Vincent Sécherre

Représentations lisses de GL( m , D ) II : β-extensions

This work is concerned with type theory for reductive groups over a non Archimedean field. Given such a field F , and a division algebra D of finite dimension over its center F , we obtain results concerning the construction of simple types for the group GL( m , D ),

Représentations lisses modulo l de GL(m,D)

m\geqslant1

Proof of the Tadić conjecture (U0) on the unitary dual of GL m (D)

. More precisely, for each simple stratum of the matrix algebra M( m , D ), we produce a set of β-extensions in the sense of Bushnell and Kutzko.

Types et contragrédientes

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 Vigneras. 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.

Smooth Representations of GLm(D) VI: Semisimple Types

Abstract Let F be a non-Archimedean local field of characteristic 0, and let D be a finite-dimensional central division algebra over F. We prove that any unitary irreducible smooth representation of a Levi subgroup of GL m (D), with m ≧ 1, induces irreducibly to GL m (D). This ends the classification of the unitary dual of GL m (D) initiated by Tadić.

Représentations lisses de GLm(D), III : types simples

Let G be a p-adic reductive group, and R an algebraically closed field. Let us consider a smooth representation of G on an R-vector space V. Fix an open compact subgroup K of G and a smooth irreducible representation of K on a finite-dimensional R-vector space W. The space of K-homomorphisms from W to V is a right module over the intertwining algebra H(G,K,W). We examine how those constructions behave when we pass to the contragredient representations of V and W, and we give conditions under which the behaviour is the same as in the case of complex representations. We take an abstract viewpoint and use only general properties of G. In the last section, we apply this to the theory of types for the group GL(n) and its inner forms over a non-Archimedean local field.