Pacific Journal of Mathematics

The index of transversally elliptic operators for locally free actions.

Jeffrey Fox and Peter Haskell

Article information

Pacific J. Math., Volume 164, Number 1 (1994), 41-85.

First available in Project Euclid: 8 December 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58G12
Secondary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22]


Fox, Jeffrey; Haskell, Peter. The index of transversally elliptic operators for locally free actions. Pacific J. Math. 164 (1994), no. 1, 41--85.

Export citation


  • [Al] N. Anh, Classification of connected unimodular Lie groups with discrete series, Ann. Inst. Fourier Grenoble, 30 (1980), 159-192.
  • [A2] N. Anh, Lie groups with square-integrable representations,Annals of Math., 104 (1976), 431-458.
  • [At1] M.Atiyah,Elliptic Operatorsand Compact Groups, Lecture Notes in Math., vol. 401,Springer-Verlag,NewYork, 1974.
  • [At2] M.Atiyah, Elliptic operators, discrete groups, and von Neumann algebras, Asterisque, 32/33 (1976), 43-72.
  • [AtS] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent Math., 42 (1977), 1-62.
  • [BaJ] S. Baaj and P. Julg, Theorie bivariante de Kasparov et operateurs non bornesdans les C*-modules Hilbertiens,C. R. Acad. Sci.Paris, 296 (1983), 875-878.
  • [Bl] B. Blackadar, K-Theory for Operator Algebras, MSRI Publications 5, Springer-Verlag, NewYork, 1986.
  • [B] A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology, 2(1963), 111-122.
  • [BH-C] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraicgroups, Annals of Math., 75 (1962),485-535.
  • [BoMa] R. Boyer and R. Martin, The group C*-algebra of the deSittergroup,Proc. Amer. Math. Soc,65 (1977), 177-184.
  • [C] A. Connes, Non-commutativedifferentialgeometry, Publ. Math. IHES; 62 (1985), 41-144.
  • [CM] A. Connes and H. Moscovici, The L2-index theorem for homogeneous spaces of Lie groups, Annals of Math., 115 (1982), 291-330.
  • [CSk] A. Connes and G. Skandalis, The longitudinal index theoremfor foliations, Publ. Res. Inst. Math. Sci., Kyoto, 20 (1984), 1139-1183.
  • [Dl] J. Dixmier, Bicontinuite dans la methode du petit de Mackey, Bull. Sci. Math., 97 (1973), 233-240.
  • [D2] J. Dixmier, C*-Algebras, North-Holland,Amsterdam, 1982.
  • [Fe] J. Fell, The dual spaceof C*-algebras, Trans. Amer. Math. Soc, 94 (1960), 365-403.
  • [FH1] J. Fox and P. Haskell, Index theory on locally homogeneous spaces, K- Theory, 4 (1991), 547-568.
  • [FH2] J. Fox and P. Haskell, K-amenability for SU( , 1), J. Funct. Anal., (to appear).
  • [FH3] J. Fox and P. Haskell, K-theory and the spectrum of discrete subgroups of Spin(4, 1) in Operator Algebras and Topology (W. B. Arveson, A. S. Mishchenko, M. Putinar, M. A. Rieffel, and S. Stratila, eds.), Pitman Res. Notes in Math., vol. 270, Longman Scientific and Technical, Harlow, England, 1992, pp. 30-44.
  • [FHRa] J. Fox, P. Haskell, and I. Raeburn, Kasparov products, KK-equivalence, and proper actions of connected reductive Lie groups, J. Operator Theory, 22 (1989), 3-29.
  • [Gr] P. Green, Square-ntegrable representationsand the dual topology,J. Funct. Anal., 35 (1980), 279-294.
  • [HiSk] M. Hilsum and G. Skandalis, Morphsmes K-orientes d'espaces de feuilles et fonctorialite en theorie de Kasparov,Ann. Sci. Ecole Norm. Sup. (4), 20 (1987), 325-390.
  • [JK] P. Julg and G. Kasparov, L'anneau KKG{C, C) pour G = SU(n, 1), C. R. Acad. Sci. Paris, 313 (1991), 259-264.
  • [JVa] P. Julg and A. Valette, K-theoreticamenability for SL2(QP), and the action on the associated tree, J. Funct. Anal., 58 (1984), 194-215.
  • [Kl] G. Kasparov, Equivariant KK-theory and the Novikov conjecture,Invent. Math., 91 (1988), 147-201.
  • [K2] G. Kasparov, An index for invariant elliptic operators, K-theory, andrepresenta- tions of Lie groups, Soviet Math. Dokl., 27 (1983), 105-109.
  • [K3] G. Kasparov,Lorentz groups: K-theoryof unitary representationsand crossedprod- ucts, Soviet Math. Dokl., 29 (1984), 256-260.
  • [K4] G. Kasparov, The operator K-functor and extensions of C*-algebras, Math.USSR Izv., 16 (1981), 513-572.
  • [Kr] H. Kraljevic, Representations of the universal covering group of the group S(n, 1), Glasnik Math., 8 (1973), 23-72.
  • [L] R. Lipsman, The dual topology for the principal and discrete series on semi-simple groups, Trans. Amer. Math. Soc, 152 (1970), 399-417.
  • [MoWo] C. Moore and J. Wolf, Square-integrablerepresentations of nilpotent Lie groups,Trans. Amer. Math. Soc, 185 (1973), 445-462.
  • [M] H. Moscovici, L2 -index of elliptic operatorson locallysymmetric spaces of finite volume, in OperatorAlgebras and K-Theory, Contemp. Math., vol. 10 (R. Douglas and C. Schochet, eds.), Amer. Math. Soc, Providence, R.I., 1982, pp. 129-137.
  • [NSt] E. Nelson and F. Stinespring, Representation of elliptic operators in an en- veloping algebra,Amer. J. Math., 81 (1959), 547-560.
  • [NeZi] A. Nestke and F. Zickermann, The index of transversallyellipticcomplexes, Rend. Circ. Mat. Palermo (2) 1985, Suppl. No. 9, 165-175.
  • [P] R. Parthasarathy, Dirac operatorsand the discrete series,Annals of Math., 96(1972), 1-30.
  • [Pe] M. Penington, K-theory and C*-algebras of Lie groups and foliations, D. Phil. Thesis, Oxford, 1983.
  • [ReSi] M. Reed and B. Simon, FourierAnalysis, Self-Adjointness, Methods of Mod- ern Mathematical Physics II, Academic Press, Orlando, 1975.
  • [R1] J. Rosenberg, Realization ofsquare-integrable representations ofunimodular Lie groups on L2-cohomology spaces,Trans. Amer. Math. So, 261 (1980), 1-32.
  • [R2] J. Rosenberg, Square-integrable factor representation of locally compact groups, Trans. Amer. Math. Soc, 237 (1978), 1-33.
  • [S] W. Schmid, L2-cohomology and the discrete series, Annals of Math., 103 (1976), 375-394.
  • [Sin] I. Singer, Future extensions of index theory and elliptic operators,inPros- pects in Mathematics, Annals of Math. Studies 70, Princeton Univ. Press, Princeton, N.J., 1971.
  • [Sk] G. Skandalis, Some remarks on Kasparovtheory,J. Funct. Anal., 56 (1984), 337-347.
  • [T] M. Taylor, Pseudodifferential Operators, Princeton Univ. Press,Princeton, N.J., 1981.
  • [Va] A. Valette, K-theoriepour certaines C*-algebres associees aux groupes de Lie, Ph.D.thesis, Universite Libre de Bruxelles, 1984.
  • [Ve] M. Vergne, Sur I'indices des operateurs transversalement elliptiques, C. R. Acad. Sci. Paris, 310 (1990), 329-332.
  • [V] D. Vogan, Representations of Real Reductive Lie Groups,Progress inMath. vol. 15, Birkhauser, Boston, 1981.
  • [VZu] D. Vogan and G. Zuckerman, Unitary representations with non-zero coho- mology, Compositio Math., 53 (1984), 51-90.
  • [W] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I and II, Springer-Verlag, New York, 1972.
  • [Wa] A. Wasserman, A proof of the Connes-Kasparovconjecturefor connected reductiveLie groups,C. R. Acad. Sci. Paris, 304 (1987), 559-562.
  • [Wil] F. Williams, Discreteseries multiplicities in L2(G/), Amer. J. Math., 106 (1984), 137-148.
  • [Wi2] F. Williams,Note on a theorem ofH. Moscovici, J. Funct. Anal., 72 (1987), 28-32.
  • [Ze] D.Zelobenko,A descriptionof the quasi-simple irreducible representationsof the groups U(n, 1) and Spin(, 1), Math. USSR Izv., 11 (1977), 31-50.
  • [Z] R. Zimmer,Ergodic Theory and Semisimple Groups, Monographs inMath., vol. 81, Birkhauser, Boston, 1984.