196,068 research outputs found
Geometric structure in smooth dual and local Langlands conjecture
No full textThis expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture
Hecke algebras for inner forms of p-adic special linear groups
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On the local Langlands correspondence for non-tempered representations
No full textLet G be a reductive p-adic group. We study how a local Langlands correspondence for irreducible tempered G-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of G. We prove that, under a natural condition involving compatibility with unramified twists, this is possible in a canonical way. To this end we introduce analytic R-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of G. We establish the basic properties of these new R-groups, which generalize Knapp–Stein R-groups
The local Langlands correspondence for inner forms of SL_n
No full textLet F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group SLn(F). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for SLn(F) enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of SLn(F) up to equivalence. An analogous result is shown in the archimedean case. For p-adic fields this is based on the work of Hiraga and Saito. To settle the case where F has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of GLn(F), when the fields are close enough compared to the depth of the representations.<br/
On L-packets and depth for SL_2(K) and its inner form
No full textWe consider the group SL_2(K), where K is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of SL_2 (K) is larger than the depth of the corresponding Langlands parameter, with equality if and only if the L-parameter is essentially tame. We also work out a classification of all L-packets for SL_2 (K) and for its non-split inner form, and we provide explicit formulae for the depths of their L-parameters.<br/
The principal series of p-adic groups with disconnected centre
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Discrete series characters for affine Hecke algebras and their formal degrees
No full textWe introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types , n=6, 7, 8) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases)
Geometric structure for the principal series of a reductive p-adic group with connected centre
No full textLet G be a split reductive p-adic group with connected centre. We show that each Bernstein block in the principal series of G admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form T//W where T is a maximal torus in the Langlands dual group of G and W is the Weyl group of G
Stable Bernstein center and Aubert-Baum-Plymen-Solleveld conjecture
No full textCette thèse s'intéresse aux liens entre la correspondance de Langlands locale et le centre de Bernstein. Pour cela, un cadre a été introduit par Vogan puis développé par Haines : le centre de Bernstein stable. Nous commençons par étendre la correspondance de Springer généralisée au groupe (non connexe) orthogonal. Ensuite, nous énonçons une conjecture concernant les paramètres de Langlands (complets) des représentations supercuspidales d'un groupe p-adique déployé que nous vérifions pour les groupes classiques et le groupe linéaire à l'aide des travaux de Moeglin, Henniart et Harris et Taylor. Nous définissons à l'aide des travaux de Lusztig sur la correspondance de Springer généralisée une application de support cuspidal pour les paramètres de Langlands complets. Avec certains résultats d'Heiermann, nous obtenons un paramétrage de Langlands des représentations irréductibles d'un groupe classique. Par ailleurs, nous énonçons une conjecture « galoisienne » analogue à la conjecture d'Aubert-Baum-Plymen-Solleveld, que nous prouvons à l'aide des résultats précédents. Ceci est une nouvelle preuve de la validité de la conjecture ABPS pour les groupes classiques et explicite ses relations avec la correspondance de Langlands. En conséquence, on obtient la compatibilité de la correspondance de Langlands avec l'induction parabolique pour les groupes classiques.This thesis focus on links between the local Langlands correspondence and the Bernstein center. A framework was introduced by Vogan and developed by Haines : the stable Bernstein center. We start by extending the generalized Springer correspondence to the orthogonal group (which is disconnected). Then we state a conjecture about (complete) Langlands parameters of supercuspidal representations of a p-adic split group and we prove it for classical and linear groups thanks to the work of M\oe glin, Henniart and Harris and Taylor. Based on the work of Lusztig on generalized Springer correspondence, we define a cuspidal support map for complete Langlands parameters. Referring to some results of Heiermann, we get a Langlands parametrization of the smooth dual of classical groups. Moreover, we state "Galois" version of the Aubert-Baum-Plymen-Solleveld conjecture and we prove that with the previous results. It gives a new proof of the validity of the ABPS conjecture for classical groups and it provides explicit relations with Langlands correspondence. As a corrolary, we obtain the compatibility of the Langlands correspondence with parabolic induction for classical groups
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