REPRÉSENTATIONS LISSES DE $\mathrm{GL}_{m}(\mathrm{D})$ IV : REPRÉSENTATIONS SUPERCUSPIDALES

Abstract

Soit F un corps commutatif localement compact non archimédien, et soit D une algèbrè a division de centre F. Nous prouvons que toute représentation irréductible supercuspidale du groupe GLm(D), de niveau non nul, est l'induite compacte d'une représentation d'un sous-groupe ouvert compact modulo le centre de GLm(D). Plus précisément, nous prouvons que de telles représentations contiennent un type simple maximal au sens de Bushnell et Kutzko.Abstract Let F be a non-Archimedean locally compact field and let D be a central F-division algebra. We prove that any positive level supercuspidal irreducible representation of the group GLm(D) is compactly induced from a representation of a compact mod centre open subgroup of GLm(D). More precisely, we prove that such representations contain a maximal simple type in the sense of Bushnell and Kutzko.Mots clés : groupe réductif p-adique ; représentation lisse supercuspidale ; type ; paire couvrante ; foncteur de Jacquet ; immeuble de Bruhat-Tits

On parabolic induction on inner forms of the general linear group over a non-archimedean local field

“…In particular, s ρ is an integer dividing the degree of D and coprime to m ([26, Theorem B.2.b] and [9, Lemma 2.4]). Alternatively, Theorem A.1 (and the information about s ρ ) is also proved in [57, §4] (which works in any characteristic as was pointed out in [4]) without the Jacquet-Langlands correspondence but using instead the theory of types developed in [21,22,[52][53][54][55][56] to reduce the problem to a question about the Hecke algebra of type A n .…”

“…However, Bernstein's method is not applicable to the case F = D since it relies crucially on the properties of the mirabolic subgroup. In the case F = D, the assertion (U0) was proved by Sécherre in [57]: 4 using the theory of types of Bushnell-Kutzko [22] for GL n (F ) and their generalization [52,[54][55][56] to GL m (D), together with some results of Barbasch and Moy [13,14] on unitarity, he was able to transfer Bernstein's result to GL m (D).…”

“…The Zelevinsky classification in the case D = F is due to Mínguez-Sécherre [42] who consider representations over fields of characteristic different from the residual characteristic of F . They use type theory which was developed by Bushnell-Kutzko in the case D = F [22] and subsequently by Sécherre and his co-authors in the general case in a series of papers ( [21,[52][53][54][55][56]). All these methods ultimately reduce the classification to the case D = F .…”

Correspondance de Jacquet–Langlands locale et congruences modulo $$\ell $$ ℓ

“…On suppose connu le langage des strates et des caractères simples [22,4]. On fixe un entier m ě 1 et on pose G " G m .…”

Local Jacquet–Langlands correspondences for simple supercuspidal representations

“…The irreducible supercuspidal representations of GL n (K) is classified in [BK] via type theory, which describes supercuspidal representations as compact inductions of representations of some open subgroups that are compact modulo center. More generally, type theory for representations of A × is developed in a series of papers [Se1], [Se2], [Se3], [SS1], [BSS] and [SS2]. So it is natural to seek a description of the LJLC via type theory.…”