Cohomology of Drinfeld symmetric spaces and Harmonic cochains
Annales de l'Institut Fourier, Volume 56 (2006) no. 3, p. 561-597

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.

Soit K un corps local non-archimédien. Ce papier donne un isomorphisme explicite entre le dual de la représentation spéciale de GL n+1 (K) et l’espace des cocycles harmoniques définis sur l’immeuble de Bruhat-Tits de GL n+1 (K), au sens de E. de Shalit [11]. Nous déduisons, en appliquant les résultats d’un papier de P. Schneider et U. Stuhler [9], qu’il existe un isomorphisme GL n+1 (K)-équivariant entre le groupe de cohomologie de l’espace symétrique de Drinfeld et l’espace des cocycles harmoniques.

DOI : https://doi.org/10.5802/aif.2194
Classification:  22E50,  20E42
Keywords: Drinfeld symmetric spaces, cohomology, Bruhat-Tits buildings, harmonic cochains, special representations
@article{AIF_2006__56_3_561_0,
     author = {A\"\i t Amrane, Yacine},
     title = {Cohomology of Drinfeld symmetric spaces and Harmonic cochains},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {3},
     year = {2006},
     pages = {561-597},
     doi = {10.5802/aif.2194},
     mrnumber = {2244224},
     zbl = {1118.22009},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_3_561_0}
}
Aït Amrane, Yacine. Cohomology of Drinfeld symmetric spaces and Harmonic cochains. Annales de l'Institut Fourier, Volume 56 (2006) no. 3, pp. 561-597. doi : 10.5802/aif.2194. http://www.numdam.org/item/AIF_2006__56_3_561_0/

[1] Aït Amrane, Y. Cohomologie des espaces symétriques de Drinfeld, cocycles harmoniques et formes automorphes, University of Toulouse 3 (2003) (Ph. D. Thesis)

[2] Aït Amrane, Y. Cohomologie des espaces symétriques de Drinfeld et cocycles harmoniques, C. R. Acad. Sci. Paris, Ser. I, Tome 338 (2004), pp. 191-196 | MR 2038322 | Zbl 1052.14023

[3] Borel, A.; Serre, J.-P. Cohomologie d’immeubles et de groupes S-arithmétiques, Topology, Tome 15 (1976), pp. 211-232 | Article | MR 447474 | Zbl 0338.20055

[4] Bourbaki, N. 4-6, Groupes et algèbres de Lie, Masson, Paris (1981) | MR 647314 | Zbl 0483.22001

[5] Brown, K. S. Buildings, Springer-Verlag, New York (1989) | MR 969123 | Zbl 0715.20017

[6] Drinfeld, V. G. Elliptic Modules, Math. USSR Sbornik, Tome 23 (1974), pp. 561-592 | Article | Zbl 0321.14014

[7] Garrett, P. Buildings and classical groups, Chapman and Hall, London (1997) | MR 1449872 | Zbl 0933.20019

[8] Van Der Put, Marius; Reversat, Marc; Gekeler, E.-U.; Van Der Put, M.; Reversat, M.; Geel, J. Van Lecture 11: Automorphic forms and Drinfeld’s reciprocity law, Drinfeld modules, modular schemes and applications, World Scientific (1997), pp. 188-223 (Proceedings of the Workshop at Alden-Biesen 9-14 sept. 1996) | MR 1630605 | Zbl 0924.11051

[9] Schneider, P.; Stuhler, U. The cohomology of p-adic symmetric spaces, Inv. Math., Tome 105 (1991), pp. 47-122 | Article | MR 1109620 | Zbl 0751.14016

[10] Schneider, P.; Teitelbaum, J. An integral transform for p-adic symmetric spaces, Duke Math. J., Tome 86 (1997), pp. 391-433 | Article | MR 1432303 | Zbl 0885.14012

[11] De Shalit, E. Residues on buildings and de Rham cohomology of p-adic symmetric domains, Duke Math. J., Tome 106 (2000), pp. 123-191 | Article | MR 1810368 | Zbl 01820775