Correspondance de Jacquet-Langlands pour les corps locaux de caract\'eristique non nulle

We prove the Jacquet-Langlands local correspondence in non-zero characteristic, using the proof by Deligne, Kazhdan and Vign\'eras for the zero characteristic case and the theory of closed fields of Kazhdan. We also prove the orthogonality relations for an inner form of $GL_n$ in non-zero characteristic.Comment: to appear in Annales Scientifiques de l'\'Ecole Normale Sup\'erieur

Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations

In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).Comment: 49 pages; Appendix by N. Grba

Un résultat de transfert et un résultat d'intégrabilité locale des caractères en caractéristique non nulle

International audienc

Legislative Documents

Also, variously referred to as: House bills; House documents; House legislative documents; legislative documents; General Court documents

On pp-adic Speh representations

International audienceThis note contains simple proofs of some known results (unitarity, character formula) on Speh representations of a group GL n (D) where D is a local non Archimedean division algebra of any characteristic. Introduction. In this note I give simple proofs of some known results on Speh representations of groups GL n (D) where D is a central division algebra of finite dimension over a local non archimedean field F of any characteristic. The new idea is to use the Moeglin-Waldspurger algorithm (MWA) for computing the dual of an irreducible representation. For the unitarizability of Speh representations , previous proofs were based either on the trace formula and the close fields theory, or on deep results of type theory. The proof here is " combinatoric " , independent of D, the characteristic , and Bernstein's (also called U0) theorem. Short general proofs using MWA are also given for other known facts. The proof is always the same: one wants to prove a relation (R) involving some representations (for example an induced representation π is irreducible). One starts by writing the " naive " relation (R') between these representations, known from standard theory, but not as strong as (R) (for example the semi simplification of π is a sum with non negative coefficients of some irreducible representations). Usually (R') has more terms that (R), because it is weaker, and one wants to prove that some terms which are supposed to appear in (R') are actually not there (for example all the subquotients of π except of the expected one have coefficient zero). The method then is to consider also the dual relation (R ") to (R'), and to play with the MWA, in order to show that the mild constraints one has on (R') and (R ") are enough to show that the extra terms are not there and (R') reduces actually to (R). All the results in this paper are already known, and here I only give new short proofs. So I kindly ask the reader, when using one of these facts, to quote, at least in first place, the original reference (see the historical notice at the end of the paper). Beside the Zelevinsky and Tadi´cTadi´c classification of the admissible dual ([Ze], [Ta2]), the proofs here rely on [Au] (dual of an irreducible representation is irreducible), [MW] and [BR2] (algorithm for computing the dual) and some easy tricks from [Ta1], [Ba3] and [CR], and do not involve any complicated technique. The idea of searching for " simple proof " for classification of unitary representations, together with a list of basic tricks to use, are due to Marko Tadi´cTadi´c ([Ta3] for example). He also was the first to formulate some properties of Speh representations (formula for ends of complimentary series, character formula, Speh representations are prime elements in the ring of representations, dual of a Speh representation is Speh) and to prove them when D = F. The starting point of my proof here of the assertion Speh representations are unitary is also due to Tadi´cTadi´c who found the simple but brilliant trick reducing the problem of unitarity to a problem of irreducibility. I would like to thank Guy Henniart who read the paper and made useful observations

Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle

International audienc

Correspondance entre GLn et ses formes intérieures en caractéristique positive

Let F be a local non archimedian field of characteristic p > 0, and let D be a central division algebra of finite rank d2 over F . Let r be a positive number and put n = rd. Then we prove that there is a Jacquet-Langlands cor­ respondence between the set of classes of irreducible essentialy square-integrable representations of GLn(F) and the set of classes of irreducible essentialy square- integrable representations of GLr(D). This correspondence leads to an isomor­ phism between the Grothendieck group of GLr(D) and a natural factor of the Grothendieck group of GLn(F), and furthermore to an isomorphism between the Hopf algebra associated á la Zelevinski with GLr(D) and a natural factor of the Hopf algebra associated by Zelevinski with GLn(F). We also prove a transfer of integral orbitals between GLn(F) and GLr(D). Consequences of these facts are the local integrability of characters, the ortho­gonality relations for square integrable representations and the irreducibility of any representation induced from a square-integrable irreducible one on GLr(D). If L is now a global field and A is a central division algebra of finite rank d2 over L, if r is a positif integer, then we also prove the finitude of automorphic cuspidal representations of GLr(A) with fixed components for almost every place.Soit F un corps local non archimédien de caractéristique p > 0, et soit D une algèbre centrale de division de rang fini d2 sur F . Soit r un nombre positif et mettons n = rd. Nous prouvons alors qu'il existe une correspondance de Jacquet-Langlands entre l'ensemble des classes de représentations irréductibles, essentiellement carrées et intégrables de GLn(F) et l'ensemble des classes de représentations irréductibles, essentiellement carrées et intégrables de GLr(D). Cette correspondance conduit à un isomorphisme entre le groupe de Grothendieck de GLr(D) et un facteur naturel du groupe de Grothendieck de GLn(F), et de plus à un isomorphisme entre l'algèbre de Hopf associée à la Zelevinski à GLr(D) et un facteur naturel de l'algèbre de Hopf associée par Zelevinski à GLn(F). Nous prouvons également un transfert d'orbitales intégrales entre GLn(F) et GLr(D).Les conséquences de ces faits sont l'intégrabilité locale des caractères, les relations d'orthogonalité pour les représentations intégrables au carré et l'irréductibilité de toute représentation induite à partir d'une représentation irréductible intégrable au carré sur GLr(D).Si L est maintenant un corps global et A une algèbre centrale de division de rang fini d2 sur L, si r est un entier positif, alors nous prouvons aussi la finitude des représentations cuspidales automorphes de GLr(A) avec des composantes fixes pour presque toutes les places