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Anis Rajhi
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.
(Cohomology with compact support of a space over the Bruhat-Tits building of ${\protect \rm GL}_n$ over a local field. Cuspidal representations of level zero.)
Confluentes Mathematici, 10 no. 1 (2018), p. 95-124, doi: 10.5802/cml.47
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Class. Math.: 22E50
Keywords: Representations of the general linear $p$-adic groups, Bruhat-Tits buildings, Cohomology with compact support.

Résumé - Abstract

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


[1] Abramenko, Peter and Brown, Kenneth S. Buildings : Theory and Applications (Graduate Texts in Mathematics), Springer, Softcover reprint of hardcover 1st ed. 2008.
[2] Brown, K.S. Cohomology of Groups (Graduate Texts in Mathematics, No. 87), Springer, 1st ed. 1982. Corr. 2nd printing 1994.
[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).
[4] Broussous, P. and Courtès, F. Distinction of the Steinberg representation, IMRN. International Mathematics Research Notices, 11, 3140–3157 (2014).
[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.
[6] Bredon, G.E. Introduction to compact transformation groups, Volume 46 (Pure and Applied Mathematics), Academic Press, 1972.
[7] Carayol, H. Représentations cuspidales du groupe linéaire, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 17, 191–225 (1984).
[8] Garrett, P.B. Buildings and Classical Groups, Chapman and Hall/CRC, 1997.
[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.
[11] Rotman, J. An Introduction to Homological Algebra (Universitext), Springer, 2008
[12] Spanier, E.H. Algebraic Topology, Springer, 1994.
[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).
[14] Tits, J. Buildings of Spherical Type and Finite BN-Pairs (Lecture Notes in Mathematics), Springer, 1986.
[15] Wagoner, J.B. Homotopy Theory for the p-adic Special Linear group,Commentarii Mathematici Helvetici, 50, 535–559 (1975).
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