We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let $\text{F}$ be a non-Archimedean locally compact field of residue characteristic $p$ , and let $\text{G}$ be an inner form of the general linear group $\text{GL}_{n}(\text{F})$ for $n\geqslant 1$ . We consider the problem of describing explicitly the local Jacquet–Langlands correspondence $\unicode[STIX]{x1D70B}\mapsto _{\text{JL}}\unicode[STIX]{x1D70B}$ between the complex discrete series representations of $\text{G}$ and $\text{GL}_{n}(\text{F})$ , in terms of type theory. We show that the congruence properties of the local Jacquet–Langlands correspondence exhibited by A. Mínguez and the first author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet–Langlands correspondence can be reduced to the case where the representations $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal with torsion number $1$ . We also give an explicit description of the Jacquet–Langlands correspondence for all essentially tame discrete series representations of $\text{G}$ , up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where  $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal.

Soit $\text{G}$ un groupe réductif $p$ -adique, et soit $\text{R}$ un corps algébriquement clos. Soit $\text{ }\!\!\pi\!\!\text{ }$ une représentation lisse de $\text{G}$ dans un espace vectoriel $\text{V}$ sur $\text{R}$ . Fixons un sous-groupe ouvert et compact $\text{K}$ de $\text{G}$ et une représentation lisse irréductible $\varrho$ de $\text{K}$ dans un espace vectoriel $\text{W}$ de dimension finie sur $\text{R}$ . Sur l'espace $\text{Ho}{{\text{m}}_{\text{K}}}(\text{W,}\,\text{V)}$ agit l'algèbre d'entrelacement $\mathcal{H}(\text{G,}\,\text{K,}\,\text{W)}$ . Nous examinons la compatibilité de ces constructions avec le passage aux représentations contragrédientes ${{\text{V}}^{\vee }}$ et ${{\text{W}}^{\vee }}$ , et donnons en particulier des conditions sur $\text{W}$ ou sur la caractéristique de $\text{R}$ pour que le comportement soit semblable au cas des représentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriétés générales de $\text{G}$ . Nous terminons par une application á la théorie des types pour le groupe $\text{G}{{\text{L}}_{n}}$ et ses formes intérieures sur un corps local non archimédien.

Let ${\rm F}$ be a non-Archimedean locally compact field of residue characteristic $p$ , let ${\rm D}$ be a finite-dimensional central division ${\rm F}$ -algebra and let ${\rm R}$ be an algebraically closed field of characteristic different from $p$ . We define banal irreducible ${\rm R}$ -representations of the group ${\rm G}={\rm GL}_{m}({\rm D})$ . This notion involves a condition on the cuspidal support of the representation depending on the characteristic of ${\rm R}$ . When this characteristic is banal with respect to ${\rm G}$ , in particular when ${\rm R}$ is the field of complex numbers, any irreducible ${\rm R}$ -representation of ${\rm G}$ is banal. In this article, we give a classification of all banal irreducible ${\rm R}$ -representations of ${\rm G}$ in terms of certain multisegments, called banal. When ${\rm R}$ is the field of complex numbers, our method provides a new proof, entirely local, of Tadić’s classification of irreducible complex smooth representations of ${\rm G}$ .

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), $m\geqslant1$. 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.