Representation theory and sheaves on the Bruhat-Tits building
type: Item
instance of: scholarly article
title: Representation theory and sheaves on the Bruhat-Tits building
zbMATH Open document ID: 0892.22012
DOI: 10.1007/BF02699536
author: Peter Schneider
author: Ulrich Stuhler
published in: Publications Mathématiques
publication date: 1998-07-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104122
review text: The Bruhat-Tits building \(\mathcal I\) of a connected reductive algebraic group \(G\) over a \(p\)-adic field \(K\) displays in a geometric way the inner structure of the locally compact group \(G(K)\) of the set of \(K\)-points of \(G\) like the classification of the maximal subgroups of \(G(K)\) or the theory of its parahoric subgroups. One might consider \(\mathcal I\) as a kind of skeleton of an analogue of a real symmetric space because of that it clearly appears as an important tool in the study of the theory of smooth representations of \(G(K)\). In the paper under review, the authors develop a systematic and conceptual theory which gives a functorial way to pass from smooth representations of \(G(K)\) to equivariant objects on \(\mathcal I\). In a cohomological theory, they also associate (again functorially) \(G(K)\)-equivariant sheaves on \(\mathcal I\) with smooth representations of \(G(K)\). Their main result in this aspect is the computation of the cohomology with compact support of the sheaves coming from an irreducible smooth representation of \(G(K)\). For any facet \(F\) of \(\mathcal I\) the authors construct certain decreasing filtrations \({\mathcal P}_F\supset U_F^{(0)}\supset U_F^{(1)}\supset\cdots\supset U_F^{(e)}\supset\cdots\) of its pointwise stabilizer \({\mathcal P}_F\) in \(G(K)\) by compact open subgroups \(U_F^{(e)}\) and establish some nice properties of that filtration (especially they study how the groups \(U_F^{(e)}\) behave if the facet \(F\) is moved along in the geodesic in the building \(\mathcal I\)). Similar filtrations appear in papers of Prasad and Raghunathan and of Moy and Prasad. Let \(e\) be a fixed ``level''. A homological functor \(\gamma_e\) is constructed from smooth representations of \(G(K)\) to \(G(K)\)-equivariant coefficient systems on \(\mathcal I\): the coefficient system \(\gamma_e(V)\) corresponding to a representation \(V\) of \(G(K)\) is formed by associating with a facet \(F\) the subspace of the \(U_G^…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 1098729
zbMATH Keywords: representations
zbMATH Keywords: \(p\)-adic groups
zbMATH Keywords: Bruhat-Tits buildings
zbMATH Keywords: sheaves
zbMATH Keywords: Zelevinsky involution
zbMATH Keywords: reductive algebraic group
Wikidata QID: Q102356438
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
cites work: Dualite Dans Le Groupe De Grothendieck De La Categorie Des Representations Lisses De Longueur Finie D'un Groupe Reductif p-Adique
cites work: On the support of Plancherel measure
cites work: Trace Paley-Wiener theorem for reductive p-adic groups
cites work: Induced representations of reductive ${\germ p}$-adic groups. I
cites work: Q3884284
cites work: Q3999894
cites work: Q4165458
cites work: Cohomologie d'immeubles et de groupes S-arithmétiques
cites work: Q3885948
cites work: Q3941610
cites work: Q3669664
cites work: Spherical buildings and the character of the Steinberg representation
cites work: Q5340151
cites work: Duality for representations of a reductive group over a finite field
cites work: Q3873183
cites work: Q5580926
cites work: Harmonic analysis on reductive \(p\)-adic groups. Notes by G. van Dijk
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Rank Element of a Projective Module
cites work: Q3999005
cites work: Duality for Representations of a Hecke Algebra
cites work: Cuspidal geometry of p-adic groups
cites work: Representations of groups over close local fields
cites work: Tamagawa numbers
cites work: Unrefined minimal \(K\)-types for \(p\)-adic groups
cites work: Topological central extensions of semi-simple groups over local fields. I, II
cites work: Q3924412
cites work: Q4029668
cites work: Q5637658
cites work: Introduction to Harmonic Analysis on Reductive P-adic Groups. (MN-23): Based on lectures by Harish-Chandra at The Institute for Advanced Study, 1971-73
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
OpenAlex ID: W1985690510
description: scientific article; zbMATH DE number 1098729