Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations


Authors: Roger E. Howe and Eng-Chye Tan
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
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [Ab] 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.
  • [AFR] R. Anderson, J. Fischer, and R. Raczka, Coupling problem for $ {U(p,q)}$ ladder representations. I, Proc. Roy. Soc. London Ser. A 302 (1968), 491-500. MR 0219659 (36:2738)
  • [Ba] V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568-640. MR 0021942 (9:133a)
  • [BB] A. Beilinson and J. Bernstein, Localisation de $ {\mathfrak{g}}$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 15-18. MR 610137 (82k:14015)
  • [BK] J-L. Brylinski, and M. Kashiwara, Demonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma, C. R. Acad. Sci. Paris Sér. I Math. 291 (1980), 373-376. MR 596075 (81k:17004)
  • [Br] T. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199-241. MR 904819 (90b:22016)
  • [BS] M. Baldoni-Silva, The unitary dual of $ {Sp(n,1)}$, $ {n \geq 2}$, Duke Math. J. 48 (1981), 549-583. MR 630585 (83e:22019)
  • [CC] L. Casian and D. Collingwood, Weight filtrations for induced representations of real reductive groups, Adv. Math. 73 (1989), 79-144. MR 979588 (90c:22046)
  • [Cl] D. Collingwood, Representations of rank one Lie groups, Pitman Res. Notes Math. Ser., vol. 137, Pitman Publishing, Boston, MA, 1985. MR 853731 (88c:22014)
  • [Cw] M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, Harmonic Analysis and Group Representations, C.I.M.E. II ciclo 1980, Liguori editore, Naples, 1982, pp. 49-128. MR 777340 (86h:22012)
  • [DGN] 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 0183410 (32:891)
  • [Di] J. Dixmier, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France 89 (1961), 9-41. MR 0140614 (25:4031)
  • [EHW] T. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, Representation Theory of Reductive Lie Groups (P. Trombi, ed.), Birkhauser, Boston, MA, 1983, pp. 97-143. MR 733809 (86c:22028)
  • [En] M. Englefield, Group theory and the Coulomb problem, Wiley-Interscience, New York, 1972. MR 0406116 (53:9908)
  • [EPWW] T. Enright, R. Parthasarathy, N. Wallach, and J. Wolf, Unitary derived functor modules with small spectrum, Acta Math. 154 (1985), 105-136. MR 772433 (86j:22026)
  • [Fa] J. Faraut, Distributions spheriques sur les espaces hyperboliques, J. Math. Pures Appl. 58 (1979), 369-444. MR 566654 (82k:43009)
  • [FJ] M. Flensted-Jensen, Analysis on non-Riemannian Symmetric Spaces, CBMS Reg. Conf. Ser. in Math., vol. 61, Amer. Math. Soc., Providence, RI, 1986, pp. 1-77. MR 837420 (87h:43013)
  • [FR] J. Fischer and R. Raczka, Degenerate representations of non-compact unitary groups. II, Comm. Math. Phys. 4 (1967), 8-21. MR 0202923 (34:2782)
  • [Fr] C. Fronsdal, Infinite multiplets and the hydrogen atom, Phys. Rev. (3) 156 (1967), 1665-1677.
  • [GN] I. M. Gel'fand and M. A. Naĭmark, Unitäre darstellungen der klassischen gruppen, Acadamie-Verlag, Berlin, 1957. MR 0085262 (19:13g)
  • [Gr] K. Gross, The dual of a parabolic subgroup and a degenerate principal series of $ {Sp(n,\mathbb{C})}$, Amer. J. Math. 93 (1971), 398-428. MR 0304558 (46:3693)
  • [Gu] A. Guillemonat, On some semi-spherical representations of a hermitian symmetric pair of tubular type, Math. Ann. 246 (1980), 93-116. MR 564679 (84m:22021b)
  • [HC1] Harish-Chandra, Infinite irreducible representations of the Lorentz group, Proc. Roy. Soc. London Ser. A 189 (1947), 372-401. MR 0021941 (9:132e)
  • [HC2] -, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185-243. MR 0056610 (15:100f)
  • [Hi1] T. Hirai, On the irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962), 258-262. MR 0191436 (32:8844)
  • [Hi2] -, Classification and the characters of irreducible representations of $ {SU(p,1)}$, Proc. Japan Acad. 42 (1966), 907-921. MR 0223491 (36:6539)
  • [Ho1] R. Howe, On some results of Strichartz and of Rallis and Schiffman, J. Funct. Anal. 32 (1979), 297-303. MR 538856 (80f:22014)
  • [Ho2] -, On a notion of rank for unitary representations of the classical groups, Harmonic Analysis and Group Representations, C.I.M.E. II ciclo 1980, Liguori editore, Naples, 1982, pp. 223-332. MR 777342 (86j:22016)
  • [Ho3] -, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570. MR 986027 (90h:22015a)
  • [Ho4] -, A century of Lie Theory, Mathematics into the Twenty-first Century (F. Browder, ed.), American Mathematical Society Centennial Publications, vol. 2, Amer. Math. Soc., Providence, RI, 1992, pp. 201-421. MR 1184608 (93d:01002)
  • [Ja] N. Jacobson, Lectures in Abstract algebra, Volume II, D. Van Nostrand, Princeton, NJ, 1953. MR 0053905 (14:837e)
  • [Jo] K. D. Johnson, Composition series and intertwining operators for the spherical principal series. II, Trans. Amer. Math. Soc. 215 (1976), 269-283. MR 0385012 (52:5882)
  • [JV] H. P. Jakobsen and M. Vergne, Wave and Dirac Operators and representations of the conformal group, J. Func. Anal. 24 (1977), 52-106. MR 0439995 (55:12876)
  • [JW] K. Johnson and N. Wallach, Composition series and intertwining operators for the spherical principal series, Trans. Amer. Math. Soc. 229 (1977), 137-173. MR 0447483 (56:5794)
  • [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)
  • [KG] A. U. Klimyk and A. M. Gavrilik, The representations of the groups $ {U(n,1)}$ and $ {SO_{0}(n,1)}$, preprint ITP-76-39 E, Institute for Theoretical Physics Kiev, USSR, 1976. MR 0579617 (58:28321)
  • [Kn] A. W. Knapp, Representation theory of semisimple groups, an overview based on examples, Princeton Univ. Press, Princeton, NJ, 1986. MR 855239 (87j:22022)
  • [Ko] B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642. MR 0245725 (39:7031)
  • [KR] S. Kudla and S. Rallis, Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), 25-45. MR 1046171 (91e:22016)
  • [KV1] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1-47. MR 0463359 (57:3311)
  • [KV2] -, Functions on the Shilov boundary of the generalised half plane, Non-commutative Harmonic Analysis, Lecture Notes in Math., vol. 728, Springer, 1979, pp. 136-176. MR 548329 (81e:22022)
  • [Ma] G. W. Mackey, Induced representations of locally compact groups II, Ann. of Math. (2) 58 (1953), 193-221. MR 0056611 (15:101a)
  • [Mo] V. F. Molcanov, Analogue of the Plancherel Formula for hyperboloids, Soviet Math. Doklady 9 (1968), 1387-1385.
  • [Na] M. A. Naĭmark, Linear representations of the Lorentz group, Amer. Math. Soc. Transl. Ser. 2, vol. 6, Amer. Math. Soc, Providence, RI, 1952, pp. 379-458. MR 0106261 (21:4995)
  • [Ni] K. Nishiyama, Algebraic structures on virtual characters of a semisimple Lie group, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, 1988, pp. 417-468. MR 1039847 (91a:22014)
  • [On] E. Onofri, Dynamical quantization of the Kepler manifold, J. Math. Phys. 17 (1976), 401-408. MR 0395530 (52:16327)
  • [Or] B. Orsted, Conformally invariant differential equations and projective geometry, J. Funct. Anal. 44 (1981), 1-23. MR 638292 (83b:22020)
  • [PS1] S. Paneitz and I. Segal, Analysis in space-time bundles, I: General considerations and the scalar bundle, J. Funct. Anal. 47 (1982), 78-142. MR 663834 (83k:22042)
  • [PS2] -, Analysis in space-time bundles, II: The spinor and form bundles, J. Funct. Anal. 49 (1982), 335-414. MR 683028 (84h:81039)
  • [P] S. Paneitz, Analysis in space-time bundles, III: Higher spin bundles, J. Funct. Anal. 54 (1983), 18-112. MR 724645 (86f:81065)
  • [Re] J. Repka, Tensor products of holomorphic discrete series and representations, Canad. J. Math., 31 (1979), 863-844. MR 540911 (82c:22017)
  • [Ro] W. Rossman, Analysis on real hyperbolic spaces, J. Funct. Anal. 30 (1978), 448-477. MR 518343 (80f:43021)
  • [RS] S. Rallis and G. Schiffmann, Discrete spectrum of the Weil representation, Bull. Amer. Math. Soc. 83 (1977), 267-270. MR 0429753 (55:2763)
  • [Sa] S. Sahi, The Capelli Identity and unitary representations, Compositio Math, (to appear). MR 1149169 (93b:22029)
  • [Spl] B. Speh, Degenerate series representations of the universal covering group of $ {SU(2,2)}$, J. Funct. Anal. 33 (1979), 95-118. MR 545386 (80j:22017)
  • [Sp2] -, The unitary dual of $ {GL(3,\mathbb{R})}$ and $ {GL(4,\mathbb{R})}$, Math. Ann. 258 (1981), 113-133. MR 641819 (83i:22025)
  • [SS] A. Salam and J. Strathdee, Relativistic $ {U(6,6)}$ Theory, Phys. Rev. 148 (1966), 1352-1358.
  • [Sc] H. Schlichtkrull, Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms, J. Funct. Anal. 70 (1987), 194-219. MR 870761 (88f:22040)
  • [Se] J. Sekiguchi, Eigenspaces of the Laplace-Beltrami operator on a hyperboloid, Nogoya Math. J. 79 (1980), 151-185. MR 587417 (82g:22010)
  • [Sta] R. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth and Cole, Monterey, CA, 1986.
  • [St1] R. S. Strichartz, Harmonic analysis on hyperboloids, J. Funct. Anal. 12 (1973), 341-383. MR 0352884 (50:5370)
  • [St2] -, Harmonic analysis as spectral theory of laplacians, J. Funct. Anal. 87 (1989), 51-148. MR 1025883 (91c:43015)
  • [SV] B. Speh and D. A. Vogan, Reducibility of generalised principal series representations, Acta Math. 145 (1980), 227-299. MR 590291 (82c:22018)
  • [SW] S. Sternberg and J. Wolf, Hermitian Lie algebras and metaplectic representations. I, Trans. Amer. Math. Soc. 238 (1978), 1-43. MR 0486325 (58:6081)
  • [Te] A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201-218. MR 0126154 (23:A3450)
  • [Tk] R. Takahashi, Sur les representations unitaire des groupes de Lorentz generalises, Bul. Soc. Math. France 91 (1963), 289-435. MR 0179296 (31:3544)
  • [Vi] N. J. Vilenkin, Special functions and the theory of group representations, Izdat Nauka Moscow 1965, Transl. Math. Monographs, vol. 22, Amer. Math. Soc, Providence, RI, 1968. MR 0229863 (37:5429)
  • [Vol] D. Vogan, The Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98. MR 0506503 (58:22205)
  • [Vo2] -, Representations of real reductive groups, Progress in Math., vol. 15, Birkhauser, Boston, MA, 1981.
  • [Vo3] -, Irreducible characters of semisimple Lie groups. III, Proof of the Kazhdan-Lusztig conjectures in the integral case, Invent. Math. 71 (1983), 381-417. MR 689650 (84h:22036)
  • [Vo4] -, Unitary representations of reductive Lie groups, Ann. Math. Stud., vol. 118, Princeton Univ. Press, Princeton, NJ, 1987. MR 908078 (89g:22024)
  • [Wa] N. Wallach, Real Reductive Groups. I, Pure Appl. Math., vol. 132, Academic Press, San Diego, CA, 1988. MR 929683 (89i:22029)
  • [Zh] D. P. Žhelobenko, Compact Lie groups and their representations, Transl. Math. Monographs, vol. 40, Amer. Math. Soc, Providence, RI, 1973. MR 0473098 (57:12776b)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 22E46, 17B10, 22-02

Retrieve articles in all journals with MSC: 22E46, 17B10, 22-02


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1993-00360-4
Keywords: Complementary series, composition series, degenerate principal series representations, K-type diagrams, light cones, unitary representations
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society