., 48 , Amer. Math. Soc., Providence, RI, 1988, pp.,61-132. · doi:10.1090/pspum/048/974332
[19] N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ., 39 , Amer. Math. Soc., Providence, RI, 1968. · Zbl 0218.17010
[20] A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École. Norm. Sup. (4), 9 (1976), 1-29. · Zbl 0346.17008
[21] S. Kaneyuki, The Sylvester’s law of inertia in simple graded Lie algebras, J. Math. Soc. Japan, 50 (1998), 593-614. · Zbl 0939.17009 · doi:10.2969/jmsj/05030593
[22] T. Kobayashi, Discrete decomposability of the restriction of \({A}_q(\lambda)\) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math., 131 (1998), 229-256. · Zbl 0907.22016 · doi:10.1007/s002220050203
[23] T. Kobayashi, Algebraic analysis of minimal representations, Publ. Res. Inst. Math. Sci., 47 (2011), 585-611, special issue in commemoration of the golden jubilee of algebraic analysis. · Zbl 1227.22015 · doi:10.2977/PRIMS/45
[24] T. Kobayashi and G. Mano, Integral formula of the unitary inversion operator for the minimal representation of \(O(p,q)\), Proc. Japan Acad. Ser. A Math. Sci., 83 (2007), 27-31, (full paper) The Schrödinger Model for the Minimal Representation of the Indefinite Orthogonal Group \(O(p,q)\), Mem. Amer. Math. Soc., 213 (2011), no.,1000. · Zbl 1230.22007 · doi:10.3792/pjaa.83.27
[25] 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 Honor of Roger E. Howe, Singapore, 2006, (eds. J.-S. Li, E.-C. Tan, N. Wallach and C.-B. Zhu), World Scientific, Hackensack, 2007, pp.,159-223. · Zbl 1390.22010 · doi:10.1142/9789812770790_0006
[26] T. Kobayashi and J. Möllers, An integral formula for \(L^2\)-eigenfunctions of a fourth-order Bessel-type differential operator, Integral Transforms Spec. Funct., 22 (2011), 521-531. · Zbl 1229.34129 · doi:10.1080/10652469.2010.533270
[27] T. Kobayashi and B. Ørsted, Analysis on the minimal representation of \({\mathrm O}(p,q)\). I. Realization via conformal geometry, Adv. Math., 180 (2003), 486-512. · Zbl 1046.22004 · doi:10.1016/S0001-8708(03)00012-4
[28] T. Kobayashi and B. Ørsted, Analysis on the minimal representation of \({\mathrm O}(p,q)\). II. Branching laws, Adv. Math., 180 (2003), 513-550. · Zbl 1049.22006 · doi:10.1016/S0001-8708(03)00013-6
[29] T. Kobayashi and B. Ørsted, Analysis on the minimal representation of \({\mathrm O}(p,q)\). III. Ultrahyperbolic equations on \(\mathbb{R}^{p-1,q-1}\), Adv. Math., 180 (2003), 551-595. · Zbl 1039.22005 · doi:10.1016/S0001-8708(03)00014-8
[30] T. Kobayashi, B. Ørsted and M. Pevzner, Geometric analysis on small unitary representations of \(GL(N,\mathbb{R})\), J. Funct. Anal., 260 (2011), 1682-1720. · Zbl 1217.22003 · doi:10.1016/j.jfa.2010.12.008
[31] T. Levasseur and J. T. Stafford, Differential operators on some nilpotent orbits, Represent. Theory, 3 (1999), 457-473 (electronic). · Zbl 0947.17004 · doi:10.1090/S1088-4165-99-00084-9
[32] G. Lion and M. Vergne, The Weil Representation, Maslov Index and Theta Series, Progr. Math., 6 , Birkhäuser, Boston, MA, 1980. · Zbl 0444.22005
[33] G. Mano, Semisimple Jordan algebras and Bessel operators, unpublished notes, 2008.
[34] J. Möllers, Minimal representations of conformal groups and generalized Laguerre functions, Ph.D. thesis, University of Paderborn, arXiv: arXiv: 1009.4549 · Zbl 1246.01050
[35] T. Okuda, Smallest complex nilpotent orbits with real points, preprint, arXiv: arXiv: 1402.6796
[36] T. Okuda, Proper actions of \(SL(2,{\bm R})\) on semisimple symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci., 87 (2011), 35-39; (full paper) Classification of semisimple symmetric spaces with proper actions, J. Differential Geom., 94 (2013), 301-342. · Zbl 1221.22020 · doi:10.3792/pjaa.87.35
[37] R. S. Palais, Real Algebraic Differential Topology. Part I, Math. Lecture Ser., 10 , Publish or Perish, Inc., Wimington, 1981. · Zbl 0477.57002
[38] M. Pevzner, Analyse conforme sur les algèbres de Jordan, J. Aust. Math. Soc., 73 (2002), 279-299. · Zbl 1019.17011 · doi:10.1017/S1446788700008831
[39] H. Sabourin, Une représentation unipotente associée à l’orbite minimale: le cas de \(so(4,3)\), J. Funct. Anal., 137 (1996), 394-465. · Zbl 0849.22016 · doi:10.1006/jfan.1996.0052
[40] S. Sahi, Explicit Hilbert spaces for certain unipotent representations, Invent. Math., 110 (1992), 409-418. · Zbl 0779.22006 · doi:10.1007/BF01231340
[41] S. Sahi, Unitary representations on the Shilov boundary of a symmetric tube domain, In: Representation Theory of Groups and Algebras, (Eds. J. Adams, R. Herb, S. Kudla, J.-S. Li, R. Lipsman and J. Rosenberg), Contemp. Math., 145 , Amer. Math. Soc., Providence, RI, 1993, pp.,275-286. · Zbl 0790.22010 · doi:10.1090/conm/145/1216195
[42] S. Sahi, Jordan algebras and degenerate principal series, J. Reine Angew. Math., 462 (1995), 1-18. · Zbl 0822.22006 · doi:10.1515/crll.1995.462.1
[43] S. P. Smith, Gelfand-Kirillov dimension of rings of formal differential operators on affine varieties, Proc. Amer. Math. Soc., 90 (1984), 1-8. · Zbl 0538.16017 · doi:10.2307/2044657
[44] J. Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture Notes in Math., 40 , Springer-Verlag, Berlin, 1967. · Zbl 0166.29703 · doi:10.1007/BFb0080324
[45] 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), 261-377. · Zbl 0941.22017 · doi:10.1215/S0012-7094-97-09009-8
[46] A. M. Vershik and M. I. Graev, Structure of the complementary series and special representations of the groups \({O}(n,1)\) and \({U}(n,1)\), Russian Math. Surveys, 61 (2006), 799-884. · Zbl 1148.22017 · doi:10.1070/rm2006v061n05abeh004356
[47] D. A. Vogan, Jr., Singular unitary representations, In: Noncommutative Harmonic Analysis and Lie Groups, Marseille, 1980, (eds. J. Carmona and M. Vergne), Lecture Notes in Math., 880 , Springer-Verlag, Berlin, 1981, pp.,506-535. · doi:10.1007/BFb0090421
[48] N. R. Wallach, Real Reductive Groups. I, Pure Appl. Math., 132 , Academic Press Inc., Boston, MA, 1988. · Zbl 0666.22002
[49] G. K. Zhang, Jordan algebras and generalized principal series representations, Math. Ann., 302 (1995), 773-786. · Zbl 0829.22023 · doi:10.1007/BF01444516
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