LEADER 08595nam 22006973 450 001 9910962220603321 005 20231110212713.0 010 $a9781470470197 010 $a1470470195 035 $a(MiAaPQ)EBC6939722 035 $a(Au-PeEL)EBL6939722 035 $a(CKB)21420567500041 035 $a(OCoLC)1309012471 035 $a(EXLCZ)9921420567500041 100 $a20220327d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aArthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples /$fClifton L.R. Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, Bin Xu 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (232 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.276 311 08$aPrint version: Cunningham, Clifton Arthur Packets for Providence : American Mathematical Society,c2022 9781470451172 327 $aCover -- 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. 327 $a6.8. 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. 327 $a15.1. 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. 330 $a"In this article we propose a geometric description of Arthur packets for padic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, including the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $ap-adic groups 606 $aRepresentations of groups 606 $aSheaf theory 606 $aNumber theory 606 $aNumber theory -- Discontinuous groups and automorphic forms -- Representation-theoretic methods; automorphic representations over local and global fields$2msc 606 $aTopological groups, Lie groups -- Lie groups -- Representations of Lie and linear algebraic groups over local fields$2msc 606 $aPartial differential equations -- General topics -- Microlocal methods; methods of sheaf theory and homological algebra in PDE$2msc 606 $aSeveral complex variables and analytic spaces -- Singularities -- Deformations of singularities; vanishing cycles$2msc 615 0$ap-adic groups. 615 0$aRepresentations of groups. 615 0$aSheaf theory. 615 0$aNumber theory. 615 7$aNumber theory -- Discontinuous groups and automorphic forms -- Representation-theoretic methods; automorphic representations over local and global fields. 615 7$aTopological groups, Lie groups -- Lie groups -- Representations of Lie and linear algebraic groups over local fields. 615 7$aPartial differential equations -- General topics -- Microlocal methods; methods of sheaf theory and homological algebra in PDE. 615 7$aSeveral complex variables and analytic spaces -- Singularities -- Deformations of singularities; vanishing cycles. 676 $a512/.22 676 $a512.22 686 $a11F70$a22E50$a35A27$a32S30$2msc 700 $aCunningham$b Clifton$f1966-$01799994 701 $aFiori$b Andrew$01799995 701 $aMoussaoui$b Ahmed$01799996 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910962220603321 996 $aArthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples$94344436 997 $aUNINA