Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples / / Clifton L.R. Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, Bin Xu

• Autore: Cunningham Clifton <1966->
• Pubblicazione: Providence : , : American Mathematical Society, , 2022; ©2022
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (232 pages)
• Disciplina: 512/.22; 512.22
• Soggetto topico: p-adic groups; Representations of groups; Sheaf theory; Number theory; Number theory -- Discontinuous groups and automorphic forms -- Representation-theoretic methods; automorphic representations over local and global fields; Topological groups, Lie groups -- Lie groups -- Representations of Lie and linear algebraic groups over local fields; Partial differential equations -- General topics -- Microlocal methods; methods of sheaf theory and homological algebra in PDE; Several complex variables and analytic spaces -- Singularities -- Deformations of singularities; vanishing cycles
• Classificazione: 11F7022E5035A2732S30
• Altri autori: FioriAndrew MoussaouiAhmed
• Nota di contenuto: Cover -- Title page -- Acknowledgments -- Chapter 1. Introduction -- 1.1. Motivation -- 1.2. Background -- 1.3. Main results -- 1.4. Conjecture -- 1.5. Examples -- 1.6. Relation to other work -- 1.7. Disclaimer -- Part 1. Arthur packets and microlocal vanishing cycles -- Chapter 2. Overview -- Chapter 3. Arthur packets and pure rational forms -- 3.1. Local Langlands group -- 3.2. L-groups -- 3.3. Semisimple, elliptic and hyperbolic elements in L-groups -- 3.4. Langlands parameters -- 3.5. Arthur parameters -- 3.6. Langlands parameters of Arthur type -- 3.7. Pure rational forms -- 3.8. Langlands packets for pure rational forms -- 3.9. Arthur packets for quasisplit symplectic or special orthogonal groups -- 3.10. Arthur packets for inner rational forms -- 3.11. Pure Arthur packets -- 3.12. Virtual representations of pure rational forms -- Chapter 4. Equivariant perverse sheaves on parameter varieties -- 4.1. Infinitesimal parameters -- 4.2. Vogan varieties -- 4.3. Parameter varieties -- 4.4. Equivariant perverse sheaves -- 4.5. Equivariant perverse sheaves on parameter varieties -- 4.6. Langlands component groups as equivariant fundamental groups -- Chapter 5. Reduction to unramified hyperbolic parameters -- 5.1. Unramification -- 5.2. Elliptic and hyperbolic parts of the image of Frobenius -- 5.3. Construction of the unramified parameter -- 5.4. Construction of the cocharacter -- 5.5. Proof of Theorem 5.1 -- 5.6. Further properties of Vogan varieties -- Chapter 6. Arthur parameters and the conormal bundle -- 6.1. Regular conormal vectors -- 6.2. Cotangent space to the Vogan variety -- 6.3. Conormal bundle to the Vogan variety -- 6.4. Orbit duality -- 6.5. Strongly regular conormal vectors -- 6.6. From Arthur parameters to strongly regular conormal vectors -- 6.7. Arthur component groups as equivariant fundamental groups. ; 6.8. Proof of Proposition 6.1 -- 6.9. Equivariant Local systems -- Chapter 7. Microlocal vanishing cycles of perverse sheaves -- 7.1. Background on vanishing cycles -- 7.2. Calculating vanishing cycles -- 7.3. Brylinski's functor Ev -- 7.4. Stalks -- 7.5. Support -- 7.6. Open orbit -- 7.7. Purity and rank on the open orbit -- 7.8. Perversity -- 7.9. Restriction to generic elements -- 7.10. Normalization of Ev -- 7.11. Remarks on stratified Morse theory and microlocalization -- Chapter 8. Arthur packets and ABV-packets -- 8.1. ABV-packets -- 8.2. Virtual representations attached to ABV-packets -- 8.3. Conjecture on Arthur packets -- 8.4. A basis for stable invariant distributions -- Part 2. Examples -- Chapter 9. Overview -- Chapter 10. Template for the examples -- 10.1. Arthur packets -- 10.2. Vanishing cycles of perverse sheaves -- 10.3. ABV-packets -- 10.4. Endoscopy and equivariant restriction of perverse sheaves -- Chapter 11. SL(2) 4-packet of quadratic unipotent representations -- 11.1. Arthur packets -- 11.2. Vanishing cycles of perverse sheaves -- 11.3. ABV-packets -- 11.4. Endoscopy and equivariant restriction of perverse sheaves -- Chapter 12. SO(3) unipotent representations, regular parameter -- 12.1. Arthur packets -- 12.2. Vanishing cycles of perverse sheaves -- 12.3. ABV-packets -- 12.4. Endoscopy and equivariant restriction of perverse sheaves -- Chapter 13. PGL(4) shallow representations -- 13.1. Arthur packets -- 13.2. Vanishing cycles of perverse sheaves -- 13.3. ABV-packets -- 13.4. Endoscopy and equivariant restriction of perverse sheaves -- Chapter 14. SO(5) unipotent representations, regular parameter -- 14.1. Arthur packets -- 14.2. Vanishing cycles of perverse sheaves -- 14.3. ABV-packets -- 14.4. Endoscopy and equivariant restriction of perverse sheaves -- Chapter 15. SO(5) unipotent representations, singular parameter. ; 15.1. Arthur packets -- 15.2. Vanishing cycles of perverse sheaves -- 15.3. ABV-packets -- 15.4. Endoscopy and equivariant restriction of perverse sheaves -- Chapter 16. SO(7) unipotent representations, singular parameter -- 16.1. Arthur packets -- 16.2. Vanishing cycles of perverse sheaves -- 16.3. ABV-packets -- 16.4. Endoscopy and equivariant restriction of perverse sheaves -- 16.5. Tables for the SO(7) example -- Bibliography -- Index -- Back Cover.
• Sommario/riassunto: "In this article we propose a geometric description of Arthur packets for padic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, including the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples"--
• Titolo autorizzato: Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples
• ISBN: 9781470470197; 1470470195
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910962220603321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society