Around Langlands Correspondences.

Cover -- Title page -- Contents -- Preface -- Change of weight theorem for pro- -Iwahori Hecke algebras -- 1. Introduction -- 2. Change of weight theorem -- 3. Description of "compact induction" for pro- -Iwahori Hecke algebra -- 4. Change of weight theorem for ℋ -- 5. Relation with Satake transform -- References -- Conjectures about -adic groups and their noncommutative geometry -- Introduction -- 1. The local Langlands correspondence -- 2. The smooth dual of a reductive -adic group -- 3. Reduction to the supercuspidal case -- 4. Topological K-theory -- References -- Introduction to the Rapid Decay property -- Overview of the situation -- 1. Short historical survey and applications -- 2. Basic definitions -- 3. The case =\Z -- 4. Equivalent definitions of property RD -- References -- A second adjoint theorem for \SL(2,ℝ) -- 1. Introduction -- 2. Categories of \SF-representations -- 3. Tempered representations -- 4. Parabolic induction and restriction -- 5. Frobenius reciprocity -- 6. The second adjoint theorem -- 7. Proof of the second adjoint theorem -- 8. Fourier transforms and intertwiners -- References -- A functoriality principle for blocks of -adic linear groups -- 1. Main statements -- 1.1. Functoriality for \oQl-blocks of groups of \GL-type -- 1.2. Functoriality for \oZl-blocks of groups of \GL-type -- 1.3. More general groups -- 2. Details and proofs -- 2.1. The centralizer and its dual -groups -- 2.2. Unipotent factorizations of a _{ }-parameter -- 2.3. Restriction of scalars -- 2.4. Groups of \GL-type -- References -- Poids de Serre dans la conjecture de Breuil-Mézard -- Introduction -- 0. Notations -- 1. Conjecture de Breuil-Mézard -- 1.1. Côté galoisien -- 1.2. Côté automorphe -- 1.3. Énoncés, interprétations et cas connus -- 1.4. Méthode de calcul des multiplicités intrinsèques.

2. Poids de Serre d'une représentation irréductible de dimension 2 -- 2.1. Rappels : congruences définissant \D(\rhobar) -- 2.2. Explicitation des formules génériques -- 2.3. Formules non génériques et poids de Serre modifiés -- 2.4. Multiplicité combinatoire -- 3. Poids de Serre d'un type modéré -- 4. Anneaux de déformations, variétés de Kisin et poids modifiés : exemples -- 4.1. En degré =2 -- 4.2. En degré =3 -- Bibliographie -- Affinoids in Lubin-Tate surfaces with exponential full level two -- Introduction -- 1. Lubin-Tate space and its formal model -- 2. Affinoid in Lubin-Tate space -- 3. Group action -- 4. On middle cohomology of the surface -- Acknowledgment -- References -- An automorphic variant of a conjecture of Deligne -- Introduction -- Basic notation -- 1. Motives and the Deligne conjecture -- 2. The automorphic variant -- References -- Paquets d'Arthur des groupes classiques complexes -- 1. Introduction -- 2. Notations et généralités sur les groupes complexes et leurs représentations -- 3. Paramètres de Langlands et d'Arthur -- 4. \GL_{ } -- 5. Les groupes classiques et leurs représentations. Paquets d'Arthur -- 6. Réduction au cas unipotent de bonne parité -- 7. Description des paquets unipotents (Barbasch-Vogan) -- 8. Lemmes de réduction -- 9. Un résultat sur les exposants -- 10. Identification des paquets de Barbasch-Vogan et d'Arthur -- 11. Démonstration du théorème 6.12 -- 12. Quelques compléments -- \frenchrefname -- Proof of the Aubert-Baum-Plymen-Solleveld conjecture for split classical groups -- Introduction -- 1. Springer correspondence -- 1.1. Ordinary Springer correspondence -- 1.2. Generalized Springer correspondence -- 1.3. Generalized Springer correspondence for orthogonal groups -- 2. Relation between the local Langlands correspondence and the Bernstein decomposition -- 2.1. Local Langlands correspondence.

2.2. Stable Bernstein centre -- 2.3. Cuspidal enhanced Langlands parameter -- 2.4. Cuspidal support -- 3. Aubert-Baum-Plymen-Solleveld conjecture for split classical groups -- 3.1. Aubert-Baum-Plymen-Solleveld conjecture -- 3.2. Galois version of ABPS conjecture -- 3.3. Proof of ABPS conjecture -- References -- From crystalline to unitary representations -- Introduction -- Part 1. -adic Hodge Theory -- 1. Big rings -- 2. Classes of geometric Galois representations -- 2.1. Crystalline Galois Representations -- Part 2. ,Γ-modules -- 3. Galois representations of fields of positive characteristic -- 4. Identifying Galois groups in characteristic and 0 -- 5. Lifting from \F_{ } to \Z_{ } -- 5.1. Coefficient rings -- 5.2. Topology -- 5.3. ,Γ-modules -- 5.4. Extending coefficients -- 6. Action of the mirabolic subgroup -- 6.1. Action of the compact mirabolic subgroup -- 6.2. Action of the mirabolic subgroup -- Part 3. The treillis of a crystalline Galois representation -- 7. Construction -- 7.1. The section on a -module -- 7.2. Boundedness on a finite free module over \cE -- 7.3. The treillis on which is surjective -- 7.4. Describing the treillis through the Wach module -- Part 4. Amice transform: From power series to function spaces -- 8. Amice transform -- 8.1. Amice Transform on \Z_{ } -- 8.2. Amice Transform on \Q_{ } -- 8.3. Action of ₀ and on \cR⁺ under the Amice transform -- 8.4. Action of on \dD under the Amice transform -- 9. The open cell of the locally analytic parabolic induction -- 9.1. The action of the Borel subgroup as locally analytic parabolic induction -- 9.2. The Amice transform of \dD as locally analytic parabolic induction -- 10. Fractional non-Archimedean differentiability -- 10.1. ^{ }-functions for a natural number -- 10.2. ^{ }-functions for ∈[0,1[ -- 10.3. ^{ }-functions for a real number ≥0.

10.4. Locally polynomial functions -- 10.5. The Amice transform -- 10.6. Order as degree of differentiability -- 11. The universal unitary norm on the locally algebraic parabolic induction -- 11.1. The greatest unitary norm -- 11.2. The universal unitary norm on the open cell -- Part 5. Intertwining -- 12. The injection into \ox\dB_{\dR}⁺ as intertwining condition -- 12.1. The intertwiner -- 12.2. The universal unitary lattice of the \K[ ]-module ^{\lr}( ) -- 12.3. The universal unitary lattice of the \K[ ]-module ^{\lr}( ) -- 13. Conclusion -- 13.1. The Universal Unitary completion as dual representation -- 13.2. Properties of the Universal Unitary completion -- References -- Representations of _{ } over finite local principal ideal rings: An overview -- 1. Introduction -- 2. Historical overview -- 3. Clifford theory for \GL_{ }(\mfoᵣ) -- 4. Regular representations, even -- 5. The constructions of Hill and Takase -- 6. The construction of Krakovski, Onn and Singla -- 7. The construction of Stasinski and Stevens -- 8. Open problems -- References -- The geometry and combinatorics of Springer fibers -- 1. Introduction -- 2. Definitions and basic examples of Schubert varieties and Springer fibers -- 3. Comparing the geometry of Schubert varieties and Springer fibers -- 4. Schubert varieties and Springer fibers in representation theory -- 5. Connecting Springer fibers with Schubert varieties -- 6. Open questions -- References -- Back Cover.

This volume contains the proceedings of the international conference "Around Langlands Correspondences", held from June 17-20, 2015, at Université Paris Sud in Orsay, France. The Langlands correspondence (nowadays called the usual Langlands correspondence), conjectured by Robert Langlands in the late 1960s and early 1970s, has recently seen some new mysterious generalizations: the modular Langlands correspondence, the p-adic Langlands correspondence, and the geometric Langlands correspondence, the last of which seems to share deep connections with the Baum-Connes conjecture. The aim of this volume is to present, through a mix of research and expository articles, some of the fascinating new directions in number theory and representation theory arising from recent developments in the Langlands program. Special emphasis is placed on nonclassical versions of the conjectural Langlands correspondences, where the underlying field is no longer the complex numbers.

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