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Mathematische Annalen

, Volume 290, Issue 1, pp 565–619 | Cite as

The Capelli identity, the double commutant theorem, and multiplicity-free actions

  • Roger Howe
  • Tôru Umeda
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References

  1. [Ba]
    Bailey, W.N.: Generalized hypergeometric series. Cambridge Math. Tracts 32 (repr. Hafner 1965). Cambridge: Cambridge Univ. Press 1935.Google Scholar
  2. [Bo]
    Boole, G.: Calculus of finite differences. New York: Chelsea 1872Google Scholar
  3. [B1]
    Borel, A.: Hermann Weyl and Lie Groups, in “Hermann Weyl 1885–1985,” (Centenary Lectures) K. Chandrasekharan (ed.), pp. 53–82. Berlin Heidelberg New York: Springer 1986Google Scholar
  4. [B2]
    Borel, A.: Linear algebraic groups. New York: Benjamin 1969Google Scholar
  5. [BW]
    Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representation of reductive groups. Princeton: Princeton Univ. Press 1980Google Scholar
  6. [Br]
    Brion, M.: Classification des espaces homogènes sphériques. Compos. Math.63, 189–208 (1987)Google Scholar
  7. [Ca1]
    Capelli, A.: Über die Zurückführung der Cayley'schen Operation Ω auf gewöhnlichen Polar-Operationen. Math. Ann.29, 331–338 (1887)Google Scholar
  8. [Ca2]
    Capelli, A.: Ricerca delle operazioni invariantive fra piu serie di variabili permutabili conogni altra operazione invariantive fra le stesse serie. Atti Sci. Fis. Mast. Napoli (2)I 1–17 (1888)Google Scholar
  9. [Ca3]
    Capelli, A.: Sur les opérations dans la théorie des formes algébriques. Math. Ann.37, 1–37 (1890)Google Scholar
  10. [CL]
    Carter, R.W., Lusztig, G.: On the modular representations of the general linear and symmetric groups. Math. Z.136, 193–242 (1974)Google Scholar
  11. [D]
    Dixmier, J.: Sur les algèbres enveloppants de sl(2, C) et af(n, C), Bull. Sci. Math. II Ser.100 57–95 (1976)Google Scholar
  12. [FZ]
    Foata, D., Zeilberger, D.: Preliminary manuscriptGoogle Scholar
  13. [G]
    Gårding, L.: Extension of a formula by Cayley to symmetric determinants. Proc. Edinb. Math. Soc. Ser. II8, 73–75 (1947)Google Scholar
  14. [Ha]
    Haris, S.J.: Some irreducible representation of exceptional algebraic groups. Am. J. Math.93, 75–106 (1971)Google Scholar
  15. [Htn]
    Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977Google Scholar
  16. [He1]
    Helgason, S.: Groups and geometric analysis. New York, London: Academic Press 1984Google Scholar
  17. [He2]
    Helgason, S.: A duality for symmetric spaces with applications. Adv. Math.5, 1–54 (1970)Google Scholar
  18. [He3]
    Helgason, S.: Some results on invariant differential operators on symmetric spaces. Preprint 1989Google Scholar
  19. [H1]
    Howe, R.: Remarks on classical invariant theory. Trans. Am. Math. Soc.313, 539–570 (1989)Google Scholar
  20. [H2]
    Howe, R.: Some highly symmetric dynamical systems. PreprintGoogle Scholar
  21. [H3]
    Howe, R.: (GL n,GL m)-duality and symmetric plethysm. Proc. Indian. Acad. Sci. Math. Sci.97, 85–109 (1987)Google Scholar
  22. [H4]
    Howe, R.: The classical groups and invariants of binary forms, in: The mathematical heritage of Hermann Weyl. Proc. Symp. Pure Math.48, 133–166 (1988)Google Scholar
  23. [H5]
    Howe, R.: Dual pairs in physics. harmonic oscillators, photons, electrons, and singletons. Lect. Appl. Math.21, 179–207 (1985)Google Scholar
  24. [H6]
    Howe, R., Tan, E-L.: Non-abelian harmonic analysis: Applications ofSL(2,∔). Berlin Heidelberg New York: Springer, to appearGoogle Scholar
  25. [Hu]
    Humphreys, J.E.: Introduction to Lie algebras and representation theory. Berlin Heidelberg New York: Springer 1972Google Scholar
  26. [I]
    Igusa, J.-I.: A classification of spinors up to dimension twelve. Am. J. Math.92, 997–1028 (1970)Google Scholar
  27. [Ja]
    Jacobson, N.: Lectures in abstract algebra II. Berlin Heidelberg New York: Springer 1952Google Scholar
  28. [Jb]
    Jimbo, M.:q-analogue ofU(gl(N+1)), Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys.11, 247–252 (1986)Google Scholar
  29. [J]
    Johnson, K.D.: On a ring of invariant polynomials on a Hermitian symmetric space. J. Algebra67, 72–81 (1980)Google Scholar
  30. [K]
    Kac, V.G.: Some remarks on nilpotent orbits. J. Algebra64, 190–213 (1980)Google Scholar
  31. [KPV]
    Kac, V.G., Popov, V.L., Vinberg, E.B.: Sur les groupes linéares algébriques dont l'algèbre des invariants est libre. C.R. Acad. Sci. Paris283, 875–878 (1976)Google Scholar
  32. [KT]
    Koike, K., Terada, I.: Young-diagrammatic methods for the representation theory of the classical groups of typeB n,C n,D n. J. Algebra107, 466–511 (1987)Google Scholar
  33. [KS]
    Kostant, B., Sahi, S.: The Capelli identity, tube domains and the generalized Laplace transform. preprint 1989Google Scholar
  34. [Kz]
    Koszul, J-L.: Les algèbre de Lie graduée de type sl(n, 1) et l'opérateur de A. Capelli. C.R. Acad. Sci. Paris292, 139–141 (1981)Google Scholar
  35. [Kr]
    Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math.38, 129–153 (1979)Google Scholar
  36. [L]
    Lang, S.: Reading, Mass. Addison-Wesley 1965Google Scholar
  37. [Mc]
    MacDonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Oxford Univ. Press 1979Google Scholar
  38. [MRS]
    Müller, I., Rübenthaler, H., Schiffmann, G.: Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées. Math. Ann.274, 95–123 (1986)Google Scholar
  39. [MF]
    Mumford, D., Fogarty, J.: Geometric invariant theory. Berlin Heidelberg New York: Springer 1982Google Scholar
  40. [Ra]
    Rais, M.: Distributions homogènes sur des espaces de matrices. (Thèse Sc. math. Paris, 1970). Bull Soc. Math. Fr. Mém.30, 1–109 (1972)Google Scholar
  41. [R]
    Ruitenburg, G.C.M.: Invariant ideals of polynomials algebras with multiplicity-free group actions. Comp. Math.71, 181–227 (1989)Google Scholar
  42. [RS]
    Rübenthaler, H., Schiffmann, G.: Opérateurs différentiels de Shimura et espace préhomogènes. Invent. Math.90, 409–442 (1987)Google Scholar
  43. [S]
    Sato, M.: The theory of prehomogeneous vector spaces, notes by T. Shintani (in Japanese), Sugaku no Ayumi15-1, 85–157 (1970)Google Scholar
  44. [SK]
    Sato, M., Kimura, T.: A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J.65, 1–155 (1977)Google Scholar
  45. [Se]
    Servedio, F.J.: Prehomogenous vector spaces and varieties. Trans. Am. Math. Soc.76, 421–444 (1973)Google Scholar
  46. [Sch1]
    Schwarz, G.: Representation of simple Lie groups with regular rings of invariants. Invent. Math.49, 167–191 (1978)Google Scholar
  47. [Sch2]
    Schwarz, G.: Representation of simple Lie groups with free module of covariants. Invent. Math.50, 1–12 (1978)Google Scholar
  48. [Sch3]
    Schwarz, G.: Lifting smooth homotopies of orbit spaces. Ins. Hautes Etud. Sci.51, 37–135 (1980)Google Scholar
  49. [Sch4]
    Schwarz, G.: Invariant theory ofG 2. Bull. Am. Math. Soc. New Ser.9, 335–338 (1983)Google Scholar
  50. [Sch5]
    Schwarz, G.: On classical invariant theory and binary cubics. Ann. Inst. Fourier37, 191–216 (1987)Google Scholar
  51. [Sch6]
    Schwarz, G.: Invariant theory ofG 2 andSpin 7. Comment. Math. Helv.63, 624–663 (1988)Google Scholar
  52. [Sh1]
    Shimura, G.: On differential operators attached to certain representations of classical groups. Invent. Math.77, 463–488 (1984)Google Scholar
  53. [Sh2]
    Shimura, G.: Invariant differential operators on Hermitian symmetric spaces. Ann. Math.132, 232–272 (1990)Google Scholar
  54. [Th]
    Thrall, R.: On symmetrized Kronecker powers and the structure of the free Lie ring. Am. J. Math.64, 371–388 (1942)Google Scholar
  55. [T1]
    Turnbull, H.W.: The theory of determinants, matrices, and invariants. New York: Dover 1960Google Scholar
  56. [T2]
    Turnbull, H.W.: Symmetric determinants and the Cayley and Capelli operators. Proc. Edinb. Math. Soc. Ser. II,8, 76–86 (1947)Google Scholar
  57. [VK]
    Vinberg, E.B., Kimelfeld, B.N.: Homogeneous domains in flag manifolds and spherical subgroups of semi-simple Lie groups. Funct. Anal. Appl.12, 12–19 (1978)Google Scholar
  58. [W]
    Weyl, H.: The classical groups, their invariants and representations. Princeton: Princeton Univ. Press 1946Google Scholar
  59. [ZS]
    Zariski, O., Samuel, P.: Commutative algebra 1. Princeton: Van Nostrand 1958Google Scholar
  60. [Zb]
    Zeilberger, D.: The method of creative telescoping. To appear, J. Symb. Comp.Google Scholar
  61. [Z]
    Zhu, C.: Thesis. Yale Univ. 1990Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Roger Howe
    • 1
  • Tôru Umeda
    • 2
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

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