Cohomology of Drinfeld symmetric spaces and Harmonic cochains

Yacine Aït Amrane[1]

  • [1] Universität Münster Mathematisches Institut Einsteinstr. 62 48149 Münster (Allemagne)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 3, page 561-597
  • ISSN: 0373-0956

Abstract

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Let K be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of G L n + 1 ( K ) and the space of harmonic cochains defined on the Bruhat-Tits building of G L n + 1 ( K ) , in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a G L n + 1 ( K ) -equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.

How to cite

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Aït Amrane, Yacine. "Cohomology of Drinfeld symmetric spaces and Harmonic cochains." Annales de l’institut Fourier 56.3 (2006): 561-597. <http://eudml.org/doc/10158>.

@article{AïtAmrane2006,
abstract = {Let $K$ be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of $GL_\{n+1\}(K)$ and the space of harmonic cochains defined on the Bruhat-Tits building of $GL_\{n+1\}(K)$, in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a $GL_\{n+1\}(K)$-equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.},
affiliation = {Universität Münster Mathematisches Institut Einsteinstr. 62 48149 Münster (Allemagne)},
author = {Aït Amrane, Yacine},
journal = {Annales de l’institut Fourier},
keywords = {Drinfeld symmetric spaces; cohomology; Bruhat-Tits buildings; harmonic cochains; special representations},
language = {eng},
number = {3},
pages = {561-597},
publisher = {Association des Annales de l’institut Fourier},
title = {Cohomology of Drinfeld symmetric spaces and Harmonic cochains},
url = {http://eudml.org/doc/10158},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Aït Amrane, Yacine
TI - Cohomology of Drinfeld symmetric spaces and Harmonic cochains
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 561
EP - 597
AB - Let $K$ be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of $GL_{n+1}(K)$ and the space of harmonic cochains defined on the Bruhat-Tits building of $GL_{n+1}(K)$, in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a $GL_{n+1}(K)$-equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.
LA - eng
KW - Drinfeld symmetric spaces; cohomology; Bruhat-Tits buildings; harmonic cochains; special representations
UR - http://eudml.org/doc/10158
ER -

References

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  1. Y. Aït Amrane, Cohomologie des espaces symétriques de Drinfeld, cocycles harmoniques et formes automorphes, (2003) 
  2. Y. Aït Amrane, Cohomologie des espaces symétriques de Drinfeld et cocycles harmoniques, C. R. Acad. Sci. Paris, Ser. I 338 (2004), 191-196 Zbl1052.14023MR2038322
  3. A. Borel, J.-P. Serre, Cohomologie d’immeubles et de groupes S -arithmétiques, Topology 15 (1976), 211-232 Zbl0338.20055MR447474
  4. N. Bourbaki, Groupes et algèbres de Lie, (1981), Masson, Paris Zbl0483.22001MR647314
  5. K. S. Brown, Buildings, (1989), Springer-Verlag, New York Zbl0715.20017MR969123
  6. V. G. Drinfeld, Elliptic Modules, Math. USSR Sbornik 23 (1974), 561-592 Zbl0321.14014
  7. P. Garrett, Buildings and classical groups, (1997), Chapman and Hall, London Zbl0933.20019MR1449872
  8. Marius van der Put, Marc Reversat, Lecture 11: Automorphic forms and Drinfeld’s reciprocity law, Drinfeld modules, modular schemes and applications (1997), 188-223, World Scientific Zbl0924.11051MR1630605
  9. P. Schneider, U. Stuhler, The cohomology of p -adic symmetric spaces, Inv. Math. 105 (1991), 47-122 Zbl0751.14016MR1109620
  10. P. Schneider, J. Teitelbaum, An integral transform for p -adic symmetric spaces, Duke Math. J. 86 (1997), 391-433 Zbl0885.14012MR1432303
  11. E. de Shalit, Residues on buildings and de Rham cohomology of p -adic symmetric domains, Duke Math. J. 106 (2000), 123-191 Zbl1103.14010MR1810368

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