Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations
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References
-
H. Abarbanel, The inverse r-squared force; an introduction to its symmetries, Essays in Honor of Valentine Bargmann (E. Lieb, B. Simon, and A. Wightman, eds.), Princeton Univ. Press, Princeton, NJ, 1976.
- R. L. Anderson, J. Fischer, and R. Rączka, Coupling problem for $\textrm {U}(p,\,q)$ ladder representations. I, Proc. Roy. Soc. London Ser. A 302 (1968), 491–500. MR 219659, DOI 10.1098/rspa.1968.0031
- V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568–640. MR 21942, DOI 10.2307/1969129
- Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
- Jean-Luc Brylinski and Masaki Kashiwara, Démonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 6, A373–A376 (French, with English summary). MR 596075
- Thomas P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), no. 2, 199–291. MR 904819, DOI 10.1016/0022-1236(87)90025-5
- M. W. Baldoni Silva, The unitary dual of $\textrm {Sp}(n,\,1)$, $n\geq 2$, Duke Math. J. 48 (1981), no. 3, 549–583. MR 630585
- Luis G. Casian and David H. Collingwood, Weight filtrations for induced representations of real reductive Lie groups, Adv. in Math. 73 (1989), no. 1, 79–146. MR 979588, DOI 10.1016/0001-8708(89)90060-1
- David H. Collingwood, Representations of rank one Lie groups, Research Notes in Mathematics, vol. 137, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 853731
- Michael Cowling, Unitary and uniformly bounded representations of some simple Lie groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 49–128. MR 777340
- Y. Dothan, M. Gell-Mann, and Y. Ne’eman, Series of hadron energy levels as representations of non-compact groups, Phys. Lett. 17 (1965), 148–151. MR 183410, DOI 10.1016/0031-9163(65)90279-9
- Jacques Dixmier, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9–41. MR 140614, DOI 10.24033/bsmf.1558
- Thomas Enright, Roger Howe, and Nolan Wallach, A classification of unitary highest weight modules, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97–143. MR 733809
- M. J. Englefield, Group theory and the Coulomb problem, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972. MR 0406116
- T. J. Enright, R. Parthasarathy, N. R. Wallach, and J. A. Wolf, Unitary derived functor modules with small spectrum, Acta Math. 154 (1985), no. 1-2, 105–136. MR 772433, DOI 10.1007/BF02392820
- J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. (9) 58 (1979), no. 4, 369–444 (French). MR 566654
- Mogens Flensted-Jensen, Analysis on non-Riemannian symmetric spaces, CBMS Regional Conference Series in Mathematics, vol. 61, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 837420, DOI 10.1090/cbms/061
- J. Fischer and R. Rączka, Degenerate representation of non-compact unitary groups. II. Continuous series, Comm. Math. Phys. 4 (1967), 8–21. MR 202923, DOI 10.1007/BF01645174 C. Fronsdal, Infinite multiplets and the hydrogen atom, Phys. Rev. (3) 156 (1967), 1665-1677.
- I. M. Gelfand and M. A. Neumark, Unitäre Darstellungen der klassischen Gruppen, Akademie-Verlag, Berlin, 1957 (German). MR 0085262
- Kenneth I. Gross, The dual of a parabolic subgroup and a degenerate principal series of $\textrm {Sp}(n,\,\textbf {C})$, Amer. J. Math. 93 (1971), 398–428. MR 304558, DOI 10.2307/2373384
- Alain Guillemonat, On some semispherical representations of an Hermitian symmetric pair of the tubular type. Study in the symmetric algebra, Manuscripta Math. 31 (1980), no. 4, 331–361. MR 586130, DOI 10.1007/BF02320699
- Harish-Chandra, Infinite irreducible representations of the Lorentz group, Proc. Roy. Soc. London Ser. A 189 (1947), 372–401. MR 21941, DOI 10.1098/rspa.1947.0047
- Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185–243. MR 56610, DOI 10.1090/S0002-9947-1953-0056610-2
- Takeshi Hirai, On irreducible representations of the Lorentz group of $n-\textrm {th}$ order, Proc. Japan Acad. 38 (1962), 258–262. MR 191436
- Takeshi Hirai, Classification and the characters of irreducible representations of $\textrm {SU}(p,\,1)$, Proc. Japan Acad. 42 (1966), 907–912. MR 223491
- Roger Howe, On some results of Strichartz and Rallis and Schiffman, J. Functional Analysis 32 (1979), no. 3, 297–303. MR 538856, DOI 10.1016/0022-1236(79)90041-7
- Roger Howe, On a notion of rank for unitary representations of the classical groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 223–331. MR 777342
- Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
- Felix E. Browder (ed.), American Mathematical Society centennial publications. Vol. II, American Mathematical Society, Providence, RI, 1992. Mathematics into the twenty-first century. MR 1184608
- Nathan Jacobson, Lectures in abstract algebra. Vol. II. Linear algebra, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0053905
- Kenneth D. Johnson, Composition series and intertwining operators for the spherical principal series. II, Trans. Amer. Math. Soc. 215 (1976), 269–283. MR 385012, DOI 10.1090/S0002-9947-1976-0385012-X
- Hans Plesner Jakobsen and Michele Vergne, Wave and Dirac operators, and representations of the conformal group, J. Functional Analysis 24 (1977), no. 1, 52–106. MR 0439995, DOI 10.1016/0022-1236(77)90005-2
- Kenneth D. Johnson and Nolan R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc. 229 (1977), 137–173. MR 447483, DOI 10.1090/S0002-9947-1977-0447483-0
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- A. U. Klimyk and A. M. Gavrilik, The representations of the groups $\textrm {U}(n,\,1)$ and $\textrm {SO}_{0}(n,\,1)$, Akad. Nauk Ukrain. SSR, Inst. Teoret. Fiz., Kiev, 1976 (English, with Russian summary). MR 0579617
- Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
- Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
- Stephen S. Kudla and Stephen Rallis, Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), no. 1, 25–45. MR 1046171, DOI 10.1007/BF02764727
- M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. MR 463359, DOI 10.1007/BF01389900
- M. Kashiwara and M. Vergne, Functions on the Shilov boundary of the generalized half plane, Noncommutative harmonic analysis (Proc. Third Colloq., Marseille-Luminy, 1978) Lecture Notes in Math., vol. 728, Springer, Berlin, 1979, pp. 136–176. MR 548329
- George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, DOI 10.2307/1969786 V. F. Molcanov, Analogue of the Plancherel Formula for hyperboloids, Soviet Math. Doklady 9 (1968), 1387-1385.
- M. A. Naĭmark, Lineĭ nye predstavleniya gruppy Lorentsa, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR 0106261
- Kyo Nishiyama, Algebraic structures on virtual characters of a semisimple Lie group, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 417–468. MR 1039847, DOI 10.2969/aspm/01410417
- Enrico Onofri, Dynamical quantization of the Kepler manifold, J. Mathematical Phys. 17 (1976), no. 3, 401–408. MR 395530, DOI 10.1063/1.522907
- Bent Ørsted, Conformally invariant differential equations and projective geometry, J. Functional Analysis 44 (1981), no. 1, 1–23. MR 638292, DOI 10.1016/0022-1236(81)90002-1
- Stephen M. Paneitz and Irving E. Segal, Analysis in space-time bundles. I. General considerations and the scalar bundle, J. Functional Analysis 47 (1982), no. 1, 78–142. MR 663834, DOI 10.1016/0022-1236(82)90101-X
- Stephen M. Paneitz and Irving E. Segal, Analysis in space-time bundles. II. The spinor and form bundles, J. Functional Analysis 49 (1982), no. 3, 335–414. MR 683028, DOI 10.1016/0022-1236(82)90073-8
- Stephen M. Paneitz, Analysis in space-time bundles. III. Higher spin bundles, J. Funct. Anal. 54 (1983), no. 1, 18–112. MR 724645, DOI 10.1016/0022-1236(83)90041-1
- Joe Repka, Tensor products of holomorphic discrete series representations, Canadian J. Math. 31 (1979), no. 4, 836–844. MR 540911, DOI 10.4153/CJM-1979-079-9
- Wulf Rossmann, Analysis on real hyperbolic spaces, J. Functional Analysis 30 (1978), no. 3, 448–477. MR 518343, DOI 10.1016/0022-1236(78)90065-4
- S. Rallis and G. Schiffmann, Discrete spectrum of the Weil representation, Bull. Amer. Math. Soc. 83 (1977), no. 2, 267–270. MR 429753, DOI 10.1090/S0002-9904-1977-14299-7
- Siddhartha Sahi, The Capelli identity and unitary representations, Compositio Math. 81 (1992), no. 3, 247–260. MR 1149169
- Birgit Speh, Degenerate series representations of the universal covering group of $\textrm {SU}(2,\,2)$, J. Functional Analysis 33 (1979), no. 1, 95–118. MR 545386, DOI 10.1016/0022-1236(79)90019-3
- Birgit Speh, The unitary dual of $\textrm {Gl}(3,\,\textbf {R})$ and $\textrm {Gl}(4,\,\textbf {R})$, Math. Ann. 258 (1981/82), no. 2, 113–133. MR 641819, DOI 10.1007/BF01450529 A. Salam and J. Strathdee, Relativistic ${U(6,6)}$ Theory, Phys. Rev. 148 (1966), 1352-1358.
- Henrik Schlichtkrull, Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms, J. Funct. Anal. 70 (1987), no. 1, 194–219. MR 870761, DOI 10.1016/0022-1236(87)90130-3
- Jir\B{o} Sekiguchi, Eigenspaces of the Laplace-Beltrami operator on a hyperboloid, Nagoya Math. J. 79 (1980), 151–185. MR 587417, DOI 10.1017/S0027763000019000 R. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth and Cole, Monterey, CA, 1986.
- Robert S. Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis 12 (1973), 341–383. MR 0352884, DOI 10.1016/0022-1236(73)90001-3
- Robert S. Strichartz, Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal. 87 (1989), no. 1, 51–148. MR 1025883, DOI 10.1016/0022-1236(89)90004-9
- Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
- Shlomo Sternberg and Joseph A. Wolf, Hermitian Lie algebras and metaplectic representations. I, Trans. Amer. Math. Soc. 238 (1978), 1–43. MR 486325, DOI 10.1090/S0002-9947-1978-0486325-5
- A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201–218. MR 126154, DOI 10.7146/math.scand.a-10610
- Reiji Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289–433 (French). MR 179296, DOI 10.24033/bsmf.1598
- N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863, DOI 10.1090/mmono/022
- David A. Vogan Jr., Gel′fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. MR 506503, DOI 10.1007/BF01390063 —, Representations of real reductive groups, Progress in Math., vol. 15, Birkhauser, Boston, MA, 1981.
- David A. Vogan, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math. 71 (1983), no. 2, 381–417. MR 689650, DOI 10.1007/BF01389104
- David A. Vogan Jr., Unitary representations of reductive Lie groups, Annals of Mathematics Studies, vol. 118, Princeton University Press, Princeton, NJ, 1987. MR 908078
- Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
- D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 28 (1993), 1-74
- MSC: Primary 22E46; Secondary 17B10, 22-02
- DOI: https://doi.org/10.1090/S0273-0979-1993-00360-4
- MathSciNet review: 1172839