PUBLICATIONS

(Les liens renvoient généralement sur le site arXiv.org vers une version préliminaire et éventuellement incomplète de l'article. Une copie de la version publiée peut être obtenue sur simple demande auprès de l'auteur.)

- (avec M. De Martino et E. Opdam) On the unramified spherical automorphic spectrum - arXiv:1512.08566v2, 24 pages, 2017.

Abstract: For a split connected reductive group G defined over a number field F, we compute the part of the spherical automorphic spectrum which is supported by the cuspidal data containing (T,1), where T is a maximal split torus and 1 is the trivial automorphic character. The proof uses the residue distributions which were introduced by the third author (in joint work with G, Heckman) in the study of graded affine Hecke algebras, and a result by M. Reeder on the weight spaces of the (anti)spherical discrete series representations of affine Hecke algebras. Note that both these ingredients are of a purely local nature. For many special cases of reductive groups G similar results have been established by various authors. The main feature of the present proof is the fact that it is uniform and general. [Warning: A substantial error has been found in the contour shift algorithm at the end – we are working on reparing it.]

- (avec D. Ciubotaru) On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras - - arXiv:1710.06223v3, 21 pages, 2018, to appear in Israel J. Math.

Abstract : Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a (quasi-split) classical p-adic group of a given type. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from π⊗σ does not depend on σ, if the supercuspidal support of π is "separated" from σ. (Here, "separated" means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ⊗σmis irreducible.) This was conjectured by E. Lapid and M. Tadi\'c. (In addition, they proved that this induced representation is always reducible if the supercuspidal support is not "separated".) More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category C(I;σ) nof smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type A, B and D and establish functoriality properties, relating categories with disjoint I's. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato's exotic geometry.

- On the value on a spherical vector of a standard intertwining operator relative to an everywhere unramified automorphic character - - arXiv:1709.09107v2, 11 pages, 2018, to appear in Proc. AMS.

Abstract : Let F be a global field, G an unramified quasi-split reductive group over F and chi an everywhere unramified automorphic character of a maximal maximally split torus of G. Using Langlands-Shahidi theory, we compute the meromorphic function defined by the action of a global standard intertwining operator associated to chi on a spherical vector and show that the ratio of its poles in the positive Weyl chamber is well behaved.

- Local Langlands Correspondence for Classical Groups and Affine Hecke Algebras – Mathematische Zeitschrift, Vol. 287, pp. 1029-1052, 2017.

Abstract: Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation withrepresentations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a split $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context. We get also parameterizations of representations of affine Hecke algebras, which seem not all to be in the literature yet.

- (avec Y. Kim) On the generic local Langlands correspondence for GSpin groups – Transactions AMS, Vol. 369, pp. 4275-4291, 2017.

Abstract: In the case of split GSpin groups, we prove an equality of L-functions between automorphic local L-functions defined by the Langlands-Shahidi method and local Artin L-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of Levi subgroups of GSpin, by constructing Langlands parameters for general generic irreducible admissible representations of GSpin from the one for generic irreducible supercuspidal representations of its Levi subgroups.

- A note on Standard modules and Vogan L-packets – Manuscripta Mathematica, Vol. 216, pp- 571-583, 2016.

Abstract: Let F be a non-Archimedean local field of characteristic 0, let G be the group of F-rational points of a connected reductive group defined over F and let G' be the group of F-rational points of its quasi-split inner form. Given standard Modules I(τ,ν) and I(τ',ν') for G and G' respectively with τ' a generic tempered representation representation, such that the Harish-Chandra's μ-functions of a representation in the supercuspidal support of τ and of a generic essentially square-integrable representation in some Jacquet module of τ' agree after a suitable identification of the underlying spaces under which ν=ν', we show that I(τ,ν) is irreducible whenever I(τ',ν') is. The conditions are satisfied if the Langlands quotients J(τ,ν) and J(τ',ν') of respectively I(τ,ν) and I(τ',ν') lie in the same Vogan L-packet (whenever this Vogan L-packet is defined), proving that, for any Vogan L-packet , all the standard modules whose Langlands quotient is equal to a member of the Vogan L-packet are irreducible, if and only if this Vogan L-packet contains a generic representation. This result for generic Vogan L-packets was proven for quasi-split orthogonal and symplectic groups by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these groups.

- (avec E. Opdam) On the tempered L-function conjecture - American Journal of Mathematics, Vol. 135, pp. 777-800, 2013.

Abstract: We give a general proof of Shahidi's tempered L-function conjecture, which has previously been known in all but one case. One of the consequences is the standard module conjecture for p-adic groups, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data generic. We have also included the result that every generic tempered representation of a p-adic group is a sub-representation of a representation parabolically induced from a generic supercuspidal representation with a non-negative real central character.

- Algèbres de Hecke avec paramètres et représentations d'un groupe p-adique classique : préservation du spectre tempéré – Journal of Algebra, Vol. 371, pp. 596-608, 2012. (published online)

Abstract: Let G be the identity component of an orthogonal or a symplectic p-adic group (not necessarily split) or an inner form of a general linear p-adic group. In a previous paper, it was shown that the Bernstein components of the category of smooth representations of G are equivalent to the category of right modules over some Hecke algebra with parameters, or more general over the semi-direct product of such an algebra with a finite group algebra. The aim of the present paper is to show that this equivalence preserves the tempered spectrum and the discrete series representations.

- Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif p-adique - le cas des groupes classiques - Selecta Mathematica, New Series, Vol. 17, Number 3, pp. 713-756, 2011.

Abstract: For G a symplectic or orthogonal p-adic group (not necessarily split), or an inner form of a general linear p-adic group, we compute the endomorphism algebras of some induced projective generators \`a la Bernstein of the category of smooth representations of G and show that these algebras are isomorphic to the semi-direct product of a Hecke algebra with parameters by a finite group algebra. Our strategy and parts of our intermediate results apply to a general reductive connected p-adic group.

- Paramètres de Langlands et Algèbres d'entrelacement, Int. Math. Res. Notices, Vol. 2010 (9), pp. 1607—1623, 2010.

Abstract: Let G be a classical p-adic group and (ψ,ε) the Langlands parameter of an irreducible supercuspidal representation of a Levi subgroup of G. Using data from (ψ,ε), we determine explicitly the intertwining algebra of the representation which is induced from the orbit of the supercuspidal representation associated to (ψ,ε).

- (avec G. Muic) Standard Modules Conjecture , Math. Zeitschr., 255, pp. 847--853, 2007.

Abstract: Let $G$ be a quasi-split $p$-adic group. Under the assumption that the local coefficients Cψ defined with respect to ψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.

- Unipotent Orbits and Local L-functions , J. reine angew. Math. 596, p. 103--114, 2006.

Abstract:In a previous article (Orbites unipotentes et pôles d'ordre maximal de la fonction μ de Harish-Chandra, to appear in Canad. J. Math.), we have assumed the existence of the local Langlands correspondence for supercuspidal representations and deduced from this a local Langlands correspondence for discrete series representations and beyond (without going into the structure of the L-packets). The aim of the present article is to show that this extension of the local Langlands correspondence for supercuspidal representations (and some of the assumptions in the article above) is compatible with the theory of L-functions due to Langlands-Shahidi.

- Une remarque sur le degré formel d'une série discrète d'un groupe linéaire général $p$-adique , Bull. Soc. Math. France, 134, pp. 165--171, 2006.

Abstract:We show in the simple case of the general linear group, how one can get from the previous work Décomposition spectrale et représentations spéciales d'un groupe réductif p-adique precise information on the formal degree of a square integrable representation of a $p$-adic group.

- Orbites unipotentes et pôles d'ordre maximal de la fonction µ de Harish-Chandra , Canad. J. Math., 58, p. 1203--1228, 2006.

Abstract:In a previous work, we have shown that a representation of a p-adic group obtained by (normalized) parabolic induction from an irreducible supercuspidal representation σ of a Levi subgroup M contains a subquotient which is square integrable, if and only if Harish-Chandra's μ-function has a pole in σ of order equal to the parabolic rank of M. The aim of the present article is to interpret this result in terms of Langlands' functoriality principle.

- Décomposition spectrale et représentations spéciales d'un groupe réductif p-adique , J. Inst. Math. Jussieu 3, p. 327--395, 2004.

Abstract:Let G be a reductiv connected p-adic group. With help of the Fourier inversion formula used in [ Une formule de Plancherel pour l'algèbre de Hecke d'un groupe réductif p-adique – V. Heiermann, Comm. Math. Helv. 76 , 388-415, 2001 ] we give a spectral decomposition on G. In particular we deduce from it essentially that a cuspidal representation of a Levi subgroup M is in the cuspidal support of a square integrable representation of G, if and only if it is a pole of Harish-Chandra's μ-function of order equal to the parabolic rank of M. These poles are of maximal order. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra's μ-function which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coefficients. These identities contain informations on the formel degree of square integrable representations and on their position in the induced representation.

- Une formule de Plancherel pour les éléments de l'algèbre de Hecke d'un groupe réductif p-adique , Comm. Math. Helv. 76, 388-415, 2001.

Résumé: Nous montrons un théorème de Paley-Wiener matriciel pour l'algèbre de Hecke d'un groupe réductif $p$-adique. La preuve est basée sur une analogue de la formule de Plancherel.

- Sur l’espace des représentations irréductibles continues complexes du groupe de Galois d’un corps local - C.R.A.S., t. 323, p. 571-576, 1996.

Résumé: Soient F un corps local non-archim\'edien, et W_{F}' son groupe de Weil- Deligne. Nous prouvons que l'application (σ, τ)→ sw (σ x τ˘)/(deg(σ) deg(τ)) est le logarithme d'une distance ultramétrique définie sur l'ensemble des classes d'équivalence de représentations indécomposables continues $\Phi $-semi- simples de W_{F}', modulo torsion par un quasi-caractère non ramifié. Ce résultat avait été conjecturé par E.-W. Zink. On montre en outre, sous une petite hypothèse sur le corps résiduel de F, qu'une représentation irréductible σ de W_F' est déterminée par l'ensemble des facteurs ε de la forme ε(σ x τ) avec deg (τ )≤ deg (σ), à équivalence près.

- De nouveaux invariants numériques pour les extensions totalement ramifiées de corps locaux - Journal of Number Theory, vol. 59, no. 1, p. 159-202, 1996.

Résumé: Nous étudions la catégorie des extensions totalement ramifiées d'un corps local à l'aide d'un certain type d'équations, definissant de telles extensions. Ce type d'équations est défini par des series formelles, et non pas par des polynômes d'Eisenstein. A cette occasion, nous ameliorons des resultats anciens de Arf et de Krasner. En particulier, nous donnons de nouveaux invariants numeriques pour les extensions totalement ramifiees d'un corps local qui ameliorent l'information donneée par la fonction de Herbrand.

- De nouveaux invariants numériques pour les extensions totalement ramifiées de corps locaux - C.R.A.S., t. 318, p. 989-993, 1994.

Résumé: Nous donnons une description de la catégorie des extensions totalement ramifiées d'un corps local à partir de séries formelles. A cette occasion, nous trouvons des invariants simples qui améliorent l'information donnée par la fonction de Herbrand.