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Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $\sigma$ be its non-trivial automorphism. We show that any $\sigma$-self-dual cuspidal representation of ${\rm GL}_n(F)$ contains a $\sigma$-self-dual Bushnell--Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai $L$-function of a ${\rm GL}_n(F_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands--Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation., Comment: 61 pages, final version to appear in JEMS

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $\sigma$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $\pi$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $\pi^{\sigma}\simeq\pi^{\vee}$ (such a representation is said to be $\sigma$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times}$. When $\pi$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $\pi$ to be ${\rm GL}_n(F_0)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $\omega$ the non-trivial $R$-character of $F_0^{\times}$ trivial on $F/F_0$-norms, we prove that any $\sigma$-selfdual supercuspidal $R$-representation is either distinguished or $\omega$-distinguished, but not both. In the modular case, that is when $\ell>0$, we give examples of $\sigma$-selfdual cuspidal non-supercuspidal representations which are not distinguished nor $\omega$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan-Kable-Tandon., Comment: 56 pages

Let F \mathrm {F} be a non-Archimedean locally compact field with residual characteristic p p , let G \mathrm {G} be an inner form of G L n ( F ) \mathrm {GL}_n(\mathrm {F}) , n ⩾ 1 n\geqslant 1 and let R \mathrm {R} be an algebraically closed field of characteristic different from p p . When R \mathrm {R} has characteristic ℓ > 0 \ell >0 , the image of an irreducible smooth R \mathrm {R} -representation π \pi of G \mathrm {G} by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G \mathrm {G} ) contains a unique irreducible term π ⋆ \pi ^\star with the same cuspidal support as π \pi . This defines an involution π ↦ π ⋆ \pi \mapsto \pi ^\star on the set of isomorphism classes of irreducible R \mathrm {R} -representations of G \mathrm {G} , that coincides with the Zelevinski involution when R \mathrm {R} is the field of complex numbers. The method we use also works for F \mathrm {F} a finite field of characteristic p p , in which case we get a similar result.

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., 15 pages, in French language, Canadian Journal of Mathematics, 2014

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 definebanalirreducible${\rm R}$-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${\rm G}$.

Let F be a non-Archimedean locally compact field of residue characteristic p and R be an algebraically closed field of characteristic l different from p. Let G be the group GLn, n ¥ 1, over F or one of its inner forms. In this article, we prove that the unramified irreducible smooth R-representations of G(F) are those representations that are irreducibly induced from an unramified character of a Levi subgroup. We deduce that any irreducible unramified Fl-representation of G(F) can be lifted to Ql, which proves a conjecture by Vignéras. Agence Nationale de la Recherche Engineering and Physical Sciences Research Council Ministerio de Ciencia e Innovación Fondo Europeo de Desarrollo Regional

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.

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.

ECHERRE Abstract. Soit F un corps commutatif localement compact non archimedien, et soit G une forme interieure de GLn(F). Nous prouvons la conjecture U1 de Tadic dans le cas ou F est de caracteristique non nulle. Ce resultat implique que la description du dual unitaire de G et le transfert de Jacquet- Langlands des representations unitaires de G, tous deux connus pour F de caracteristique nulle, sont vrais en caracteristique quelconque.

Let $\F$ be a non-Archimedean locally compact field, $q$ be the cardinality of its residue field, and $\R$ be an algebraically closed field of characteristic $\ell$ not dividing $q$.We classify all irredu\-cible smooth $\R$-representations of $\GL\_n(\F)$ having a nonzero $\GL\_{n-1}(\F)$-inva\-riant linear form, when $q$ is not congruent to $1$ mod $\ell$.Partial results in the case when $q$ is $1$ mod $\ell$ show that, unlike the complex case, the space of $\GL\_{n-1}(\F)$-invariant linear forms has dimension $2$ for certain irreducible representations.

Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).

We prove that the affine Yokonuma-Hecke algebra defined by Chlouveraki and Poulain d'Andecy is a particular case of the pro-$p$-Iwahori-Hecke algebra defined by Vign\'eras.

Resume Soit F un corps local non archimedien, soit D une F-algebre a division et soit G le groupe GL m ( D ) , avec m ⩾ 1 . Dans cet article, nous poursuivons le travail entrepris dans [V. Secherre, Representations lisses de GL ( m , D ) , I : caracteres simples, Bull. Soc. Math. France 132 (3) (2004) 327–396] et [V. Secherre, Representations lisses de GL ( m , D ) , II : β-extensions, Compositio Math. 141 (2005) 1531–1550] visant a generaliser les resultats dus a Bushnell et Kutzko [C.J. Bushnell, P.C. Kutzko, The Admissible Dual of GL ( N ) via Compact Open Subgroups, Princeton University Press, Princeton, NJ, 1993] concernant le groupe deploye GL n ( F ) . Pour tout entier r ⩾ 1 divisant m, et pour certaines representations lisses irreductibles supercuspidales Π 0 de G 0 = GL m / r ( D ) , nous construisons un type pour la classe inertielle [ G 0 r , Π 0 ⊗ r ] G . Nous determinons la structure de son algebre de Hecke qui, comme dans le cas deploye, est une algebre de Hecke affine de type A r − 1 .

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), . 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.

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 Vign\'eras. 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., Comment: in French

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., 37 pages

For a nonarchimedean local field F, let GL(n):= GL(n, F) and GL(n−1) be embedded in GL(n) via g ↦ ( 0 1 0 ). Let π be an irreducible admissible representation of GL(n) for n ≥ 3. We prove that π is GL(n − 1)-distinguished if and only if the Langlands parameter L(π) associated to π by the Local Langlands Correspondence has a subrepresentation L(11 n−2) of dimension n−2 corresponding to the trivial representation of GL(n−2) such that the two-dimensional quotient L(π)/L(11 n−2) corresponds either to an infinite-dimensional representation or the one-dimensional representations $\nu ^{ \pm (\tfrac{{n - 2}}{2})} $ of GL(2). We also prove that, for a parabolic subgroup P of GL(n) and an irreducible admissible representation ρ of the Levi subgroup of P, $\dim _\mathbb{C} (Hom_{GL(n - 1)} [ind_P^{GL(n)} (\rho ),\mathbb{I}_{n - 1} ]) \leqslant 2$ . For the standard Borel subgroup B n of GL(n) and characters x i of GL(1), we classify all representations ξ of the form $ind_{B_n }^{GL(n)} (\chi _1 \otimes \cdots \otimes \chi _n )$ for which $\dim _\mathbb{C} (Hom_{GL(n - 1)} [\xi ,\mathbb{I}_{n - 1} ]) = 2$ .

Let k be a nonarchimedean locally compact field of residue characteristic p, let G be a connected reductive group defined over k, let σ be an involutive k-automorphism of G, and H an open k-subgroup of the fixed points group of σ. We denote by G k and H k the groups of k-points of G and H. We obtain an analogue of the Cartan decomposition for the reductive symmetric space H k \Gk in the case where G is k-split and p is odd. More precisely, we obtain a decomposition of G k as a union of (H k , K)-double cosets, where K is the stabilizer of a special point in the Bruhat-Tits building of G over k. This decomposition is related to the H k -conjugacy classes of maximal σ-antiinvariant k-split tori in G. In a more general context, Benoist and Oh obtained a polar decomposition for any p-adic reductive symmetric space. In the case where G is k-split and p is odd, our decomposition makes more precise that of Benoist and Oh, and generalizes results of Offen for GL n .

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 representation of a Levi subgroup of GL(m,D), with m a positive integer, induces irreducibly to GL(m,D). This ends the classification of the unitary dual of GL(m,D) initiated by Tadic.