Annals of K-Theory

A fixed point theorem on noncompact manifolds

Peter Hochs and Hang Wang

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/akt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We generalise the Atiyah–Segal–Singer fixed point theorem to noncompact manifolds. Using K K -theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah–Segal–Singer’s result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant K -theory and K -homology, and a nonlocalised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.

Article information

Source
Ann. K-Theory, Volume 3, Number 2 (2018), 235-286.

Dates
Received: 14 October 2016
Revised: 19 September 2017
Accepted: 4 October 2017
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.akt/1522807259

Digital Object Identifier
doi:10.2140/akt.2018.3.235

Mathematical Reviews number (MathSciNet)
MR3781428

Zentralblatt MATH identifier
06861674

Subjects
Primary: 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]
Secondary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 22E46: Semisimple Lie groups and their representations

Keywords
equivariant index fixed point formula noncompact manifold $KK$-theory

Citation

Hochs, Peter; Wang, Hang. A fixed point theorem on noncompact manifolds. Ann. K-Theory 3 (2018), no. 2, 235--286. doi:10.2140/akt.2018.3.235. https://projecteuclid.org/euclid.akt/1522807259


Export citation

References

  • N. Anghel, “On the index of Callias-type operators”, Geom. Funct. Anal. 3:5 (1993), 431–438.
  • M. F. Atiyah and R. Bott, “A Lefschetz fixed point formula for elliptic complexes, II: Applications”, Ann. of Math. $(2)$ 88 (1968), 451–491.
  • M. Atiyah and F. Hirzebruch, “Spin-manifolds and group actions”, pp. 18–28 in Essays on topology and related topics (Mémoires dédiés à Georges de Rham), Springer, 1970.
  • M. F. Atiyah and G. B. Segal, “The index of elliptic operators, II”, Ann. of Math. $(2)$ 87 (1968), 531–545.
  • M. F. Atiyah and I. M. Singer, “The index of elliptic operators, I”, Ann. of Math. $(2)$ 87 (1968), 484–530.
  • M. F. Atiyah and I. M. Singer, “The index of elliptic operators, III”, Ann. of Math. $(2)$ 87 (1968), 546–604.
  • P. Baum, A. Connes, and N. Higson, “Classifying space for proper actions and $K$-theory of group $C^\ast$-algebras”, pp. 240–291 in $C^\ast$-algebras: 1943–1993 (San Antonio, TX, 1993), edited by R. S. Doran, Contemporary Math. 167, American Mathematical Society, Providence, RI, 1994.
  • N. Berline and M. Vergne, “The Chern character of a transversally elliptic symbol and the equivariant index”, Invent. Math. 124:1-3 (1996), 11–49.
  • N. Berline and M. Vergne, “L'indice équivariant des opérateurs transversalement elliptiques”, Invent. Math. 124:1-3 (1996), 51–101.
  • N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren der Math. Wissenschaften 298, Springer, 1992.
  • B. Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications 5, Cambridge University Press, 1998.
  • M. Braverman, “Index theorem for equivariant Dirac operators on noncompact manifolds”, $K$-Theory 27:1 (2002), 61–101.
  • M. Braverman, “The index theory on non-compact manifolds with proper group action”, J. Geom. Phys. 98 (2015), 275–284.
  • M. Braverman and P. Shi, “Cobordism invariance of the index of Callias-type operators”, Comm. Partial Differential Equations 41:8 (2016), 1183–1203.
  • U. Bunke, “A $K$-theoretic relative index theorem and Callias-type Dirac operators”, Math. Ann. 303:2 (1995), 241–279.
  • C. Callias, “Axial anomalies and index theorems on open spaces”, Comm. Math. Phys. 62:3 (1978), 213–234.
  • A. L. Carey, V. Gayral, A. Rennie, and F. A. Sukochev, Index theory for locally compact noncommutative geometries, vol. 231, Memoirs of the American Mathematical Society 1085, American Mathematical Society, Providence, RI, 2014.
  • I. Dell'Ambrogio, H. Emerson, and R. Meyer, “An equivariant Lefschetz fixed-point formula for correspondences”, Doc. Math. 19 (2014), 141–194.
  • H. Emerson, “Duality, correspondences and the Lefschetz map in equivariant $K\!K$-theory: a survey”, pp. 41–78 in Perspectives on noncommutative geometry (Toronto, 2008), edited by M. Khalkhali and G. Yu, Fields Institute Communications 61, American Mathematical Society, Providence, RI, 2011.
  • M. Gromov and H. B. Lawson, Jr., “Positive scalar curvature and the Dirac operator on complete Riemannian manifolds”, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196.
  • V. Guillemin, V. Ginzburg, and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs 98, American Mathematical Society, Providence, RI, 2002.
  • Harish-Chandra, “Discrete series for semisimple Lie groups, II: Explicit determination of the characters”, Acta Math. 116 (1966), 1–111.
  • N. Higson and J. Roe, Analytic $K$-homology, Oxford University Press, 2000.
  • P. Hochs and V. Mathai, “Geometric quantization and families of inner products”, Adv. Math. 282 (2015), 362–426.
  • P. Hochs and Y. Song, “An equivariant index for proper actions I”, J. Funct. Anal. 272:2 (2017), 661–704.
  • P. Hochs and Y. Song, “Equivariant indices of ${\rm Spin}^c$-Dirac operators for proper moment maps”, Duke Math. J. 166:6 (2017), 1125–1178.
  • P. Hochs and H. Wang, “A fixed point formula and Harish-Chandra's character formula”, Proc. London Math. Soc. (online publication August 2017).
  • G. Kasparov, “Elliptic and transversally elliptic index theory from the viewpoint of $K\!K$-theory”, J. Noncommut. Geom. 10:4 (2016), 1303–1378.
  • A. W. Knapp, Representation theory of semisimple groups, Princeton University Press, 2001.
  • D. Kucerovsky, “A short proof of an index theorem”, Proc. Amer. Math. Soc. 129:12 (2001), 3729–3736.
  • H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton University Press, 1989.
  • X. Ma and W. Zhang, “Geometric quantization for proper moment maps: the Vergne conjecture”, Acta Math. 212:1 (2014), 11–57.
  • R. S. Palais, “On the existence of slices for actions of non-compact Lie groups”, Ann. of Math. $(2)$ 73 (1961), 295–323.
  • P.-E. Paradan, “${\rm Spin}^c$-quantization and the $K$-multiplicities of the discrete series”, Ann. Sci. École Norm. Sup. $(4)$ 36:5 (2003), 805–845.
  • P.-E. Paradan, “Formal geometric quantization, II”, Pacific J. Math. 253:1 (2011), 169–211.
  • J. Rosenberg, “The $G$-signature theorem revisited”, pp. 251–264 in Tel Aviv Topology Conference: Rothenberg Festschrift (Tel Aviv, 1998), edited by M. Farber et al., Contemporary. Math. 231, American Mathematical Society, Providence, RI, 1999.
  • W. Schmid, “$L\sp{2}$-cohomology and the discrete series”, Ann. of Math. $(2)$ 103:2 (1976), 375–394.
  • B.-L. Wang and H. Wang, “Localized index and $L^2$-Lefschetz fixed-point formula for orbifolds”, J. Differential Geom. 102:2 (2016), 285–349.