Pacific Journal of Mathematics

Coherent states, holomorphic extensions, and highest weight representations.

Karl-Hermann Neeb

Article information

Pacific J. Math., Volume 174, Number 2 (1996), 497-542.

First available in Project Euclid: 6 December 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
Secondary: 22A25: Representations of general topological groups and semigroups 58F06 81R30: Coherent states [See also 22E45]; squeezed states [See also 81V80]


Neeb, Karl-Hermann. Coherent states, holomorphic extensions, and highest weight representations. Pacific J. Math. 174 (1996), no. 2, 497--542.

Export citation


  • [Bou90] N. Bourbaki, Groupes et algebres de Lie, Chapitres 7 et 8, Masson, Paris, 1990.
  • [Bou82] N. Bourbaki, Groupes et algebres de Lie, Chapitre 9, Masson, Paris, 1982.
  • [BK80] M. Brunet and P. Kramer, Complex extensions of the representation of the syn- plectic group associated with the canonical commutation relations. Reports on Math. Physics, 17 (1980), 205-215.
  • [Br85] M. Brunet, The metaplecticsemigroupand related topics,Reports on Math. Physics, 22 (1985), 149-170.
  • [Dix64] J. Dixmier, Les C*-algebres et leurs representations,Gauthier-Villars, Paris, 1964.
  • [DoNa88] J. Dorfmeister and K. Nakajima, The fundamental conjecture for homogeneous Khler manifolds, Acta. Math., 161 (1988), 23-70.
  • [EHW83] T.J. Enright, R. Howe and N. Wallach, A classification of unitary highestweight modules,Proc. Representation theory of reductive groups (Park City, UT, 1982), pp. 97-149; Progr. Math., 40 (1983), 97-143.
  • [Fo89] G.B. Folland, Harmonic Analysis in Phase Space,Princeton University Press, Prin- ceton, New Jersey, 1989.
  • [GS84] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge Uni- versity Press, Cambridge, 1984.
  • [HC55] Harish-Chandra, Representations of semi-simple Lie groups,IV, Amer. J. Math., 77 (1955), 743-777.
  • [HC56] Harish-Chandra, Representations of semi-simpleLie groups,VI, Amer. J. Math., 78 (1956), 564-628.
  • [Hi89] J. Hilgert, A note on Howe's oscillator semigroup, Annales de l'institut Fourier, 39 (1989), 663-688.
  • [HHL89] J. Hilgert, K.H. Hofmann and J.D. Lawson, Lie Groups, Convex Cones, and Semi- groups,Oxford University Press, 1989.
  • [HiNe91] J. Hilgert and K.-H. Neeb, Lie-Gruppenund Lie-Algebren, Vieweg, Braunschweig, 1991.
  • [HiNe93] J. Hilgert and K.-H. Neeb, Lie semigroupsand their applications, Lecture Notes in Math., 1552, Sprin- ger, 1993.
  • [HNP93] J. Hilgert, K.-H. Neeb and W. Plank, Symplectic convexity theorems andcoadjoint orbits,Comp. Math., 94 (1994), 129-180.
  • [HiO192] J. Hilgert and G. Olafsson, Analytic continuations of representations, thesolvable case,Jap. Journal of Math., 18 (1992), 213-290.
  • [How88] R. Howe, The Oscillator semigroup,in "The Mathematical Heritage of Hermann Weyl", Proc. Symp. Pure Math., 48, R. O. Wells ed., AMS Providence, 1988.
  • [Ja79] J.C. Jantzen, Moduln mit einem hchsten Gewicht,Lecture Notes in Math., 750, Springer, 1979.
  • [Jak83] H.P. Jakobsen, Hermitean symmetric spaces and their unitary highest weightmod- ules, J. Funct. Anal., 52 (1983), 385-412.
  • [La94] J.D. Lawson, Polar and OVshanskidecompositions, J. fur Reine Ang. Math., 448 (1994), 191-219.
  • [Li90] W. Lisiecki, Kdhler coherent state orbits for representations of semisimple Lie groups,Ann. Inst. Henri Poincare, 53(2) (1990), 245-258.
  • [Li91] W. Lisiecki, A classification of coherentstate representations of unimodular Liegroups, Bull, of the AMS, 25(1) (1991), 37-43.
  • [Ne92] K.-H. Neeb, On the fundamental group of a Lie semigroup,Glasgow Math. J., 34 (1992), 379-394.
  • [Ne93a] K.-H. Neeb, Invariant subsemigroups of Lie groups,Memoirs of the AMS, 499 (1993).
  • [Ne93b] K.-H. Neeb, Holomorphic representation theory and coadjoint orbits of convexity type, Habilitationsschrift, Technische Hochschule Darmstadt, January, 1993.
  • [Ne94a] K.-H. Neeb, Contraction semigroups and representations, Forum Math., 6 (1994), 233- 270.
  • [Ne94b] K.-H. Neeb, A convexity theorem for semisimple symmetric spaces, Pacific Journal of Math., 162(2) (1994), 305-349.
  • [Ne94c] K.-H. Neeb, On closedness and simple connectedness of adjoint and coadjoint orbits, Manuscripta Math., 82 (1994), 51-65.
  • [Ne94d] K.-H. Neeb, Holomorphic representation theory!, Math. Annalen, 301 (1995), 155-181.
  • [Ne94e] K.-H. Neeb, Holomorphic representation theory II,Acta Math., 173 (1994), 103-133.
  • [Ne94f] K.-H. Neeb, Khler structures and convexity properties of coadjoint orbits, Forum Math., 7 (1995), 349-384.
  • [Ne94g] K.-H. Neeb, On the convexity of the moment mapping for a unitary highest weight representation, J. Funct. Anal., 127(2) (1995), 301-325.
  • [0182] G.I.Ol'shanski,Invariant cones in Lie algebras, Lie semigroups, and the holomor- phic discrete series, Funct. Anal, and Appl., 15 (1982), 275-285.
  • [Pe86] A. Perelomov, Generalized coherent states and their applications, Texts and Mono- graphs in Physics, Springer, 1986.
  • [Sa71] I. Satake, Unitary representations of a semi-direct product of Lie groups on d- cohomology spaces, Math. Ann., 190 (1971), 177-202.
  • [Sta86] R.J. Stanton, Analytic Extension of the holomorphic discrete series, Amer. J. Math, 108 (1986), 1411-1424.
  • [Wal88] N.R. Wallach, Real reductive groups I, Academic Press Inc., Boston, New York, Tokyo, 1988.
  • [Wal92] N.R. Wallach, Real reductive groups II, Academic Press Inc.,Boston, New York, Tokyo, 1992.
  • [Wa72] G. Warner, Harmonic analysis on semisimple Lie groups I, Springer, Berlin, Hei- delberg, New York, 1972.
  • [We76] J. Weidmann, Lineare Operatoren in Hilbertrdumen, Teubner, Stuttgart, 1976.