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Modular Springer Correspondence and Decomposition Matrices Daniel Juteau Modular Springer correspondence and decomposition matrices Daniel Juteau To cite this version: Daniel Juteau. Modular Springer correspondence and decomposition matrices. Representation Theory [math.RT]. Université Paris-Diderot - Paris VII, 2007. English. tel-00355559 HAL Id: tel-00355559 https://tel.archives-ouvertes.fr/tel-00355559 Submitted on 23 Jan 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UFR de math´ematiques Ecole´ Doctorale de Math´ematiques de Paris centre Universit´eParis 7 – Denis Diderot These` de Doctorat pr´esent´ee par Daniel Juteau pour obtenir le grade de Docteur de Math´ematiques de l’Universit´eParis 7 – Denis Diderot Correspondance de Springer modulaire et matrices de d´ecomposition — Modular Springer correspondence and decomposition matrices Th`ese soutenue le 11 d´ecembre 2007, devant le jury compos´e de : C´edric Bonnafe´ (co-directeur) Charg´ede Recherche `al’Universit´ede Franche-Comt´e Michel Brion Directeur de Recherche `al’Universit´ede Grenoble 1 Michel Broue´ Professeur `al’Universit´eParis 7 Bernard Leclerc Professeur `al’Universit´ede Caen Jean Michel Directeur de Recherche `al’Universit´eParis 7 Rapha¨el Rouquier (co-directeur) Directeur de Recherche `al’Universit´eParis 7 Professeur `al’Universit´ed’Oxford Wolfgang Soergel (rapporteur) Professeur `al’Universit´eAlbert-Ludwigs de Freiburg Tonny Springer Professeur `al’Universit´ed’Utrecht Rapporteur non pr´esent `ala soutenance : Bao Chˆau Ngoˆ Professeur `al’Universit´eParis-Sud (Paris 11) Membre de l’Institute for Advanced Study `aPrinceton Table des mati`eres Remerciements 5 Introduction (fran¸cais) 7 Introduction 23 1 Perverse sheaves over K, O, F 37 1.1 Context ..................................... 37 1.2 t-categories ................................... 40 1.3 Torsion theories and t-structures ....................... 42 1.4 Recollement................................... 44 1.5 Torsion theories and recollement . 48 1.6 Complements on perverse extensions . 49 1.6.1 Top and socle of perverse extensions . 49 1.6.2 Perverse extensions and multiplicities . ..... 51 1.7 Perverse t-structures .............................. 51 1.8 Modular reduction and truncation functors . ...... 53 1.9 Modular reduction and recollement . 55 1.10 Decompositionnumbers . 56 1.11Equivariance .................................. 58 2 Examples 61 2.1 Semi-smallmorphisms . .. .. .. .. .. .. .. .. .. .. 61 2.2 Equivalentsingularities . 62 2.3 Cones ...................................... 63 2.4 E-smoothness.................................. 63 2.4.1 Definitionandremarks. 63 2.4.2 Astabilityproperty . .. .. .. .. .. .. .. .. .. 64 2.4.3 Cone over a smooth projective curve . 65 2.5 Simplesingularities. 65 2.5.1 Rationaldoublepoints. 66 2.5.2 Symmetries on rational double points . 67 2.5.3 Perverse extensions and decomposition numbers . ..... 68 1 Table des mati`eres 3 Cohomology of the minimal nilpotent orbit 73 3.1 Long roots and distinguished coset representatives . .......... 75 3.1.1 Rootsystems .............................. 75 3.1.2 Basis, positive roots, height . 76 3.1.3 Length.................................. 77 3.1.4 Highestroot .............................. 79 3.1.5 Orders.................................. 81 3.2 Resolution of singularities, Gysin sequence . ......... 85 3.2.1 Line bundles on G/B, cohomology of G/B ............. 86 3.2.2 Parabolic invariants . 87 ∗ 3.2.3 Cohomology of a C -fiber bundle on G/PI .............. 88 3.2.4 Resolution of singularities . 89 3.3 Case-by-caseanalysis. 90 3.3.1 Type An−1 ............................... 91 3.3.2 Type Bn ................................. 92 3.3.3 Type Cn ................................. 93 3.3.4 Type Dn ................................ 94 3.3.5 Type E6 ................................. 96 3.3.6 Type E7 ................................. 97 3.3.7 Type E8 ................................. 99 3.3.8 Type F4 .................................102 3.3.9 Type G2 .................................103 3.4 Another method for type A ..........................103 4 Some decomposition numbers 105 4.1 Subregularclass................................. 105 4.1.1 Case Γ = Bn ..............................107 4.1.2 Case Γ = Cn ..............................107 4.1.3 Case Γ = F4 ...............................107 4.1.4 Case Γ = G2 ..............................107 4.2 Minimalclass ..................................108 4.3 Rowsandcolumns ...............................109 4.4 Specialclasses.................................. 111 5 Fourier-Deligne transform 113 5.1 Preliminaries ..................................114 5.2 Definition, first properties and examples . 116 5.2.1 Definition ................................116 5.2.2 Firstproperties.............................117 5.2.3 Examples ................................125 5.3 Fourier-Deligne transform and duality . 128 2 Table des mati`eres 6 Springer correspondence and decomposition matrices 131 6.1 Thegeometriccontext ............................. 131 6.1.1 Notation.................................131 6.1.2 Thefinitequotientmap . .131 6.1.3 Theadjointquotient . .131 6.1.4 Springer’s resolution of the nilpotent variety . ........132 6.1.5 Grothendieck’s simultaneous resolution of the adjoint quotient . 132 6.2 Springer correspondence for EW .......................133 6.2.1 The perverse sheaves Krs, K and KN .................133 6.2.2 Springer correspondence for KW by restriction . 134 6.2.3 The Fourier-Deligne transform of EK .................135 6.2.4 Springer correspondence by Fourier-Deligne tranform........137 6.3 Decompositionmatrices . 138 6.3.1 Comparison of e maps.........................138 6.3.2 Comparison of d maps.........................138 6.4 Modular Springer correspondence for GLn ..................139 7 Tables 141 7.1 Type A1 .....................................142 7.2 Type A2 .....................................143 7.3 Type A3 .....................................144 7.4 Type B2 .....................................145 7.5 Type B3 .....................................146 7.6 Type C3 .....................................148 7.7 Type G2 .....................................150 8 Character sheaves on sl2 151 Bibliography 155 3 Table des mati`eres 4 Remerciements Tout d’abord, je suis infiniment reconnaissant `aC´edric Bonnaf´eet Rapha¨el Rouquier pour la qualit´eexceptionnelle de leur encadrement. J’ai grandement appr´eci´eleurs im- menses qualit´es humaines et math´ematiques. Je mesure la chance que j’ai eue de les avoir comme directeurs. Ils ont tous deux ´et´ed’une tr`es grande disponibilit´e, mˆeme quand la distance nous s´eparait. Je les remercie pour le sujet qu’ils m’ont donn´e, leurs conseils, leurs encouragements, leur soutien, et la libert´equ’ils m’ont accord´ee. Je pense que c’´etait parfois ´eprouvant de m’avoir comme ´etudiant, notamment quand il a fallu relire mon manuscrit jusqu’`ala derni`ere minute ! Je les remercie aussi, ainsi que leurs ´epouses Anne et Meredith, pour les moments de convivialit´e que nous avons partag´es. Je remercie tr`es sinc`erement les deux rapporteurs, Bao Chˆau Ngˆoet Wolfgang Soergel, pour le temps qu’ils ont pass´e`alire ce travail et `a´ecrire les rapports. Plus particuli`e- rement, je remercie Bao Chˆau Ngˆopour avoir sugg´er´ed’´etudier les faisceaux pervers en caract´eristique positive sur les courbes modulaires et les vari´et´es de Shimura en vue d’applications arithm´etiques (j’esp`ere acqu´erir le bagage n´ecessaire `acette fin) ; et je re- mercie Wolfgang Soergel pour sa lecture tr`es minutieuse et ses questions qui ont permis d’am´eliorer le manuscrit. Je remercie vivement chaque membre du jury pour avoir accept´ed’y participer. C’est un tr`es grand honneur pour moi de pouvoir r´eunir un jury constitu´ede math´ematiciens de tout premier plan, dans les domaines de la g´eom´etrie et des repr´esentations. La pr´esence de chacun d’entre eux me fait tr`es plaisir. Les nombreux voyages que j’ai dˆufaire pendant ma th`ese m’ont permis de rencontrer des personnes qui m’ont beaucoup apport´e, humainement et math´ematiquement. Je les remercie pour nos discussions, et les moments que nous avons partag´es, dans les diff´erents endroits o`uj’ai s´ejourn´e(Paris, Besan¸con, Yale, Lausanne, Oxford, Nancy et Hendaye), que ce soit dans le milieu math´ematique ou non. De mˆeme pour les conf´erences auxquelles j’ai particip´e, notamment `aLuminy et `aOberwolfach. Merci aussi `atoutes les personnes qui m’ont donn´el’occasion de pr´esenter mes r´esultats dans diff´erents s´eminaires. Merci `aceux qui m’ont h´eberg´e, et pour la chaleur de leur accueil. Enfin, merci `atous ceux qui se seront d´eplac´es pour ma soutenance, et `aceux qui auront contribu´e`amon pot de th`ese. Je pr´ef`ere remercier chacun de vive voix, plutˆot que citer une longue liste ici. Je remercie les diff´erentes institutions qui ont permis tous ces d´eplacements : l’Institut de Math´ematiques de Jussieu, l’Ecole´ Doctorale de Sciences Math´ematiques de Paris centre, le Laboratoire de Math´ematiques de Besan¸con, l’Universit´ed’Oxford via le r´eseau europ´een Liegrits,
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