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On the Connes–Kasparov isomorphism, I

The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual

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Japanese Journal of Mathematics Aims and scope

Abstract

This is the first of two papers dedicated to the detailed determination of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and a new and very explicit proof of the Connes–Kasparov conjecture for these groups using representation theory. In this part we shall give details of the C*-algebraic Morita equivalence and then explain how the Connes–Kasparov morphism in operator K-theory may be computed using what we call the Matching Theorem, which is a purely representation-theoretic result. We shall prove our Matching Theorem in the sequel, and indeed go further by giving a simple, direct construction of the components of the tempered dual that have non-trivial K-theory using David Vogan’s approach to the classification of the tempered dual.

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References

  1. A. Afgoustidis and A.-M. Aubert, C*-blocks and crossed products for classical p-adic groups, Int. Math. Res. Not. IMRN, 2022 (2022), no. 22, 17849–17908.

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  3. P. Baum, A. Connes and N. Higson, Classifying space for proper actions and K-theory of group C*-algebras, In: C*-Algebras: 1943–1993, San Antonio, 1993, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291.

    Google Scholar 

  4. J. Chabert, S. Echterhoff and R. Nest, The Connes–Kasparov conjecture for almost connected groups and for linear p-adic groups, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 239–278.

    Article  MathSciNet  Google Scholar 

  5. P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis, 12 (1973), 401–414.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. P. Clare, N. Higson and Y. Song, On the Connes–Kasparov isomorphism, II: The Vogan classification of essential components in the tempered dual, Jpn. J. Math., 19 (2024), 111–141.

    Google Scholar 

  8. M. Cowling, U. Haagerup and R. Howe, Almost L2 matrix coefficients, J. Reine Angew. Math., 387 (1988), 97–110.

    MathSciNet  Google Scholar 

  9. J. Dixmier, C*-Algebras. Translated from the French by Francis Jellett, North-Holland Math. Library, 15, North-Holland Publishing Co., 1977.

  10. S. Echterhoff and O. Pfante, Equivariant K-theory of finite dimensional real vector spaces, Münster J. Math., 2 (2009), 65–94.

    MathSciNet  Google Scholar 

  11. M.P. Gomez Aparicio, P. Julg and A. Valette, The Baum–Connes conjecture: an extended survey, In: Advances in Noncommutative Geometry—On the Occasion of Alain Connes’ 70th Birthday, Springer-Verlag, 2019, pp. 127–244.

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

    Article  MathSciNet  Google Scholar 

  13. N. Higson, A primer on KK-theory, In: Operator Theory: Operator Algebras and Applications, Part 1, Durham, 1988, Proc. Sympos. Pure Math., 51, Amer. Math. Soc., Providence, RI, 1990, pp. 239–283.

    Google Scholar 

  14. J.-S. Huang and P. Pandžić, Dirac Operators in Representation Theory, Math. Theory Appl., Birkhäuser Boston, Inc., Boston, MA, 2006.

    Google Scholar 

  15. G.G. Kasparov, The operator K-functor and extensions of C*-algebras, Math. USSR-Izv., 16 (1981), 513–572.

    Article  Google Scholar 

  16. G.G. Kasparov, Index for invariant elliptic operators, K-theory and representations of Lie groups, Dokl. Akad. Nauk SSSR, 268 (1983), 533–537.

    MathSciNet  Google Scholar 

  17. G.G. Kasparov, Operator K-theory and its applications: elliptic operators, group representations, higher signatures, C*-extensions, In: Proceedings of the International Congress of Mathematicians. Vol. 1, 2, Warsaw, 1983, PWN, Warsaw, 1984, pp. 987–1000.

    Google Scholar 

  18. G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math., 91 (1988), 147–201.

    Article  ADS  MathSciNet  Google Scholar 

  19. A.W. Knapp, Commutativity of intertwining operators for semisimple groups, Compositio Math., 46 (1982), 33–84.

    MathSciNet  Google Scholar 

  20. A.W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples, Princeton Math. Ser., 36, Princeton Univ. Press, Princeton, NJ, 1986.

    Book  Google Scholar 

  21. A.W. Knapp, Lie Groups beyond an Introduction. 2nd ed., Progr. Math., 140, Birkhäuser Boston, Inc., Boston, MA, 2002.

    Google Scholar 

  22. A.W. Knapp and E.M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2), 93 (1971), 489–578.

    Article  MathSciNet  Google Scholar 

  23. A.W. Knapp and E.M. Stein, Intertwining operators for semisimple groups. II, Invent. Math., 60 (1980), 9–84.

    Article  ADS  MathSciNet  Google Scholar 

  24. A.W. Knapp and D.A. Vogan, Jr., Cohomological Induction and Unitary Representations, Princeton Math. Ser., 45, Princeton Univ. Press, Princeton, NJ, 1995.

    Book  Google Scholar 

  25. V. Lafforgue, Banach KK-theory and the Baum–Connes conjecture, In: Proceedings of the International Congress of Mathematicians. Vol. II, Beijing, 2002, Higher Education Press, 2002, pp. 795–812.

    Google Scholar 

  26. V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes, Invent. Math., 149 (2002), 1–95.

    Article  ADS  MathSciNet  Google Scholar 

  27. E.C. Lance, Hilbert C*-Modules, London Math. Soc. Lecture Note Ser., 210, Cambridge Univ. Press, Cambridge, 1995.

    Book  Google Scholar 

  28. R.L. Lipsman, The dual topology for the principal and discrete series on semi-simple groups, Trans. Amer. Math. Soc., 152 (1970), 399–417.

    Article  MathSciNet  Google Scholar 

  29. E. Meinrenken, Clifford Algebras and Lie Theory, Ergeb. Math. Grenzgeb. (3), 58, Springer-Verlag, 2013.

  30. J.S. Milne, Algebraic Groups. The Theory of Group Schemes of Finite Type over a Field, Cambridge Stud. Adv. Math., 170, Cambridge Univ. Press, Cambridge, 2017.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. G.K. Pedersen, C*-Algebras and Their Automorphism Groups, London Math. Soc. Monogr., 14, Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, London-New York, 1979.

    Google Scholar 

  33. M.G. Penington and R.J. Plymen, The Dirac operator and the principal series for complex semisimple Lie groups, J. Funct. Anal., 53 (1983), 269–286.

    Article  MathSciNet  Google Scholar 

  34. I. Raeburn and D.P. Williams, Morita Equivalence and Continuous-Trace C*-Algebras, Math. Surveys Monogr., 60, Amer. Math. Soc., Providence, RI, 1998.

    Book  Google Scholar 

  35. M.A. Rieffel, Actions of finite groups on C*-algebras, Math. Scand., 47 (1980), 157–176.

    Article  MathSciNet  Google Scholar 

  36. M. Rørdam, F. Larsen and N. Laustsen, An Introduction to K-Theory for C*-Algebras, London Math. Soc. Stud. Texts, 49, Cambridge Univ. Press, Cambridge, 2000.

    Book  Google Scholar 

  37. J. Rosenberg, Group C*-algebras and topological invariants, In: Operator Algebras and Group Representations. Vol. II, Neptun, 1980, Monogr. Stud. Math., 18, Pitman, Boston, MA, 1984, pp. 95–115.

    Google Scholar 

  38. A. Valette, K-theory for the reduced C*-algebra of a semisimple Lie group with real rank 1 and finite centre, Quart. J. Math. Oxford Ser. (2), 35 (1984), 341–359.

    Article  MathSciNet  Google Scholar 

  39. A. Valette, Dirac induction for semisimple Lie groups having one conjugacy class of Cartan subgroups, In: Operator Algebras and Their Connections with Topology and Ergodic Theory, Buşteni, 1983, Lecture Notes in Math., 1132, Springer-Verlag, 1985, pp. 526–555.

  40. D.A. Vogan, Jr., Representations of Real Reductive Lie Groups, Progr. Math., 15, Birkhäuser, Boston, MA, 1981.

    Google Scholar 

  41. A. Wassermann, Une démonstration de la conjecture de Connes–Kasparov pour les groupes de Lie linéaires connexes réductifs, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 559–562.

    MathSciNet  Google Scholar 

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Acknowledgements

This research was supported by NSF grants DMS-1952669 (NH), DMS-1800667, DMS-1952557 (YS), DMS-1800666 and DMS-1952551 (XT). Part of the research was carried out within the online Research Community on Representation Theory and Noncommutative Geometry sponsored by the American Institute of Mathematics.

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Authors and Affiliations

  1. College of William & Mary, P.O. Box 8795, Williamsburg, VA, 23187-8795, USA

    Pierre Clare

  2. Penn State University, University Park, PA, 16802, USA

    Nigel Higson

  3. Washington University in St. Louis, Campus Box 1089, One Brookings Drive, St. Louis, MO, 63130-4899, USA

    Yanli Song & Xiang Tang

Authors
  1. Pierre Clare
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  2. Nigel Higson
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  3. Yanli Song
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  4. Xiang Tang
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Correspondence to Pierre Clare, Nigel Higson, Yanli Song or Xiang Tang.

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Communicated by: Toshiyuki Kobayashi

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Clare, P., Higson, N., Song, Y. et al. On the Connes–Kasparov isomorphism, I. Jpn. J. Math. 19, 67–109 (2024). https://doi.org/10.1007/s11537-024-2220-2

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  • DOI: https://doi.org/10.1007/s11537-024-2220-2

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