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Homological algebra for Schwartz algebras of reductive p-adic groups

  • Ralf Meyer
Part of the Aspects of Mathematics book series (ASMA)

Abstract

Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of G to the derived category of all smooth representations of G is fully faithful. Here we consider representations on bornological vector spaces. As a consequence, if G is semi-simple, V and W are tempered irreducible representations of G, and V or W is square-integrable, then Ext G n (V, W) ≅ 0 for all n ≥ 1. We use this to prove in full generality a formula for the formal dimension of square-integrable representations due to Schneider and Stuhler.

Keywords

Full Subcategory Projective Resolution Homological Algebra Compact Open Subgroup Smooth Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Anne-Marie Aubert and Roger Plymen, Plancherel measure for GL(n, F) and GL(m, D): explicit formulas and Bernstein decomposition, J. Number Theory 112 (2005), 26–66. MR2131140MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Hyman Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155–196. MR0432781 (55 #5764)MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. N. Bernstein, Le “centre” de Bernstein, 1–32, Edited by P. Deligne. MR771671 (86e:22028)Google Scholar
  4. [4]
    Martin R. Bridson and Andre Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999, ISBN 3-540-64324-9. MR1744486 (2000k:53038)Google Scholar
  5. [5]
    F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972), 5–251. MR0327923 (48 #6265) (French)Google Scholar
  6. [6]
    Heath Emerson and Ralf Meyer, Dualizing the coarse assembly map, J. Inst. Math. Jussieu (2004), http://arxiv.org/math.OA/0401227 (to appear).Google Scholar
  7. [7]
    Alexander Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., vol. 16, 1955. MR0075539 (17,763c) (French)Google Scholar
  8. [8]
    Henri Hogbe-Nlend, Complétion, tenseurs et nucléarité en bornologie, J. Math. Pures Appl. (9) 49 (1970), 193–288. MR0279557 (43 #5279) (French)MATHMathSciNetGoogle Scholar
  9. [9]
    _____, Bornologies and functional analysis, North-Holland Mathematics Studies, vol. 26, North-Holland Publishing Co., Amsterdam, 1977, ISBN 0-7204-0712-5. MR0500064 (58 #17774)Google Scholar
  10. [10]
    James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990, ISBN 0-521-37510-X. MR1066460 (92h:20002)Google Scholar
  11. [11]
    Bernhard Keller, Derived categories and their uses, 1996, pp. 671–701. MR1421815 (98h:18013)Google Scholar
  12. [12]
    Wolfgang Lück and Holger Reich, The Baum-Connes and the Farrell-Jones Conjectures in K-and L-theory, Preprintreihe SFB 478 324 (2004), Universität Münster.Google Scholar
  13. [13]
    Ralf Meyer, Analytic cyclic cohomology, Ph.D. Thesis, Westfälische Wilhelms-Universität Münster, 1999.Google Scholar
  14. [14]
    _____, Bornological versus topological analysis in metrizable spaces, 249–278. MR2097966Google Scholar
  15. [15]
    _____, Smooth group representations on bornological vector spaces, Bull. Sci. Math. 128 (2004), 127–166. MR2039113 (2005c:22013) (English, with English and French summaries)Google Scholar
  16. [16]
    _____, Combable groups have group cohomology of polynomial growth (2004), http://arxiv.org/math.KT/0410597 (to appear in Q. J. Math.)Google Scholar
  17. [17]
    _____, Embeddings of derived categories of bornological modules (2004), http://arxiv.org/math.FA/0410596 (eprint).Google Scholar
  18. [18]
    A. Yu. Pirkovskii, Stably flat completions of universal enveloping algebras (2003), http://arxiv.org/math.FA/0311492 (eprint).Google Scholar
  19. [19]
    Jonathan Rosenberg, Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994, ISBN 0-387-94248-3. MR1282290 (95e:19001)Google Scholar
  20. [20]
    Peter Schneider and Ulrich Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Inst. Hautes Études Sci. Publ. Math. (1997), 97–191. MR1471867 (98m:22023)Google Scholar
  21. [21]
    P. Schneider and E.-W. Zink, K-types for the tempered components of a p-adic general linear group, J. Reine Angew. Math. 517 (1999), 161–208, With an appendix by P. Schneider and U. Stuhler. MR1728541 (2001f:22029)MATHMathSciNetGoogle Scholar
  22. [22]
    Allan J. Silberger, Introduction to harmonic analysis on reductive p-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J., 1979, ISBN 0-691-08246-4, Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR544991 (81m:22025)MATHGoogle Scholar
  23. [23]
    J. Tits, Reductive groups over local fields, 29–69. MR546588 (80h:20064)Google Scholar
  24. [24]
    François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967. MR0225131 (37 #726)MATHGoogle Scholar
  25. [25]
    Marie-France Vignéras, On formal dimensions for reductive p-adic groups, 225–266. MR1159104 (93c:22034)Google Scholar
  26. [26]
    J.-L. Waldspurger, La formule de Plancherel pour les groupes p-adiques (d–après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), 235–333. MR1989693 (2004d:22009) (French) E-mail address: rameyer@math.uni-muenster.deMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Ralf Meyer
    • 1
  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany

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