Available in electronic format
Available in print format
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)
     

Homogeneous Functions on Light Cones: the Infinitesimal Structure of some Degenerate Principal Series Representations

Author(s): Roger E. Howe; Eng-Chye Tan
Journal: Bull. Amer. Math. Soc. 28 (1993), 1-74.
MSC (2000): Primary 22E46; Secondary 17B10
MathSciNet review: 1172839
Retrieve article in: PDF

References | Similar articles | Additional information

References:

[Ab]
{Ab} H. Abarbanel, {\it The inverse $r$-squared force; anintroduction to its symmetries}, Essays in Honor of Valentine Bargmann(E. Lieb, B. Simon, and A. Wightman, eds.), Princeton Univ. Press,Princeton, NJ, 1976. MR
[AFR]
{AFR} R. Anderson, J. Fischer, and R. Raczka, {\it Couplingproblem for $\RMU(p,q)$ ladder representations}. I, Proc. Roy. Soc.London Ser. A {\bf302} (1968),491--500. MR 219659
[Ba]
{Ba} V. Bargmann, {\it Irreducible unitary representations ofthe Lorentz group}, Ann. of Math. (2) {\bf48} (1947), 568--640. MR 21942
[BB]
{BB} A. Beilinson and J. Bernstein, {\itLocalisation de ${\frak g}$-modules}, C. R. Acad.Sci. Paris S\'er. I Math. {\bf292} (1981), 15--18. MR 610137
[BK]
{BK} J-L. Brylinski, and M. Kashiwara, {\itDemonstration de la conjecture de Kazhdan-Lusztigsur les modules de Verma}, C. R. Acad. Sci. ParisS\'er. I Math. {\bf291} (1980), 373--376. MR 596075
[Br]
{Br} T. Branson, {\it Group representations arising from Lorentzconformal geometry}, J. Funct. Anal. {\bf74} (1987), 199--241. MR 904819
[BS]
{BS} M. Baldoni-Silva, {\it The unitary dual of$\SP(n,1)$, $n \ge 2$}, Duke Math. J. {\bf48} (1981), 549--583. MR 630585
[CC]
{CC} L. Casian and D. Collingwood, {\itWeight filtrations for induced representations ofreal reductive groups}, Adv. Math. {\bf73} (1989), 79--144. MR 979588
[Cl]
{Cl} D. Collingwood, {\it Representations of rank one Lie groups},Pitman Res. Notes Math. Ser., vol. 137, Pitman Publishing, Boston, MA, 1985. MR 853731
[Cw]
{Cw} M. Cowling, {\it Unitary and uniformly boundedrepresentations of some simple Lie groups}, Harmonic Analysis andGroup Representations, C.I.M.E. II ciclo 1980, Liguori editore,Naples, 1982, pp. 49--128. MR 777340
[DGN]
{DGN} Y. Dothan, M. Gell-mann, and Y. Ne'eman, {\itSeries of hadron energy levels as representations of non-compactgroups}, Phys. Lett. {\bf17} (1965), 148--151. MR 183410
[Di]
{Di} J. Dixmier, {\it Repr\'{e}sentationsint\'{e}grables du groupe de De Sitter}, Bull. Soc. Math.France {\bf89} (1961), 9--41. MR 140614
[EHW]
{EHW} T. Enright, R. Howe, andN. Wallach, {\it A classification of unitary highestweight modules}, Representation Theory of ReductiveLie Groups (P. Trombi, ed.), Birkhauser, Boston, MA, 1983, pp. 97--143. MR
[En]
{En} M. Englefield, {\it Group theory and the Coulomb problem},Wiley-Interscience, New York, 1972. MR 406116
[EPWW]
{EPWW} T. Enright, R. Parthasarathy, N. Wallach, and J.Wolf, {\it Unitary derived functor modules with small spectrum}, ActaMath. {\bf154} (1985), 105--136. MR 772433
[Fa]
{Fa} J. Faraut, {\it Distributions spheriquessur les espaces hyperboliques}, J. Math.Pures Appl. {\bf58} (1979), 369--444. MR 566654
[FJ]
{FJ} M. Flensted-Jensen, {\it Analysis on non-RiemannianSymmetric Spaces}, CBMS Reg. Conf. Ser. in Math., vol. 61, Amer.Math. Soc., Providence, RI, 1986, pp. 1--77. MR 837420
[FR]
{FR} J. Fischer and R. Raczka, {\it Degenerate representationsof non-compact unitary groups}. II, Comm. Math. Phys.{\bf4} (1967), 8--21. MR 202923
[Fr]
{Fr} C. Fronsdal, {\it Infinite multiplets and the hydrogen atom},Phys. Rev. (3) {\bf156} (1967), 1665--1677. MR
[GN]
{GN} I. M. Gel'fand and M. A. Na\u{i}mark, {\it Unit\"{a}redarstellungen der klassischen gruppen}, Acadamie-Verlag, Berlin, 1957. MR
[Gr]
{Gr} K. Gross, {\it The dual of a parabolic subgroup and adegenerate principal series of $\SP(n,\C)$}, Amer. J. Math. {\bf93}(1971), 398--428. MR 304558
[Gu]
{Gu} A. Guillemonat, {\it On some semi-sphericalrepresentations of a hermitian symmetric pair of tubular type}, Math.Ann. {\bf246} (1980), 93--116. MR 564679
[HC1]
{HC1} Harish-Chandra, {\it Infinite irreduciblerepresentations of the Lorentz group}, Proc. Roy. Soc. London Ser. A{\bf189} (1947), 372--401. MR 21941
[HC2]
{HC2} \bysame, {\it Representations of asemisimple Lie group on a Banach space. I}, Trans.Amer. Math. Soc. {\bf75} (1953), 185--243. MR 56610
[Hi1]
{Hi1} T. Hirai, {\it On the irreducible representationsof the Lorentz group of $n$-th order}, Proc. Japan Acad. {\bf38}(1962), 258--262. MR 191436
[Hi2]
{Hi2} \bysame, {\it Classification and the characters ofirreducible representations of $\SU(p,1)$}, Proc. Japan Acad. {\bf42}(1966), 907--921. MR 223491
[Ho1]
{Ho1} R. Howe, {\it On some results of Strichartz and ofRallis and Schiffman}, J. Funct. Anal. {\bf32} (1979), 297--303. MR 538856
[Ho2]
{Ho2} \bysame, {\it On a notion of rank for unitaryrepresentations of the classical groups}, Harmonic Analysisand Group Representations, C.I.M.E. II ciclo 1980,Liguori editore, Naples, 1982, pp. 223--332. MR
[Ho3]
{Ho3} \bysame, {\it Remarks on classical invarianttheory}, Trans. Amer. Math. Soc. {\bf313} (1989), 539--570. MR 986027
[Ho4]
{Ho4} \bysame, {\it A century of Lie Theory},Mathematics into theTwenty-first Century (F. Browder, ed.),American Mathematical Society CentennialPublications, vol. 2, Amer. Math. Soc.,Providence, RI, 1992, pp. 201--421. MR 1184617
[Ja]
{Ja} N. Jacobson, {\it Lectures in Abstract algebra}, Volume II,D. Van Nostrand, Princeton, NJ, 1953. MR 53905
[Jo]
{Jo} K. D. Johnson, {\it Composition series andintertwining operators for the spherical principal series. II}, Trans.Amer. Math. Soc. {\bf215} (1976), 269--283. MR 385012
[JV]
{JV} H. P. Jakobsen and M. Vergne, {\it Wave and DiracOperators and representations of the conformal group}, J. Func. Anal.{\bf24} (1977), 52--106. MR 439995
[JW]
{JW} K. Johnson and N. Wallach, {\it Composition series andintertwining operators for the spherical principal series},Trans. Amer. Math. Soc. {\bf229} (1977), 137--173. MR 447483
[KL]
{KL} D. Kazhdan and G. Lusztig, {\itRepresentations of Coxeter groups and Heckealgebras}, Invent. Math. {\bf53} (1979), 165--184. MR 560412
[KG]
{KG} A. U. Klimyk and A. M. Gavrilik, {\itThe representations of the groups $\RMU(n,1)$ and$\SO_0(n,1)$}, preprint ITP-76-39 E, Institute forTheoretical Physics Kiev, USSR, 1976. MR 579617
[Kn]
{Kn} A. W. Knapp, {\em Representation theoryof semisimple groups, an overview based on examples},Princeton Univ. Press, Princeton, NJ, 1986. MR 855239
[Ko]
{Ko} B. Kostant, {\it On the existence and irreducibilityof certain series of representations}, Bull. Amer. Math. Soc. {\bf75}(1969), 627--642. MR 245725
[KR]
{KR} S. Kudla and S. Rallis, {\it Degenerate principal series andinvariant distributions}, Israel J. Math. {\bf69} (1990), 25--45. MR 1046171
[KV1]
{KV1} M. Kashiwara and M. Vergne, {\it On the Segal-Shale-Weilrepresentations and harmonic polynomials}, Invent.Math. {\bf44} (1978), 1--47. MR 463359
[KV2]
{KV2} \bysame, {\it Functions on the Shilov boundaryof the generalised half plane}, Non-commutative Harmonic Analysis,Lecture Notes in Math., vol. 728, Springer, 1979, pp. 136--176. MR 548329
[Ma]
{Ma} G. W. Mackey, {\it Induced representations of locallycompact groups} II, Ann. of Math. (2) {\bf58} (1953), 193--221. MR 56611
[Mo]
{Mo} V. F. Molcanov, {\it Analogue of the Plancherel Formula forhyperboloids}, Soviet Math. Doklady {\bf9} (1968), 1387--1385. MR
[Na]
{Na} M. A. Na\u\i mark, {\it Linear representations of theLorentz group}, Amer. Math. Soc. Transl. Ser. 2, vol. 6, Amer. Math.Soc., Providence, RI, 1952, pp. 379--458. MR
[Ni]
{Ni} K. Nishiyama, {\it Algebraic structures on virtualcharacters of a semisimple Lie group}, Representations of Lie groups,Kyoto, Hiroshima, 1986,Adv. Stud. Pure Math., vol.~14,1988, pp. 417--468. MR 1039847
[On]
{On} E. Onofri, {\it Dynamical quantization of the Keplermanifold}, J. Math. Phys. {\bf17} (1976), 401--408. MR 395530
[Or]
{Or} B. Orsted, {\it Conformally invariant differentialequations and projective geometry}, J. Funct. Anal. {\bf44} (1981),1--23. MR 638292
[PS1]
{PS1} S. Paneitz and I. Segal, {\it Analysis inspace-time bundles, \RM{I:} General considerations and the scalarbundle}, J. Funct. Anal. {\bf 47} (1982), 78--142. MR 663834
[PS2]
{PS2} \bysame, {\it Analysis inspace-time bundles, \RM{II:} The spinor and form bundles}, J. Funct.Anal. {\bf 49} (1982), 335--414. MR 683028
[P]
{P} S. Paneitz, {\it Analysis inspace-time bundles, \RM{III:} Higher spin bundles}, J. Funct. Anal.{\bf 54} (1983), 18--112. MR 724645
[Re]
{Re} J. Repka, {\it Tensor products of holomorphicdiscrete series and representations}, Canad. J. Math., {\bf31} (1979),863--844. MR 540911
[Ro]
{Ro} W. Rossman, {\it Analysis on real hyperbolic spaces},J. Funct. Anal. {\bf30} (1978), 448--477. MR 518343
[RS]
{RS} S. Rallis and G. Schiffmann, {\it Discrete spectrum of theWeil representation}, Bull. Amer. Math. Soc. {\bf83} (1977), 267--270. MR 429753
[Sa]
{Sa} S. Sahi, {\it The Capelli Identity and unitaryrepresentations}, Compositio Math. (to appear). MR
[Sp1]
{Sp1} B. Speh, {\it Degenerate series representations ofthe universal covering group of $\SU(2,2)$}, J. Funct. Anal. {\bf33}(1979), 95--118. MR 545386
[Sp2]
{Sp2} \bysame, {\it The unitary dual of $\GL(3,\R)$ and$\GL(4,\R)$}, Math. Ann. {\bf258} (1981), 113--133. MR 641819
[SS]
{SS} A. Salam and J. Strathdee, {\it Relativistic$\RMU(6,6)$ Theory}, Phys. Rev. {\bf148} (1966), 1352--1358. MR
[Sc]
{Sc} H. Schlichtkrull, {\it Eigenspaces of the Laplacianon hyperbolic spaces: composition series and integral transforms}, J.Funct. Anal. {\bf70} (1987), 194--219. MR 870761
[Se]
{Se} J. Sekiguchi, {\it Eigenspaces of theLaplace-Beltrami operator on a hyperboloid}, Nogoya Math. J. {\bf79}(1980), 151--185. MR 587417
[Sta]
{Sta} R. Stanley, {\it Enumerative Combinatorics}, vol. 1,Wadsworth and Cole, Monterey, CA, 1986. MR
[St1]
{St1} R. S. Strichartz, {\it Harmonic analysis onhyperboloids}, J. Funct. Anal. {\bf12} (1973), 341--383. MR 352884
[St2]
{St2} \bysame, {\it Harmonic analysis asspectral theory of laplacians}, J. Funct. Anal. {\bf87} (1989), 51--148. MR 1025883
[SV]
{SV} B. Speh and D. A. Vogan, {\it Reducibility of generalisedprincipal series representations}, Acta Math. {\bf145} (1980), 227--299. MR 590291
[SW]
{SW} S. Sternberg and J. Wolf, {\it Hermitian Lie algebras andmetaplectic representations. I}, Trans. Amer. Math. Soc. {\bf238} (1978),1--43. MR 486325
[Te]
{Te} A. Tengstrand, {\it Distributions invariant under anorthogonal group of arbitrary signature}, Math. Scand. {\bf8} (1960),201--218. MR 126154
[Tk]
{Tk} R. Takahashi, {\it Sur les representations unitairedes groupes de Lorentz generalises}, Bul. Soc. Math. France {\bf91}(1963), 289--435. MR 179296
[Vi]
{Vi} N. J. Vilenkin, {\em Special functions and the theoryof group representations}, Izdat Nauka Moscow 1965, Transl. Math.Monographs, vol. 22, Amer. Math. Soc., Providence, RI, 1968. MR 229863
[Vo1]
{Vo1} D. Vogan, {\it The Gelfand-Kirillov dimension forHarish-Chandra modules}, Invent. Math. {\bf48} (1978), 75--98. MR 506503
[Vo2]
{Vo2} \bysame, {\it Representations of real reductive groups},Progress in Math., vol. 15, Birkhauser, Boston, MA, 1981. MR
[Vo3]
{Vo3} \bysame, {\it Irreducible characters of semisimpleLie groups. III}, Proof of the Kazhdan-Lusztig conjectures in theintegral case, Invent. Math. {\bf71} (1983), 381--417. MR 689650
[Vo4]
{Vo4} \bysame, {\it Unitary representations of reductive Liegroups}, Ann. Math. Stud., vol. 118, Princeton Univ. Press, Princeton, NJ,1987. MR
[Wa]
{Wa} N. Wallach, {\it Real Reductive Groups}. I, Pure Appl. Math.,vol. 132, Academic Press, San Diego, CA, 1988. MR 929683
[Zh]
{Zh} D. P. \v{Z}helobenko, {\em Compact Lie groups and theirrepresentations}, Transl. Math. Monographs, vol. 40, Amer. Math. Soc.,Providence, RI, 1973. MR

Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 22E46, 17B10

Retrieve articles in all Journals with MSC (2000): 22E46, 17B10


Additional Information:

DOI: 10.1090/S0273-0979-1993-00360-4
PII: S 0273-0979(1993)00360-4