arXiv subjects
Publications and source records attributed to Vincent Sécherre.
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let l be a prime number different from p. We develop a theory of l-modular types for the group GL(m,D), m\>1, in preparation of the study of the l-modular smooth representations of this group.
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. We classify all smooth irreducible representations of GL(m,D) with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadic and Vignéras. We prove that any irreducible R-representation of GL(m,D) has a unique supercuspidal support, and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal support, and one by aperiodic multisegments, classifying representations with a given cuspidal support. These constructions are made in a purely local way, with a substantial use of type theory.
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. To any irreducible smooth representation of G=GL(m,D) with coefficients in R, we can attach a uniquely determined inertial class of supercuspidal pairs of G. This provides us with a partition of the set of all isomorphism classes of irreducible representations of G. We write R(G) for the category of all smooth representations of G with coefficients in R. To any inertial class O of supercuspidal pairs of G, we can attach the subcategory R(O) made of smooth representations all of whose irreducible subquotients are in the subset determined by this inertial class. We prove that R(G) decomposes into the product of the R(O), where O ranges over all possible inertial class of supercuspidal pairs of G, and that each summand R(O) is indecomposable.
Soit F un corps commutatif localement compact non archimédien de caractéristique résiduelle p, soit D une F-algèbre à division centrale de dimension finie et soit R un corps algébriquement clos de caractéristique différente de p. Nous définissons la notion de R-représentation irréductible banale de G=GL(m,D), notion qui généralise celle de caractéristique banale pour G, et qui repose sur une condition portant sur le support cuspidal de la représentation et dépendant de la caractéristique de R. Lorsque le corps R est de caractéristique banale, en particulier lorsque R est le corps des nombres complexes, toute R-représentation irréductible de G est banale. Nous donnons dans cet article une classification des R-représentations irréductibles banales de G en termes de multisegments dits banals. Lorsque R est le corps des nombres complexes, notre méthode fournit une nouvelle preuve, entièrement locale, de la classification de Tadic des représentations lisses irréductibles complexes de G.
Let F be a p-adic field with residue class field k. We investigate the structure of certain mod p universal modules for GL(3,F) over the corresponding Hecke algebras. To this end, we first study the structure of some mod p universal modules for the finite group GL(n,k) as modules over the corresponding Hecke algebras. We then relate this finite case to the p-adic one by using homological coefficient systems on the the affine Bruhat-Tits building of GL(3). Suppose now that k has cardinality p. We prove that the mod p universal module of GL(3,F) relative to the Iwahori subroup is flat and projective over the Iwahori-Hecke algebra. When replacing the Iwahori subgroup of GL(3,F) by its pro-p-radical, we prove that the corresponding module is flat over the pro-p Iwahori-Hecke algebra if and only if p=2.
We give a complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact nonarchimedean local field. More precisely, for each inertial class in the Bernstein spectrum, we construct a type and compute its Hecke algebra. The Hecke algebras that arise are all naturally isomorphic to products of affine Hecke algebras of type A. We also prove that, for cuspidal classes, the simple type is unique up to conjugacy.
Let F be a locally compact nonarchimedean local field. In this article, we extend to any inner form of GL(n) over F the notion of endo-class introduced by Bushnell and Henniart for GL(n,F). We investigate the intertwining relations of simple characters of these groups, in particular their preservation properties under transfer. This allows us to associate to any discrete series representation of an inner form of GL(n,F) an endo-class over F. We conjecture that this endo-class is invariant under the local Jacquet-Langlands correspondence.