Cohomologie à support compact d’un espace au-dessus de l’immeuble de Bruhat-Tits de GL n sur un corps local. Représentations cuspidales de niveau zéro.
[Cohomology with compact support of a space over the Bruhat-Tits building of GL n over a local field. Cuspidal representations of level zero.]
Confluentes Mathematici, Volume 10 (2018) no. 1, pp. 95-124.

Let G the group GL n (F), where F is a non-archimedean locally compact field, and 𝔅(G) its Bruhat-Tits building. We construct a simplicial complex 𝒲 ˜, equipped with an action of G and with a G-equivariant proper simplicial projection p:𝒲 ˜𝔅(G). We prove that the cohomology with compact support in higher dimensions H c n-1 (𝒲 ˜,) contains as subquotients all irreducible cuspidal level zero representations.

Soit G le groupe GL n (F), où F est un corps localement compact non-archimédien, et 𝔅(G) son immeuble de Bruhat-Tits. Nous construisons un complexe simplicial 𝒲 ˜, doté d’une action de G et d’une projection propre simpliciale G-équivariante p:𝒲 ˜𝔅(G). Nous démontrons qu’en dimension supérieure la cohomologie à support compact H c n-1 (𝒲 ˜,) contient comme sous-quotient toutes les représentations cuspidales irréductibles de niveau zéro.

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Accepted:
Published online:
DOI: 10.5802/cml.47
Classification: 22E50
Keywords: Representations of the general linear $p$-adic groups, Bruhat-Tits buildings, Cohomology with compact support.
Rajhi, Anis 1

1 Université de Sousse ; École supérieure des sciences et technologies de Hammam-Sousse, Hammam Sousse 4011, Tunisie
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Rajhi, Anis. Cohomologie à support compact d’un espace au-dessus de l’immeuble de Bruhat-Tits de ${\protect \rm GL}_{n}$ sur un corps local. Représentations cuspidales de niveau zéro.. Confluentes Mathematici, Volume 10 (2018) no. 1, pp. 95-124. doi : 10.5802/cml.47. http://www.numdam.org/articles/10.5802/cml.47/

[1] Abramenko, Peter and Brown, Kenneth S. Buildings : Theory and Applications (Graduate Texts in Mathematics), Springer, Softcover reprint of hardcover 1st ed. 2008. | DOI

[2] Brown, K.S. Cohomology of Groups (Graduate Texts in Mathematics, No. 87), Springer, 1st ed. 1982. Corr. 2nd printing 1994. | DOI

[3] Broussous, P. Representations of PGL(2) of a local field and harmonic cochains on graphs, Annales de la Faculté des Sciences de Toulouse, vol XVIII, 541–559 (2009). | DOI | MR | Zbl

[4] Broussous, P. and Courtès, F. Distinction of the Steinberg representation, IMRN. International Mathematics Research Notices, 11, 3140–3157 (2014). | DOI | MR | Zbl

[5] Borel, A. and Serre, J.P. Cohomologie à support compacts des immeubles de Bruhat-Tits, applications à la cohomologie des groupes S-arithmétiques, C.R.Acad.sc.Paris, 1971. | Zbl

[6] Bredon, G.E. Introduction to compact transformation groups, Volume 46 (Pure and Applied Mathematics), Academic Press, 1972. | DOI | Zbl

[7] Carayol, H. Représentations cuspidales du groupe linéaire, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 17, 191–225 (1984). | DOI | Numdam | Zbl

[8] Garrett, P.B. Buildings and Classical Groups, Chapman and Hall/CRC, 1997. | DOI | Zbl

[9] Munkers, J.R. Elements Of Algebraic Topology, Westview Press, 1996.

[10] Murnaghan, F. Representations of reductive p-adic groups, "http://www.math.toronto.edu/murnaghan/courses/mat1197/", 2009. | Zbl

[11] Rotman, J. An Introduction to Homological Algebra (Universitext), Springer, 2008 | DOI

[12] Spanier, E.H. Algebraic Topology, Springer, 1994. | DOI

[13] Schneider, P. and Stuhler, U. Representation theory and sheaves on the Bruhat-Tits building, Publications mathématiques de l’I.H.E.S, 85, 97-191 (1997). | DOI | Numdam | Zbl

[14] Tits, J. Buildings of Spherical Type and Finite BN-Pairs (Lecture Notes in Mathematics), Springer, 1986. | DOI | Zbl

[15] Wagoner, J.B. Homotopy Theory for the p-adic Special Linear group,Commentarii Mathematici Helvetici, 50, 535–559 (1975). | DOI | MR | Zbl

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