Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-11T05:55:01.642Z Has data issue: false hasContentIssue false
  • English
  • Français

The Guillemin–Sternberg conjecture for noncompact groups and spaces

Published online by Cambridge University Press:  11 February 2008

P. Hochs  and
N.P. Landsman
P. Hochs
Affiliation:
peter@hondsrug.netRadboud University Nijmegen, Institute for Mathematics, Astrophysics, and Particle Physics, Toernooiveld 1, 6525 ED NIJMEGENThe Netherlands
N.P. Landsman
Affiliation:
landsman@math.ru.nlRadboud University Nijmegen, Institute for Mathematics, Astrophysics, and Particle Physics, Toernooiveld 1, 6525 ED NIJMEGEN, The Netherlands
Get access

Abstract

The Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.

Keywords

Guillemin–Sternberg conjecturequantisationreduction
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 3 , June 2008 , pp. 473 - 533
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abraham, R. and Marsden, J.E.. Foundations of Mechanics, 2nd ed.Addison Wesley, Redwood City, 1985Google Scholar
2.Atiyah, M. Elliptic operators, discrete groups and von Neumann algebras. Soc. Math. France Astérisque 32–33: 43–72, 1976Google Scholar
3.Atiyah, M. and Schmid, W.. A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42: 1–62, 1977Google Scholar
4.Baum, P. and Connes, A.. Geometric K-theory for Lie groups and foliations. Enseign. Math. (2) 46: 3–42, 2000. Preprint from 1982Google Scholar
5.Baum, P. Connes, A., and Higson, N.. Classifying space for proper actions and K-theory of group C*-algebras. Contemporary Mathematics 167: 241–291, 1994Google Scholar
6.Beals, R. Characterization of pseudodifferential operators and applications. Duke Math. J. 44: 45–57, 1977Google Scholar
7.Blackadar, B. K-theory For Operator Algebras . Cambridge University Press, Cambridge, 1998Google Scholar
8.Bourbaki, N. Éléments de mathématique, livre VI: Intégration . Hermann, Paris, 1963Google Scholar
9.Chabert, J. and Echterhoff, S.. Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions. K-Theory 23: 157–200, 2001Google Scholar
10.Chabert, J. Echterhoff, S., and Nest, R.. The Connes-Kasparov conjecture for almost connected groups. Publ. Math. Inst. Hautes Études Sci. 97: 239–278, 2003CrossRefGoogle Scholar
11.Connes, A. Noncommutative Geometry . Academic Press, London, 1994Google Scholar
12.Connes, A. and Moscovici, H.. The L 2-index theorem for homogeneous spaces of Lie groups. Ann. Math. (2) 115: 291–330, 1982Google Scholar
13.Dirac, P.A.M.Lectures on Quantum Mechanics . Belfer School of Science, Yeshiva University, New York, 1964Google Scholar
14.Dixmier, J. C*-algebras. North-Holland, Amsterdam, 1977Google Scholar
15.Dixmier, J. and Douady, A.. Champs continus d'espaces hilbertiens et de C*-algèbres. Bulletin de la SMF 91: 227–284, 1963Google Scholar
16.Duistermaat, J.J.The Heat Kernel Lefschetz Fixed Point Theorem for the Spinc Dirac Operator . Birkhäuser, Basel, 1996Google Scholar
17.Dunford, N. and Schwartz, J. Linear Operators I. Wiley Interscience, New York, 1958Google Scholar
18.Duval, C., Elhadad, J., Gotay, M.J.Śniatycki, J., and Tuynman, G.M.. Quantization and bosonic BRST theory. Ann. Phys. (N.Y.) 206: 1–26, 1991CrossRefGoogle Scholar
19.Duval, C.Elhadad, J., and Tuynman, G.M.. The BRS method and geometric quantization: some examples. Comm. Math. Phys. 126: 535–557, 1990CrossRefGoogle Scholar
20.Friedrich, T. Dirac Operators in Riemannian Geometry. American Mathematical Society, Providence, 2000Google Scholar
21.Gotay, M.J.Constraints, reduction, and quantization. J. Math. Phys. 27: 2051–2066, 1986CrossRefGoogle Scholar
22.Gracia-Bondía, J.M.Várilly, J.C., and Figueroa, H.. Elements of Noncommutative Geometry. Birkhäuser, Boston, 2001CrossRefGoogle Scholar
23.Green, P. C*-algebras of transformation groups with smooth orbit space. Pacific J. Math. 72: 71–97, 1977Google Scholar
24.Griffiths, P. and Harris, J.. Principles of Algebraic Geometry. Wiley, New York, 1978Google Scholar
25.Guillemin, V.Ginzburg, V., and Karshon, Y.. Moment Maps, Cobordisms, and Hamiltonian Group Actions. American Mathematical Society, Providence, 2002CrossRefGoogle Scholar
26.Guillemin, V. and Sternberg, S. Geometric quantization and multiplicities of group representations. Invent. Math. 67: 515–538, 1982CrossRefGoogle Scholar
27.Hawkins, E. Quantization of multiply connected manifolds. Comm. Math. Phys. 255, no. 3: 513–575, 2005CrossRefGoogle Scholar
28.Heckman, G.J. and Hochs, P.. Proc. MRI Spring School 2004: Lie Groups in Analysis, Geometry and Mechanics , van den Ban, E. et al. (eds.), to appearGoogle Scholar
29.Henneaux, M. and Teitelboim, C.. Quantization of Gauge Systems. Princeton University Press, Princeton, 1992Google Scholar
30.Higson, N. Categories of fractions and excision in KK-theory. J. Pure Appl. Algebra 65: 119–138, 1990CrossRefGoogle Scholar
31.Higson, N. and Roe, J.. Analytic K-Homology. Oxford University Press, Oxford, 2000Google Scholar
32.Hochs, P. Quantisation commutes with reduction at discrete series representations of semisimple Lie groups. arXiv:0705.2956Google Scholar
33.Jeffrey, L.C. and Kirwan, F.C.. Localization and the quantization conjecture. Topology 36: 647–693, 1997CrossRefGoogle Scholar
34.Kasparov, G.G.The operator K-functor and extensions of C*-algebras. Math. USSR Izvestija 16: 513–572, 1981Google Scholar
35.Kasparov, G.G.Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91: 147–201, 1988CrossRefGoogle Scholar
36.Lafforgue, V. K-Théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math. 149: 2002, 1–95CrossRefGoogle Scholar
37.Lafforgue, V. Banach KK-theory and the Baum-Connes conjecture. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 795–812. Higher Ed. Press, Beijing, 2002Google Scholar
38.Landsman, N.P.Rieffel induction as generalized quantum Marsden-Weinstein reduction. J. Geom. Phys. 15: 285–319, 1995CrossRefGoogle Scholar
39.Landsman, N.P.Mathematical Topics Between Classical and Quantum Mechanics. Springer, New York, 1998Google Scholar
40.Landsman, N.P. Quantized reduction as a tensor product. Ref. [44], 137–180, 2001. arXiv:math-ph/0008004Google Scholar
41.Landsman, N.P.Quantization as a functor. Contemporary Mathematics 315: 9–24, 2002. arXiv:math-ph/0107023CrossRefGoogle Scholar
42.Landsman, N.P.Deformation quantization and the Baum-Connes conjecture. Comm. Math. Phys. 237: 87–103, 2003. arXiv:math-ph/0210015CrossRefGoogle Scholar
43.Landsman, N.P.Functorial quantization and the Guillemin–Sternberg conjecture. Twenty Years of Bialowieza: A Mathematical Anthology, 23–45. Ali, S.T., Emch, G.G., Odzijewicz, A., Schlichenmaier, M., and Woronowicz, S.L. (eds). World Scientific, Singapore, 2005. arXiv:math-ph/0307059CrossRefGoogle Scholar
44.Landsman, N.P.Pflaum, M., and Schlichenmaier, M. (eds.). Quantization of Singular Symplectic Quotients. Birkhäuser, Basel, 2001CrossRefGoogle Scholar
45.Lusztig, G. Novikov's higher signature and families of elliptic operators. J. Differential Geometry 7: 229–256, 1972Google Scholar
46.Marsden, J.E., Misiołek, G., Ortega, J.-P.Perlmutter, M., and Ratiu, T.S.. Hamiltonian Reduction by Stages. To appearGoogle Scholar
47.Marsden, J.E. and Ratiu, T.S.. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer-Verlag, New York, 1994Google Scholar
48.Marsden, J.E. and Weinstein, A.. Reduction of symplectic manifolds with symmetry. Rep. Math. Physics 5: 121–130, 1974Google Scholar
49.Marsden, J.E. and Weinstein, A.. Comments on the history, theory, and applications of symplectic reduction. Ref [44], 1–19, 2001CrossRefGoogle Scholar
50.Meinrenken, E. Symplectic surgery and the Spinc-Dirac operator. Adv. Math. 134: 240–277, 1998Google Scholar
51.Meinrenken, E. and Sjamaar, R.. Singular reduction and quantization. Topology 38: 699–761, 1999Google Scholar
52.Metzler, D.S.A K-theoretic note on geometric quantization. Manuscripta Math. 100: 277–289, 1999CrossRefGoogle Scholar
53.Mislin, G. and Valette, A.. Proper Group Actions and the Baum–Connes Conjecture. Birkhäuser, Basel, 2000Google Scholar
54.Paradan, P.-E.Localization of the Riemann-Roch character. J. Funct. Anal. 187: 442–509, 2001Google Scholar
55.Paradan, P.-E.Spinc-quantization and the K-multiplicities of the discrete series. Ann. Sci. École Norm. Sup. 36, no. 5: 805–845, 2003Google Scholar
56.Parthasarathy, R. Dirac operator and the discrete series. Ann. Math. (2) 96: 1–30, 1972Google Scholar
57.Pedersen, G.K.C*-Algebras and their Automorphism Groups. Academic Press, London, 1979Google Scholar
58.Pierrot, F. Une généralisation en K-théorie du théorème d'indice L 2 d'Atiyah Mém. Soc. Math. France 89, 2002Google Scholar
59.Reed, M. and Simon, B.. Methods of Modern Mathematical Physics, Vol I: Functional Analysis. Academic Press, New York, 1972Google Scholar
60.Rieffel, M.A.Induced representations of C*-algebras. Adv. Math. 13: 176–257, 1974Google Scholar
61.Rieffel, M.A.Deformation quantization of Heisenberg manifolds. Comm. Math. Phys. 122: 531–562, 1989Google Scholar
62.Rieffel, M.A.Quantization and C*-algebras. Contemporary Mathematics 167: 66–97, 1994CrossRefGoogle Scholar
63.Rørdam, M.Larsen, F., and Laustsen, N.. An Introduction to K-Theory for C*-Algebras. Cambridge University Press, Cambridge, 2000Google Scholar
64.Rosenberg, J. K-theory of group C*-algebras, foliation C*-algebras, and crossed products. Contemporary Mathematics 70: 251–301, 1986CrossRefGoogle Scholar
65.Sjamaar, R. Symplectic reduction and Riemann-Roch formulas for multiplicities. Bull. Amer. Math. Soc. (N.S.) 33: 327–338, 1996Google Scholar
66.Smale, S. Topology and mechanics. I. Invent. Math. 10: 305–331, 1970CrossRefGoogle Scholar
67.Sundermeyer, K. Constrained Dynamics. Springer-Verlag, Berlin, 1982Google Scholar
68.Takahashi, A. A duality between Hilbert modules and fields of Hilbert spaces. Rev. Colombiana Mat. 13: 93–120, 1979Google Scholar
69.Taylor, M. Pseudodifferential Operators. Princeton University Press, Princeton, 1982Google Scholar
70.Tian, Y. and Zhang, W.. An analytic proof of the geometric quantisation conjecture of Guillemin–Sternberg. Invent. Math. 132: 229–259, 1998Google Scholar
71.Valette, A. Introduction to the Baum–Connes Conjecture. Birkhäuser, Basel, 2002Google Scholar
72.Vergne, M. Quantification géométrique et réduction symplectique. Astérisque 282: 249–278, 2002Google Scholar
73.Wegge-Olsen, N.E.K-theory and C*-algebras. Oxford University Press, Oxford, 1993Google Scholar
74.Weinstein, A. The local structure of Poisson manifolds. J. Diff. Geom. 18: 523–557, 1983Google Scholar
75.Wolf, J. Essential self-adjointness for the Dirac operator and its square. Indiana Univ. Math. J. 22: 611–640, 1972Google Scholar
76.Xu, P. Morita equivalence of Poisson manifolds. Comm. Math. Phys. 142: 493–509, 1991CrossRefGoogle Scholar