Representations of Reductive Groups

Description

This volume contains the proceedings of the Conference on Representation Theory and Algebraic Geometry, held in honor of Joseph Bernstein, from June 11 16, 2017, at the Weizmann Institute of Science and The Hebrew University of Jerusalem. The topics reflect the decisive and diverse impact of Bernstein on representation theory in its broadest scope. The themes include representations of p-adic groups and Hecke algebras in all characteristics, representations of real groups and supergroups, theta correspondence, and automorphic forms. Cover Title page Contents Preface Character values and Hochschild homology 1. Introduction 2. Weightless functions and invariant distributions 2.1. The conjecture 2.2. Almost elliptic orbits 2.3. The case of 𝑃𝐺𝐿(2) 3. The compactified category of smooth modules 3.1. Definition of the compactified category 3.2. Compactified center and a spectral description of the compactified category 3.3. The spectral description of \Smb 3.4. Compactified category and filtered modules 4. Hochschild homology and character values 5. 𝑆𝐿(2) calculations 5.1. Explicit complexes for Hochschild homology 5.2. Calculation of 𝐻𝐻₀ 5.3. Cocycles for Chern character and Euler characteristic 5.4. Tate cocycle and weighted orbital integral 5.5. Proof of part (b) of Theorem 5.1 References Schwartz space of parabolic basic affine space and asymptotic Hecke algebras 1. Introduction and statement of the results 2. Tempered representations and Harish-Chandra algebra 3. Paley-Wiener theorems and the definition of the algebra \calJ(𝐺) 4. Intertwining operators 5. Intertwining operators in the cuspidal corank 1 case 6. Proof of \reft{inter-schwartz} 7. Some further questions References Explicit local Jacquet-Langlands correspondence: The non-dyadic wild case Introduction 1. The Lagrangian subgroup 2. Extensions of simple characters 3. Change of group and endo-classes 4. Transfer via completion 5. The basic character relation 6. Consequences References On the Casselman-Jacquet functor Introduction 0.1. The Casselman-Jacquet functor 0.2. Functorial interpretation of the Casselman-Jacquet functor 0.3. The pseudo-identity functor and the ULA condition 0.4. The “2nd adjointness” conjecture 0.5. Organization of the paper 0.6. Conventions and notation 0.7. How to get rid of DG categories? 0.8. Acknowledgements 1. Recollections 1.1. Groups acting on categories: a reminder 1.2. Localization theory 1.3. Translation functors 1.4. The long intertwining operator 2. Casselman-Jacquet functor as averaging 2.1. Casselman-Jacquet functor in the abstract setting 2.2. Casselman-Jacquet functor as completion 2.3. The Casselman-Jacquet functor for 𝐴-modules 2.4. The Casselman-Jacquet functor for \fg-modules 2.5. ULA vs finite-generation 3. The pseudo-identity functor 3.1. The pseudo-identity functor: recollections 3.2. Pseudo-identity, averaging and the ULA property 3.3. A variant 3.4. First applications 3.5. Transversality and the proof of \propref{p:equiv ULA} 4. The case of a symmetric pair 4.1. Adjusting the previous framework 4.2. The Casselman-Jacquet functor for (\fg,𝐾)-modules 4.3. Proof of \thmref{t:J exact on K} 4.4. The “2nd adjointness” conjecture References Periods and theta correspondence 1. Introduction 2. Shalika and Linear Periods 3. Type II Dual Pairs 4. Big Theta Lift 5. A Theorem of Tunnell and Saito References Generalized and degenerate Whittaker quotients and Fourier coefficients 1. Introduction Acknowledgments 1.1. Notation 2. Degenerate Whittaker models and Fourier coefficients 2.1. Definitions 2.2. Fourier coefficients 2.3. Comparison between different Whittaker pairs 2.4. The Slodowy slice 3. Admissible and quasi-admissible orbits 3.1. Definitions 3.2. On the proof of Theorem B 4. Wave-front sets 4.1. Definition 4.2. On the proof of Theorem A 4.3. Archimedean case References Geometric approach to the fermionic Fock space, via flag varieties and representations of algebraic (super)groups Introduction 1. Basic setting 2. Fock space 3. Duality between geometric induction and restriction 4. Two functors on ℱ 5. Link with the Clifford algebra 6. Translation functors References Representations of a 𝑝-adic group in characteristic 𝑝 I. Introduction II. Some general algebra II.1. Review on scalar extension II.2. A bit of ring and module theory II.3. Proof of the decomposition theorem (Thm.I.1 and Cor.I.2) II.4. Proof of the lattice theorems (Thm. I.3, I.5 and Cor. I.4, I.6) III. Classification theorem for 𝐺 III.1. Admissibility, 𝐾-invariants, and scalar extension III.2. Decomposition Theorem for 𝐺 III.3. The representations 𝐼_{𝐺}(𝑃,𝜎,𝑄) III.4. Supersingular representations III.5. Classification of irreducible admissible 𝑅-representations of 𝐺 IV. Classification theorem for 𝐻(𝐺) IV.1. Pro-𝑝 Iwahori Hecke ring IV.2. Parabolic induction \Ind_{𝑃}^{𝐻(𝐺)} IV.3. The 𝐻(𝐺)_{𝑅}-module \St_{𝑄}^{𝐻(𝐺)}(\cV) IV.4. The module 𝐼_{𝐻(𝐺)}(𝑃,\cV,𝑄) IV.5. Classification of simple modules over the pro-𝑝 Iwahori Hecke algebra V. Applications V.1. Vanishing of the smooth dual V.2. Lattice of submodules (Proof of Theorem I.10) V.3. Proof of Theorem I.12 VI. Appendix: Eight inductions \Mod_{𝑅}(𝐻(𝑀))→\Mod_{𝑅}(𝐻(𝐺)) References On the support of matrix coefficients of supercuspidal representations of the general linear group over a local non-archimedean field 1. Introduction 2. A variant for Whittaker functions 3. Proof of main result References On the generalized Springer correspondence Introduction Contents 1. Preliminaries 2. A trace computation 3. Computations in certain groups of semisimple rank \le5 4. Euler characteristic computations 5. The main result 6. Final comments References The modular pro-𝑝 Iwahori-Hecke 𝐸𝑥𝑡-algebra 1. Introduction 2. Notations and preliminaries 2.1. Elements of Bruhat-Tits theory 2.2. The pro-𝑝 Iwahori-Hecke algebra 2.3. Supersingularity 3. The \Ext-algebra 3.1. The definition 3.2. The technique 3.3. The cup product 4. Representing cohomological operations on resolutions 4.1. The Shapiro isomorphism 4.2. The Yoneda product 4.3. The cup product 4.4. Conjugation 4.5. The corestriction 4.6. Basic properties 5. The product in 𝐸* 5.1. A technical formula relating the Yoneda and cup products 5.2. Explicit left action of 𝐻 on the Ext-algebra 5.3. Appendix 6. An involutive anti-automorphism of the algebra 𝐸* 7. Dualities 7.1. Finite and twisted duals 7.2. Duality between 𝐸ⁱ and 𝐸^{𝑑-𝑖} when 𝐼 is a Poincaré group 8. The structure of 𝐸^{𝑑} References Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density 1. Introduction 2. The conjecture of Hiraga, Ichino and Ikeda 2.1. The decomposition of the trace 2.2. Normalization of Haar measure 2.3. Local Langlands parameters 2.4. 𝐿-functions and 𝜀 factors 2.5. A conjectural tempered local Langlands correspondence 2.6. The conjectures of Hiraga, Ichino and Ikeda 2.7. Known results and further comments 3. The Plancherel formula for affine Hecke algebras 3.1. The Bernstein center 3.2. Types, Hecke algebras and Plancherel measure 3.3. Affine Hecke algebras as Hilbert algebras 3.4. A formula for the trace of an affine Hecke algebra 3.5. Spectral decomposition of 𝜏 3.6. Residual cosets and their properties 3.7. Deformation of discrete series and the computation of 𝑑_{\Hc,𝛿} 3.8. Central characters and Langlands parameters 4. Lusztig’s representations of unipotent reduction and spectral transfer maps. Main result. 4.1. Unipotent types and unipotent affine Hecke algebras 4.2. Langlands parameters and residual cosets 4.3. Spectral transfer maps 4.4. Lusztig’s geometric-arithmetic correspondences and STMs 4.5. Main Theorem References Limiting cycles and periods of Maass forms 1. Introduction 2. Proof References On vector-valued twisted conjugation invariant functions on a group; with an appendix by Stephen Donkin 1. Introduction 2. Filtered vector spaces and Rees modules 3. Filtration on representations 4. Vector-valued twisted conjugation invariant functions 5. Chevalley groups with an automorphism 6. The determinant of the pairing \bfJ(𝑉)⊗\bfJ(𝑉*)→\bfJ References Appendix: A remark on freeness References Local theta correspondence and nilpotent invariants 1. Introduction 2. Dual pairs and local theta correspondence: a brief review 3. Correspondence of generalized Whittaker models 4. Correspondence of associated characters References Back Cover