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An index theorem for higher orbital integrals

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Abstract

Recently, two of the authors of this paper constructed cyclic cocycles on Harish–Chandra’s Schwartz algebra of linear reductive Lie groups that detect all information in the K-theory of the corresponding group \(C^*\)-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions.

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References

  1. Abels, H.: Parallelizability of proper actions, global \(K\)-slices and maximal compact subgroups. Math. Ann. 212, 1–19 (1974/75)

  2. Arthur, J.: A theorem on the Schwartz space of a reductive Lie group. Proc. Natl. Acad. Sci. USA 72(12), 4718–4719 (1975)

    Article  MathSciNet  Google Scholar 

  3. Atiyah, M., Schmid, W.: A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42, 1–62 (1977)

    Article  MathSciNet  Google Scholar 

  4. Atiyah, M., Segal, G.: The index of elliptic operators. II. Ann. Math. 2(87), 531–545 (1968)

    Article  MathSciNet  Google Scholar 

  5. Atiyah, M., Singer, I.: The index of elliptic operators. III. Ann. Math. 2(87), 546–604 (1968)

    Article  MathSciNet  Google Scholar 

  6. Barbasch, D., Moscovici, H.: \(L^{2}\)-index and the Selberg trace formula. J. Funct. Anal. 53(2), 151–201 (1983)

    Article  MathSciNet  Google Scholar 

  7. Baum, P., Connes, A., Higson, N.: Classifying space for proper actions and \(K\)-theory of group \(C^\ast \)-algebras. In \(C^\ast \)-algebras: 1943–1993 (San Antonio, TX, 1993), volume 167 of Contemp. Math., pp. 240–291. American Mathematical Society, Providence, RI (1994)

  8. Baum, P.F., van Erp, E.: \(K\)-homology and Fredholm operators II: elliptic operators. Pure Appl. Math. Q. 12(2), 225–241 (2016)

    Article  MathSciNet  Google Scholar 

  9. Bismut, J.-M.: Hypoelliptic Laplacian and Orbital Integrals. Annals of Mathematics Studies, vol. 177. Princeton University Press, Princeton, NJ (2011)

  10. Clare, P.: Hilbert modules associated to parabolically induced representations. J. Oper. Theory 69(2), 483–509 (2013)

    Article  MathSciNet  Google Scholar 

  11. Clare, P., Crisp, T., Higson, N.: Parabolic induction and restriction via \(C^*\)-algebras and Hilbert \(C^*\)-modules. Compos. Math. 152(6), 1286–1318 (2016)

    Article  MathSciNet  Google Scholar 

  12. Connes, A.: Noncommutative Geometry. Academic Press Inc, San Diego, CA (1994)

    MATH  Google Scholar 

  13. Connes, A., Moscovici, H.: The \(L^{2}\)-index theorem for homogeneous spaces of Lie groups. Ann. Math. (2) 115(2), 291–330 (1982)

    Article  MathSciNet  Google Scholar 

  14. Connes, A., Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29(3), 345–388 (1990)

    Article  MathSciNet  Google Scholar 

  15. Fukumoto, Y.: \(G\)-homotopy invariance of the analytic signature of proper co-compact \(G\)-manifolds and equivariant Novikov conjecture. J. Noncommun. Geom. (to appear). arXiv:1709.05884

  16. Gong, S.: Finite part of operator \(K\)-theory for groups with rapid decay. J. Noncommun. Geom. 9(3), 697–706 (2015)

    Article  MathSciNet  Google Scholar 

  17. Harish-Chandra.: Spherical functions on a semi-simple Lie group. II. Am. J. Math. 80, 553–613 (1958)

  18. Harish-Chandra: Discrete series for semisimple Lie groups. II. Explicit determination of the characters. Acta Math. 116, 1–111 (1966)

  19. Nigel, H., John, R.: Analytic \(K\)-homology. Oxford Mathematical Monographs. Oxford University Press, Oxford, Oxford Science Publications (2000)

  20. Hochs, P.: Quantisation commutes with reduction at discrete series representations of semisimple groups. Adv. Math. 222(3), 862–919 (2009)

    Article  MathSciNet  Google Scholar 

  21. Hochs, P., Landsman, K.: The Guillemin–Sternberg conjecture for noncompact groups and spaces. J. K-Theory 1(3), 473–533 (2008)

    Article  MathSciNet  Google Scholar 

  22. Hochs, P., Mathai, V.: Quantising proper actions on Spin\(^c\)-manifolds. Asian J. Math. 21(4), 631–685 (2017)

    Article  MathSciNet  Google Scholar 

  23. Hochs, P., Song, Y.: An equivariant index for proper actions. III. The invariant and discrete series indices. Differ. Geom. Appl. 49, 1–22 (2016)

    Article  MathSciNet  Google Scholar 

  24. Hochs, P., Wang, H.: A fixed point formula and Harish–Chandra’s character formula. Proc. Lond. Math. Soc. 00(3), 1–32 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Hochs, P., Wang, H.: Orbital integrals and \({K}\)-theory classes. Ann. \({K}\)-theory 2(4), 185–209 (2019)

  26. Knapp, A.W: Representation theory of semisimple groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1986)

  27. Knapp, A.W., Zuckerman, G.J.: Classification of irreducible tempered representations of semisimple groups. Ann. Math. (2) 116(2), 389–455 (1982)

    Article  MathSciNet  Google Scholar 

  28. Knapp, A.W., Zuckerman, G.J.: Classification of irreducible tempered representations of semisimple groups. II. Ann. Math. (2) 116(3), 457–501 (1982)

    Article  MathSciNet  Google Scholar 

  29. Kobayashi, T.: Discrete decomposability of the restriction of \(A_{{q}}(\lambda )\) with respect to reductive subgroups. II. Micro-local analysis and asymptotic \(K\)-support. Ann. Math. (2) 147(3), 709–729 (1998)

    Article  MathSciNet  Google Scholar 

  30. Landsman, K.: Functorial quantization and the Guillemin–Sternberg conjecture. In: Twenty years of Bialowieza: a mathematical anthology, volume 8 of World Sci. Monogr. Ser. Math., pp. 23–45. World Sci. Publ., Hackensack, NJ (2005)

  31. Mesland, B., Sengun, M.H., Wang, H.: A \(K\)-theoretic selberg trace formula. In: Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology, Ronald G. Douglas memorial volume. Operator Theory: Advances and Applications, vol. 278. Birkhäuser, Springer, Cham (2020)

  32. Mislin, G., Valette, A.: Proper group actions and the Baum–Connes conjecture. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2003)

    Book  Google Scholar 

  33. Parthasarathy, R.: Dirac operator and the discrete series. Ann. Math. 2(96), 1–30 (1972)

    Article  MathSciNet  Google Scholar 

  34. Pflaum, M., Posthuma, H., Tang, X.: The transverse index theorem for proper cocompact actions of Lie groupoids. J. Differ. Geom. 99(3), 443–472 (2015)

    Article  MathSciNet  Google Scholar 

  35. Piazza, P., Posthuma, H.B.: Higher genera for proper actions of Lie groups. Ann. K-Theory 4(3), 473–504 (2019)

    Article  MathSciNet  Google Scholar 

  36. Samurkaş, S.K.: Bounds for the rank of the finite part of operator \(K\)-theory. J. Noncommut. Geom. 14(2), 413–439 (2020)

  37. Song, Y., Tang, X.: Higher orbit integrals, cyclic cocyles, and \({K}\)-theory of reduced group \({C^*}\)-algebra. arXiv:1910.00175 (2019)

  38. Wang, H.: \(L^2\)-index formula for proper cocompact group actions. J. Noncommun. Geom. 8(2), 393–432 (2014)

    Article  Google Scholar 

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Acknowledgements

Hochs is partially supported by Discovery Project DP200100729 from the Australian Research Council; Song is partially supported by NSF Grant DMS-1800667; Tang is partially supported by NSF Grants DMS-1363250 and DMS-1800666.

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Authors and Affiliations

  1. Radboud University, Heyendaalseweg 135, 6525, AJ Nijmegen, The Netherlands

    Peter Hochs

  2. Washington University in St. Louis, St. Louis, MO, USA

    Yanli Song & Xiang Tang

Authors
  1. Peter Hochs
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  2. Yanli Song
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  3. Xiang Tang
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Correspondence to Yanli Song.

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Communicated by Thomas Schick.

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Hochs, P., Song, Y. & Tang, X. An index theorem for higher orbital integrals. Math. Ann. 382, 169–202 (2022). https://doi.org/10.1007/s00208-021-02233-3

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