Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

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Hlavní autoři: Cunningham, Clifton, Fiori, Andrew, Moussaoui, Ahmed, Mracek, James, Xu, Bin
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, RI American Mathematical Society 2022
Vydání:1
Témata:
Number theory
Number theory > Discontinuous groups and automorphic forms > Representation-theoretic methods; automorphic representations over local and global fields. msc
p-adic groups
Partial differential equations > General topics > Microlocal methods; methods of sheaf theory and homological algebra in PDE. msc
Representations of groups
Several complex variables and analytic spaces > Singularities > Deformations of singularities; vanishing cycles. msc
Sheaf theory
Topological groups, Lie groups > Lie groups > Representations of Lie and linear algebraic groups over local fields. msc
ISBN:1470451174, 9781470451172
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Obsah:
  • 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
  • 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