Tilting Modules and the p-canonical Basis 285629880X, 9782856298800

Table of contents :
Chapter 1. Introduction
1.1. Overview
1.2. The ``categorical'' conjecture
1.3. Tilting modules and the antispherical module
1.4. Tilting characters
1.5. The case of the group GLn()
1.6. The diagrammatic Hecke category and parity sheaves
1.7. Variants
1.8. Simple characters
1.9. Comparison with Lusztig's conjecture
1.10. Acknowledgements
1.11. Organization of the book
Part I. General conjecture
Chapter 2. Tilting objects and sections of the -flag
2.1. Highest weight categories
2.2. Canonical -flags
2.3. Tilting objects and sections of the -flag
Chapter 3. Regular and subregular blocks of reductive groups
3.1. Definitions
3.2. Translation functors
3.3. Sections of the -flag and translation to a wall
3.4. Sections of the -flag and translation from a wall
3.5. Morphisms between ``Bott-Samelson type'' tilting modules
Chapter 4. Diagrammatic Hecke category and the antispherical module
4.1. The affine Hecke algebra and the antispherical module
4.2. Diagrammatic Soergel bimodules
4.3. Two lemmas on DBS
4.4. Categorified antispherical module
4.5. Morphisms in the categorified antispherical module
Chapter 5. Main conjecture and consequences
5.1. Statement of the conjecture
5.2. Tilting modules and antispherical Soergel bimodules
5.3. Surjectivity
5.4. Dimensions of morphism spaces
5.5. Proof of Theorem 5.2.1
5.6. Graded form of `39`42`"613A``45`47`"603ARep0(G)
5.7. Integral form of Tilt(`39`42`"613A``45`47`"603ARep0(G))
5.8. Integral form of `39`42`"613A``45`47`"603ARep0(G)
Part II. The case of GLn()
Chapter 6. Representations of GLn in characteristic p as a 2-representation of gl"0362glp
6.1. The affine Lie algebra gl"0362glN
6.2. The natural representation of gl"0362glN
6.3. Realization of n natp as a Grothendieck group
6.4. `39`42`"613A``45`47`"603ARep(G) as a 2-representation
Chapter 7. Restriction of the representation to
7.1. Combinatorics
7.2. Categorifying the combinatorics
7.3. First relations
7.4. Restriction of the 2-representation to U(gl"0362gln)
Chapter 8. From categorical gl"0362gln-actions to DBS-modules
8.1. Strategy
8.2. Preliminary lemmas
8.3. Polynomials
8.4. One color relations
8.5. Cyclicity
8.6. Jones-Wenzl relations
8.7. Two color associativity
8.8. Zamolodchikov (three color) relations
Part III. Relation to parity sheaves
Chapter 9. Parity complexes on flag varieties
9.1. Reminder on Kac-Moody groups and their flag varieties
9.2. Partial flag varieties
9.3. Derived categories of sheaves on X and Xs
9.4. Parity complexes on flag varieties
9.5. Sections of the !-flag
9.6. Sections of the !-flag and pushforward to Xs
9.7. Sections of the !-flag and pullback from Xs
9.8. Morphisms between ``Bott-Samelson type'' parity complexes
Chapter 10. Parity complexes and the Hecke category
10.1. Diagrammatic category associated with G
10.2. More on Bott-Samelson parity complexes
10.3. Statement of the equivalences
10.4. Construction of the functor BS
10.5. Verification of the relations
10.6. Fully-faithfulness of BS
10.7. The case of the affine flag variety
Chapter 11. Whittaker sheaves and antispherical diagrammatic categories
11.1. Definition of Whittaker sheaves
11.2. Whittaker parity complexes
11.3. Sections of the !-flag for Whittaker parity complexes
11.4. Surjectivity
11.5. Description of the antispherical diagrammatic category in terms of Whittaker sheaves
11.6. Application to the light leaves basis in the antispherical category
11.7. Iwahori-Whittaker sheaves on the affine flag variety
List of notation
Bibliography

Citation preview

397

ASTÉRISQUE 2018

TILTING MODULES AND THE p-CANONICAL BASIS Simon Riche & Geordie Williamson

SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

Astérisque est un périodique de la Société Mathématique de France. Numéro 397, 2018

Comité de rédaction Ahmed Abbes Hélène Esnault Viviane Baladi Philippe Eyssidieux Laurent Berger Michael Harris Philippe Biane Alexandru Oancea Nicolas Burq Fabrice Planchon Damien Calaque Éric Vasserot (dir.) Diffusion

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ISSN : 0303-1179 (print) 2492-5926 (electronic) ISBN 978-2-85629-880-0 Directeur de la publication : Stéphane Seuret

397

ASTÉRISQUE 2018

TILTING MODULES AND THE p-CANONICAL BASIS Simon Riche & Geordie Williamson

SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

Simon Riche Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France. [email protected] Geordie Williamson School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia. [email protected]

Classification mathématique par sujet (2010). — 17B10, 20G15, 14F05. Mots-clefs. — Modules basculants, groupes algébriques, faisceaux à parité, catégorie de Hecke. S.R. was partially supported by ANR Grant No. ANR-13-BS01-0001-01. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677147).

TILTING MODULES AND THE p-CANONICAL BASIS by Simon RICHE & Geordie WILLIAMSON

Abstract. — In this book we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the antispherical quotient of the Hecke category. We prove our conjecture for GLn (k) using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding KacMoody group. Résumé — Dans cet ouvrage nous proposons une nouvelle approche à l’étude des modules basculants pour les groupes algébriques réductifs sur des corps de caractéristique positive. Nous conjecturons que les foncteurs de translation induisent une action de la catégorie de Hecke (diagrammatique) du groupe de Weyl affine sur le bloc principal. Cette conjecture implique des formules de caractères pour les modules simples et les modules basculants en termes de la base p-canonique, ainsi qu’une description du bloc principal comme le quotient anti-sphérique de la catégorie de Hecke. Nous démontrons notre conjecture pour le groupe GLn (k) en utilisant la théorie des représentations des algèbres 2-Kac-Moody. Enfin, nous prouvons que la catégorie de Hecke diagrammatique d’un groupe de Coxeter cristallographique général peut être décrite en termes de faisceaux à parité sur la variété de drapeaux du groupe de Kac-Moody correspondant.

© Astérisque 397, SMF 2018

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The “categorical” conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Tilting modules and the antispherical module . . . . . . . . . . . . . . . . . . . . . . . 1.4. Tilting characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. The case of the group GLn (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. The diagrammatic Hecke category and parity sheaves . . . . . . . . . . . . . . . 1.7. Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Simple characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. Comparison with Lusztig’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11. Organization of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 8 9 10 11 15 16 17

Part I. General conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2. Tilting objects and sections of the ∇-flag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Highest weight categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Canonical ∇-flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Tilting objects and sections of the ∇-flag . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 23

3. Regular and subregular blocks of reductive groups . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Translation functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Sections of the ∇-flag and translation to a wall . . . . . . . . . . . . . . . . . . . . . 3.4. Sections of the ∇-flag and translation from a wall . . . . . . . . . . . . . . . . . . . 3.5. Morphisms between “Bott-Samelson type” tilting modules . . . . . . . . . .

27 27 29 30 33 36

4. Diagrammatic Hecke category and the antispherical module . . . . . . . . . . . . . . 4.1. The affine Hecke algebra and the antispherical module . . . . . . . . . . . . . . 4.2. Diagrammatic Soergel bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Two lemmas on DBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Categorified antispherical module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Morphisms in the categorified antispherical module . . . . . . . . . . . . . . . . .

39 39 41 45 46 47

5. Main conjecture and consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.

Statement of the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tilting modules and antispherical Soergel bimodules . . . . . . . . . . . . . . . . Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensions of morphism spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graded form of Rep0 (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral form of Tilt(Rep0 (G)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral form of Rep0 (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 56 59 60 61 64 66

Part II. The case of GLn (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

bp . . . . . 6. Representations of GLn in characteristic p as a 2-representation of gl bN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. The affine Lie algebra gl bN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. The natural representation of gl Vn 6.3. Realization of natp as a Grothendieck group . . . . . . . . . . . . . . . . . . . . . 6.4. Rep(G) as a 2-representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 79 84

bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Restriction of the representation to gl 7.1. Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Categorifying the combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. First relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b n) . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Restriction of the 2-representation to U (gl

93 93 94 95 100

b n -actions to DBS -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. From categorical gl 8.1. Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. One color relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Cyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Jones-Wenzl relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Two color associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8. Zamolodchikov (three color) relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 105 107 108 109 110 112 117

Part III. Relation to parity sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

9. Parity complexes on flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Reminder on Kac-Moody groups and their flag varieties . . . . . . . . . . . . 9.2. Partial flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Derived categories of sheaves on X and X s . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Parity complexes on flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Sections of the !-flag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. Sections of the !-flag and pushforward to X s . . . . . . . . . . . . . . . . . . . . . . . 9.7. Sections of the !-flag and pullback from X s . . . . . . . . . . . . . . . . . . . . . . . . . 9.8. Morphisms between “Bott-Samelson type” parity complexes . . . . . . . .

127 127 130 132 133 136 137 140 142

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10. Parity complexes and the Hecke category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Diagrammatic category associated with G . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. More on Bott-Samelson parity complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Statement of the equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Construction of the functor ∆BS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Verification of the relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6. Fully-faithfulness of ∆BS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7. The case of the affine flag variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 146 147 148 152 154 158

11. Whittaker sheaves and antispherical diagrammatic categories . . . . . . . . . . . . 11.1. Definition of Whittaker sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Whittaker parity complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Sections of the !-flag for Whittaker parity complexes . . . . . . . . . . . . . . . 11.4. Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5. Description of the antispherical diagrammatic category in terms of Whittaker sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6. Application to the light leaves basis in the antispherical category . . 11.7. Iwahori-Whittaker sheaves on the affine flag variety . . . . . . . . . . . . . . .

161 161 163 165 167 168 169 170

List of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 1 INTRODUCTION

1.1. Overview In this book we give new conjectural character formulas for simple and indecomposable tilting modules for a connected reductive algebraic group in characteristic p, (1) and we prove our conjectures in the case of the group GLn (k) when n ≥ 3. These conjectures are formulated in terms of the p-canonical basis of the corresponding affine Hecke algebra. They should be regarded as evidence for the philosophy that KazhdanLusztig polynomials should be replaced by p-Kazhdan-Lusztig polynomials in modular representation theory. From this point of view several conjectures (Lusztig’s conjecture, James’ conjecture, Andersen’s conjecture) become the question of agreement between canonical (or Kazhdan-Lusztig) and p-canonical bases. In the general setting we prove that the new character formulas follow from a very natural conjecture of a more categorical nature, which has remarkable structural consequences for the representation theory of reductive algebraic groups. It is a classical observation that wall-crossing functors provide an action of the affine Weyl group W on the Grothendieck group of the principal block; in this way, the principal block gives a categorification of the antispherical module for W . We conjecture that this action can be categorified: namely, that the action of wall-crossing functors on the principal block gives rise to an action of the diagrammatic Bott-Samelson Hecke category attached to W as in [31]. From this conjecture we deduce the following properties. 1. The principal block is equivalent (as a module category over the diagrammatic Bott-Samelson Hecke category) to a categorification of the antispherical module defined by diagrammatic generators and relations. 2. The principal block admits a grading. Moreover, this graded category arises via extension of scalars from a category defined over the integers. Thus the principal block of any reductive algebraic group admits a “graded integral form”. 1. This book is written under the assumption that p is (strictly) larger than the Coxeter number, and so p cannot be too small. However there is a variant of our conjectures for any p involving singular variants of the Hecke category. In particular, it seems likely that p-Kazhdan-Lusztig polynomials give correct character formulas for tilting modules in any characteristic (see Conjecture 1.4.3). We hope to return to this subject in a future work.

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CHAPTER 1. INTRODUCTION

3. The (graded) characters of the simple and tilting modules are determined by the p-canonical basis in the antispherical module for the Hecke algebra of W . From (1) one may describe the principal block in terms of parity sheaves on the affine flag variety, which raises the possibility of calculating simple and tilting characters topologically. Point (2) gives a strong form of “independence of p”. Finally, point (3) implies Lusztig’s character formula for large p. We prove this “categorical” conjecture (hence in particular the character formulas) for the groups GLn (k) using the Khovanov-Lauda-Rouquier theory of 2-Kac-Moody algebra actions. We view this, together with the agreement with Lusztig’s conjecture for large p and character formulas of Soergel and Lusztig in the context of quantum groups (see § 1.7 for details) as strong evidence for our conjecture.

1.2. The “categorical” conjecture Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p with simply connected derived subgroup. We assume that p > h, where h is the Coxeter number of G. Let T ⊂ B ⊂ G be a maximal torus and a Borel subgroup in G. Denote by X := X ∗ (T ) the lattice of characters of T and by X+ ⊂ X the subset of dominant weights. Consider a regular block Rep0 (G) of the category of finite-dimensional algebraic G-modules, corresponding to a weight λ0 ∈ X in the fundamental alcove, with its natural highest weight structure. If Φ is the root system of (G, T ), Wf = NG (T )/T is the corresponding Weyl group, and W := Wf n ZΦ is the affine Weyl group, then the simple, standard, costandard and indecomposable tilting objects in the highest weight category Rep0 (G) are all parametrized by W r λ0 ∩ X+ . (Here “ r ” denotes “p-dilated dot action of W of X,” see § 3.1.) If f W ⊂ W is the subset of elements w which are minimal in their coset Wf w, then there is a natural bijection ∼ f W − → W r λ0 ∩ X+ sending w to w r λ0 . In this way we can parametrize the simple, standard, costandard and indecomposable tilting objects in Rep0 (G) by f W , and denote them by L(x r λ0 ), ∆(x r λ0 ), ∇(x r λ0 ) and T(x r λ0 ) respectively. In particular, on the level of Grothendieck groups we have M (1.2.1) [Rep0 (G)] = Z[∇(x r λ0 )]. x∈f W

Let S ⊂ W denote the simple reflections. To any s ∈ S one can associate a “wallcrossing” functor Ξs by translating to and from an s-wall of the fundamental alcove. Consider the “antispherical” right Z[W ]-module Zε ⊗Z[Wf ] Z[W ], where Zε denotes the sign module for the finite Weyl group Wf (viewed as a right Z[Wf ]-module). This module has a basis (as a Z-module) consisting of the elements 1 ⊗ w with w ∈ f W . Then we can reformulate (1.2.1) as an isomorphism (1.2.2)

ASTÉRISQUE 397

∼

φ : Zε ⊗Z[Wf ] Z[W ] − → [Rep0 (G)]

1.2. THE “CATEGORICAL” CONJECTURE

3

defined by φ(1 ⊗ w) = [∇(w r λ0 )] for w ∈ f W . The advantage of this formulation over (1.2.1) is that φ becomes an isomorphism of right Z[W ]-modules if we let 1 + s act on the right by [Ξs ] for any s ∈ S (as follows easily from standard translation functors combinatorics). Let H be the Hecke algebra of (W, S) over Z[v, v −1 ], with standard basis {Hw : w ∈ W } and Kazhdan-Lusztig basis {H w : w ∈ W }. (Our normalization is as in [70].) Let D denote the diagrammatic Hecke category over k: this is an additive monoidal k-linear category introduced by B. Elias and the second author in [31]. This category is defined by diagrammatic generators and relations. If k is a field of characteristic zero, then (a variant of) D is equivalent to the category of Soergel bimodules for W [74]. In the modular case (and for affine Weyl groups as here) Soergel bimodules are not expected to be well behaved; but D provides a convenient replacement for these objects. (Some piece of evidence for this idea is provided by the fact that D can also be described in topological terms using parity sheaves on an affine flag variety; see § 1.6 below.) The fundamental properties of D, which generalize well-known properties of Soergel bimodules, are the following (see [31]): 1. 2. 3. 4.

is graded with shift functor h1i; is idempotent complete and Krull-Schmidt; D is generated as a graded monoidal category by some objects Bs for s ∈ S; we have a canonical isomorphism of Z[v, v −1 ]-algebras

D

D

(1.2.3)

H

∼

− → [D] : H s 7→ [Bs ]

where [D] denotes the split Grothendieck group of D (a Z[v, v −1 ]-module via v · [M ] := [M h1i]); 5. for each w ∈ W there exists an object Bw ∈ D (well-defined up to isomorphism) such that the map (w, n) 7→ Bw hni gives an identification between W × Z and the isomorphism classes of indecomposable objects in D. In particular, (4) tells us that D provides a categorification of H . Let us note also that D is defined as the Karoubi envelope of the additive hull of a category DBS (which we will call the diagrammatic Bott-Samelson Hecke category) which can be obtained by base change to k from a category DBS,Z0 defined over a ring Z0 which is either Z or Z[ 21 ]. (In other words, DBS has a natural integral form.) The following conjecture says that the above shadow of translation functors on the Grothendieck group can be upgraded to a categorical action. Conjecture 1.2.1 (Rough version). — The assignment of Bs to a wall-crossing functor Ξs for all s ∈ S induces a right action of the diagrammatic Bott-Samelson Hecke category DBS on Rep0 (G). The reason that this formulation of the conjecture is only a rough statement is that we also need to define the images of certain generating morphisms in DBS . We require that most of these morphisms arise from fixed choices of adjunctions between

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translation functors. See Conjecture 5.1.1 for a precise statement, and Remark 5.1.2 for comments. 1.3. Tilting modules and the antispherical module We now explain the main results of the first part of this book. In short, we prove that Conjecture 1.2.1 leads to an explicit description of the regular block Rep0 (G) in terms of the diagrammatic Hecke category D, which provides a character formula for tilting modules and allows us to construct graded and integral forms of Rep0 (G). Let H be the Hecke algebra of W as above, and let Hf ⊂ H be the Hecke algebra of Wf (i.e., the subalgebra with basis {Hw : w ∈ Wf }). Let M

asph

:= sgn ⊗Hf H

be the corresponding antispherical (right) H -module. (Here sgn = Z[v, v −1 ], with Hs acting via multiplication by −v for simple reflections s in Wf ; the module M asph is denoted N 0 in [70].) This module has a standard basis {Nw : w ∈ f W } and a Kazhdan-Lusztig basis {N w : w ∈ f W }; by definition we have Nw = 1 ⊗ Hw , and it is not difficult to check that N w = 1 ⊗ H w . The antispherical H -module M asph is a quantization of the antispherical Z[W ]-module considered in § 1.2 in the sense that the specialization v 7→ 1 gives a canonical isomorphism Z ⊗Z[v,v−1 ] M asph ∼ = Zε ⊗Z[W ] Z[W ]. f

asph

Now define D as the quotient of D by the morphisms which factor through a sum of indecomposable objects of the form Bw hni with w ∈ / f W . Then D asph is naturally a right D-module category and we have a canonical isomorphism of right H -modules (1.3.1)

M

asph ∼

− → [D asph ].

In particular, D asph provides a categorification of M asph . The categorified antispherical module D asph has the following properties: 1. D asph is a graded category (with shift functor h1i) and its indecomposable objects (up to isomorphism) are the images B w hni of the objects Bw hni for w ∈ f W and n ∈ Z; 2. D asph may be obtained as the Karoubi envelope of the additive hull of a right DBS -module category obtained by base change from a category defined over an explicit localization R of Z0 . asph Let us denote by Ddeg the “degrading” of D asph , i.e., the category with the same asph objects as D but with morphisms given by M HomD asph (M, N ) := HomD asph (M, N hni). deg

n∈Z asph The map on Grothendieck groups induced by the obvious functor D asph → Ddeg is the specialization v 7→ 1.

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1.3. TILTING MODULES AND THE ANTISPHERICAL MODULE

5

Let Tilt(Rep0 (G)) denote the additive category of tilting modules in Rep0 (G). Because translation functors preserve tilting modules, the wall-crossing functors preserve Tilt(Rep0 (G)); hence if Conjecture 1.2.1 holds (or in fact its more precise formulation in Conjecture 5.1.1) then the diagrammatic Bott-Samelson Hecke category DBS acts on Tilt(Rep0 (G)). Our first main result is the following. Theorem 1.3.1. — Suppose that Conjecture 5.1.1 holds. Then we have an equivalence of additive right DBS -module categories asph ∼

Ddeg

− → Tilt(Rep0 (G))

which sends B w to [T(w r λ0 )] for any w ∈ f W . In rough terms, this result says that once we have the action of DBS , we automatically know that Tilt(Rep0 (G)) “is” the (categorified) antispherical module. Or in other words, we can upgrade the isomorphism between Grothendieck groups deduced from (1.2.2) and (1.3.1) to an equivalence of categories. asph In view of the above properties of Ddeg , the following corollary is immediate. (See Theorem 5.7.4 for a more precise version.) Corollary 1.3.2. — Suppose that p > N ≥ h and that Conjecture 5.1.1 holds in any characteristic p0 > N . Then Tilt(Rep0 (G)) admits D asph as a graded enhancement. Moreover, this graded enhancement may be obtained as the Karoubi envelope of the additive hull of a graded DBS -module category coming via base change from a category defined over Z[ N1 ! ]. Because tilting modules have no higher extensions and generate Rep0 (G) the inclusion functor gives an equivalence of triangulated categories: (2) ∼

K b (Tilt(Rep0 (G))) − → Db (Rep0 (G)). The results above tell us that Tilt(Rep0 (G)) has a graded Z[ N1 ! ]-form, and hence so does K b (Tilt(Rep0 (G))). We show (see Theorems 5.6.5 and 5.8.12 below) that the t-structure defining Rep0 (G) ⊂ Db (Rep0 (G)) may be lifted to the graded integral form of K b (Tilt(Rep0 (G))). Hence (under the assumption of our conjecture in any characteristic p > N ) there exists a graded abelian category from which Rep0 (G) is obtained by extension of scalars and “degrading” for any field k of characteristic p > N . This gives a strong form of the “independence of p” property of [8]. Remark 1.3.3. — In [49] Libedinsky and the second author study the analog of the category D asph for any Coxeter system and any subset of simple reflections. For such categories, structural results analogous to those of this book are proved. (Indeed, some of the statements were motivated by the current work.) The techniques are different and do not rely on geometry or representation theory. In particular the authors deduce the positivity of Deodhar’s signed Kazhdan-Lusztig polynomials. 2. Indeed, it is from this property that tilting modules derive their name.

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1.4. Tilting characters The preceding theorem giving a description of the category Tilt(Rep0 (G)) has a “combinatorial shadow,” namely a character formula for tilting modules. First, recall that the p-canonical basis {pH w : w ∈ W } of H is defined as the inverse image under (1.2.3) of the basis of [D] consisting of the classes of the objects {Bw : w ∈ W }. This basis has many favorable properties similar to the usual Kazhdan-Lusztig basis {H w : w ∈ W } of H , see [40]. In particular: — for each w ∈ W , we have pH w = H w if p  0; — the elements pH w can be computed algorithmically, though this algorithm is much more complicated than the algorithm for computing the usual KazhdanLusztig basis. (Note that our category D does not coincide with the category used to define the p-canonical basis in [40]; however the two choices lead to the same basis of H ; see Remark 10.7.2 for details.) Similarly, the p-canonical basis {pN w : w ∈ f W } of M asph can be defined as the inverse image under (1.3.1) of the basis of [D asph ] consisting of the classes of the objects B w . By construction, for any w ∈ f W we have p

N w = 1 ⊗ pH w .

In particular, this basis is easy to compute if we know the p-canonical basis of H . We can then define the antispherical p-Kazhdan-Lusztig polynomials pnw,y (for w, y ∈ f W ) via the formula X p p Nw = ny,w Ny . y∈f W

As an immediate corollary of Theorem 1.3.1 we obtain the following result. Corollary 1.4.1. — Assume that Conjecture 5.1.1 holds. Then for any w ∈ f W the isomorphism φ of (1.2.2) satifies (1.4.1)

φ−1 ([T(w r λ0 )]) = 1 ⊗ pN w .

In other words, for any w, y ∈ f W we have (1.4.2)

(T(w r λ0 ) : ∇(y r λ0 )) = pny,w (1).

Note that the characters of the costandard modules are known, and given by the Weyl character formula. Hence the Formula (1.4.2) for multiplicities provides a character formula for tilting modules; this is the conjectural character formula for tilting modules referred to in § 1.1. Of course, there already exists a conjecture, due to Andersen, for the multiplicities of indecomposable tilting modules whose weights lie in the lowest p2 -alcove; see [6]. Recall that {N w : w ∈ f W } is the Kazhdan-Lusztig basis of M asph . Then Andersen’s conjecture can be expressed as follows: (1.4.3)

ASTÉRISQUE 397

if w ∈ f W and ∀α ∈ Φ+ , hw r λ0 + ρ, α∨ i < p2 , then φ−1 ([T(w r λ0 )]) = 1 ⊗ N w .

1.4. TILTING CHARACTERS

7

(Here Φ+ ⊂ Φ denotes the set of positive roots, and ρ is the halfsum of positive roots.) It is known that if p ≥ 2h − 2 and if this conjecture is true over k, then Lusztig’s conjecture [50] holds over k; see § 1.8 below for details. Hence the counterexamples to the expected bound in Lusztig’s conjecture found by the second author [79] show that this conjecture does not hold, except perhaps if p is very large. There are two important differences between the multiplicity Formula (1.4.1) and Andersen’s Conjecture (1.4.3). The first one is that the Kazhdan-Lusztig basis has been replaced by the p-canonical basis. The second one is that our formula applies to all tilting modules in the principal block, and not only to those in the lowest p2 -alcove. Remark 1.4.2. — 1. To see a concrete example where (1.4.2) differs from Andersen’s formula, one can consider the case G = SL(2) and p = 3. If s and s0 are the unique elements in Sf := S ∩ Wf and S \ Sf respectively, one can easily check that (T(s0 ss0 s•0) : ∇(s0 s•0)) = 1. On the other hand, the coefficient of N s0 ss0 s on Ns0 s is 0, while 3ns0 s,s0 ss0 s = 1 (see [40, § 5.3]). Of course, s0 ss0 s • 0 does not satisfy the condition in (1.4.3). 2. As noted several times already, Andersen’s conjecture is only concerned with weights in the lowest p2 -alcove. There exists a “tensor product theorem” for tilting modules, but with assumptions which are different from the assumptions of Steinberg’s tensor product theorem for simples modules. In particular, whereas one can obtain the characters of all simple modules if one knows the characters of restricted simple modules, one cannot deduce from Andersen’s formula character formulas for all indecomposable tilting modules in Rep0 (G), not even when p is large; see [53] for an investigation of which characters can be obtained in this way. 3. For fixed w ∈ f W , since pH w = H w for large p, comparing our definition of pN w with [70, Proposition 3.4] we see that also pN w = N w for large p. However, this observation alone does not imply Andersen’s conjecture for large p as the set {w ∈ f W | w r λ0 lies in the lowest p2 -alcove} grows with p. Hence as p grows we would need the equality pN w = N w for an increasing number of elements w. Therefore, even for large p, Andersen’s conjecture would imply a highly non-trivial stability property of the p-canonical basis. (Note that in the setting of affine Weyl groups it is known that there is no p such that pH w = H w for all w ∈ W .) 4. Conjecture 1.2.1 provides a very direct way to obtain the multiplicity formula (1.4.1). However this conjecture might be difficult to prove in general (see in particular Remark 1.5.2 below), and there might be other, more indirect, ways to prove this formula. In fact, after this book was made public, a proof has been obtained in [1] (a joint work with P. Achar and S. Makisumi), building on the results of Part III and those obtained by P. Achar and the first author in [5].

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Corollary 1.4.1 (contingent on our Conjecture 1.2.1) has a reformulation which makes sense for any p and which we would like to state as a conjecture. So let us temporarily allow p to be arbitrary. Fix a weight λ0 ∈ X in the closure of the fundamental alcove. (Note that λ0 will be neither regular nor dominant in general.) Let I ⊂ S be such that WI is the stabilizer of λ0 under the action λ0 7→ w r λ0 of W . We have a bijection ∼ f WI − → W r λ0 ∩ X+ where f W I denotes the subset of elements w ∈ W which both belong to f W and are maximal in their coset wWI . The bijection is given by w 7→ w r λ0 . Conjecture 1.4.3. — For any w, y ∈ f W I we have (1.4.4)

(T(w r λ0 ) : ∇(y r λ0 )) = pny,w (1).

Remark 1.4.4. — In the case when regular weights exist (i.e., when p ≥ h), the general case of Conjecture 1.4.3 follows from the case when λ0 is regular (in which case it coincides with (1.4.2)) using [7, Proposition 5.2]. 1.5. The case of the group GLn (k) In the second part of the book we restrict to the case G = GLn (k) (in which case h = n), and λ0 is the weight (n, . . . , n) (under the standard identification X = Zn ). The main result of this part, proved in § 8.1, is the following. Theorem 1.5.1. — Let n ∈ Z≥3 . If G = GLn (k) and λ0 = (n, . . . , n), then Conjecture 5.1.1 holds. Our proof of this result uses in a crucial way the Khovanov-Lauda-Rouquier 2-category U (h) associated with a Kac-Moody Lie algebra h. In a sense, to check the relations between wall-crossing functors we decompose them in terms of “simpler” functors appearing in the KLR 2-category, and use the known relations between these “simpler functors” to prove the desired relations. More precisely, our proof of Theorem 1.5.1 consists of 3 steps: b p ) on the category Rep(G) of finite dimensional algebraic 1. define an action of U (gl G-modules; b n ); 2. show that one can restrict this action to an action of U (gl b 3. show that the categorical action of U (gln ) induces an action of DBS on an appropriate weight space equal to Rep0 (G). Each of these steps is known: (1) is due to Chuang-Rouquier [24], (2) follows from a result of Maksimau [57], and (3) follows from results of Mackaay-Stošić-Vaz and Mackaay-Thiel, see [54, 55, 56]. However, as these authors do not all use the same conventions, and for the benefit of readers not used to the KLR formalism (like the authors), we give a detailed account of each step. (Our proof of step (3) is slightly different from the proof in [54, 55, 56], which does not use our step (2).) We also

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1.6. THE DIAGRAMMATIC HECKE CATEGORY AND PARITY SHEAVES

9

make an essential use of a recent result of Brundan [21] proving that the 2-categories defined by Khovanov-Lauda and by Rouquier are equivalent. In particular, this theorem implies that the character Formula (1.4.1) is proved for the group G = GLn (k) when p > n ≥ 3. This case is especially interesting because, due to work of Donkin [27], Erdmann [32] and Mathieu [60], the knowledge of characters of indecomposable tilting modules for all groups GLn (k) provides (in theory) a dimension formula for the irreducible representations of all the symmetric groups Sm over k. To use this idea in practice we would need a character formula valid for all characteristics, not only when p > n. But our formula might still have interesting applications in this direction (which we have not investigated yet). Remark 1.5.2. — 1. Theorem 1.5.1 is also trivially true in case n = 1. The case n = 2 can be proved by methods similar to those we use in Part II. However, since the extended Dynkin diagram looks differently in this case, some arguments have to be modified, and for simplicity we decided to exclude this case. (In fact, in this case there are only “one color relations,” so that one only needs to adapt the considerations of § 8.4.) In any case, the tilting modules for the group GL2 (k) are essentially the same as for SL2 (k), and in the latter case they are well understood thanks to work of Donkin [27]. 2. From the above discussion it should be clear that the extra structure provided by the categorical Kac-Moody action on Rep(GLn (k)) is absolutely central to our proof in this case. There has been recent work by several authors on possible replacements for Kac-Moody actions in other classical types (see in particular [14]); however it is unclear whether these works will open the way to a proof of our conjecture in these cases similar to our proof for GLn (k).

1.6. The diagrammatic Hecke category and parity sheaves As explained above, in this book a central role is played by the diagrammatic Hecke category D, viewed as a monoidal category defined by certain diagrammatic generators and relations. In the third part of the book we show that D is equivalent to the “geometric” (or “topological”) Hecke category, i.e., the additive category of Iwahori-equivariant parity sheaves on the Langlands dual affine flag variety. This provides an alternative and more intrinsic description of D, and also allows us (under the assumption that Conjecture 5.1.1 holds) to relate Rep0 (G) to parity sheaves. In fact, in view of other expected applications, we consider more generally the diagrammatic Hecke category DK (G ) associated with the Weyl group W of a KacMoody group G , with coefficients in a Noetherian (commutative) complete local ring K (assuming that 2 is invertible in K in some cases, see § 10.1). We denote by B ⊂ G the Borel subgroup, and by X := G /B the associated flag variety. Then we can consider the category ParityB (X , K) of B-equivariant parity complexes on X with coefficients in K (in the sense of [41]). This category is a full subcategory in the B-equivariant

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b derived category DB (X , K) which is closed under the convolution product ?B and is graded, with shift functor [1]. The main result of the third part of the book is the following.

Theorem 1.6.1. — There exists an equivalence of monoidal graded additive categories D

K

∼

(G ) − → ParityB (X , K).

Coming back to the setting of § 1.2, we can define G∧ as the simply connected cover of the derived subgroup of the complex reductive
group which is Langlands dual to G, and consider an Iwahori subgroup I ∧ ⊂ G∧ C[[z]] and the associated affine flag
∧ variety Fl := G∧ C((z)) /I ∧ . Then we have a monoidal category ParityI ∧ (Fl∧ , k) of parity complexes on Fl∧ with coefficients in k. The same methods as for Theorem 1.6.1 provide a “topological” description of the category D of § 1.2, in the form of an equivalence of monoidal additive graded categories D

∼

− → ParityI ∧ (Fl∧ , k).

One can also deduce a topological description of D asph in terms of IwahoriWhittaker (étale) sheaves on Fl∧ (or rather a version of Fl∧ over an algebraically closed field of positive characteristic different from p); see Theorem 11.7.1 for details. Remark 1.6.2. — As explained above, if Conjecture 5.1.1 holds, combining Theorem 1.3.1 with the above “topological” description of D asph one obtains a description of Rep0 (G) in terms of constructible sheaves on Fl∧ . This relation is not the same as the relation conjectured by Finkelberg-Mirković in [35]; in fact it is Koszul dual (see [5] and [1] for details). 1.7. Variants Although we will not treat this in detail, methods similar to those of this book apply in the following contexts: 1. regular block of category O of a reductive complex Lie algebra; 2. regular block of the category of finite-dimensional representations of Lusztig’s quantum groups at a root of unity; 3. Soergel’s modular category O . (In cases (1) and (3) one needs to replace the affine Weyl group W by Wf , the antispherical module M asph by the regular right module Hf , and the affine flag variety by the finite flag variety; in cases (1) and (2) one needs to replace k by the appropriate field of characteristic 0.) In each case one can formulate an analog of our Conjecture 5.1.1 and show that this conjecture implies a description of the category of tilting objects as in Theorem 1.3.1, and a character formula for these tilting objects as in Corollary 1.4.1. In case (1) one can show (using a Radon transform as in [16], or an algebraic analog) that this formula is equivalent to the Kazhdan-Lusztig conjecture proved by Be˘ılinson-Bernstein and Brylinski-Kashiwara. In case (2) this formula is Soergel’s conjecture [70] proved by

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1.8. SIMPLE CHARACTERS

11

Soergel [71], which implies in particular Lusztig’s conjecture on characters of simple representations of quantum groups at a root of unity [51], see § 1.9 below. And in case (3) these statements are equivalent to the main results of [72]. In cases (1) and (2), the appropriate variant of our results for the group G = GLn (k) also apply. (In case (1) one uses the action of U (gl∞ ) on the category O constructed b ` where ` is the order of the root of unity in [24]. In case (2) one uses an action of gl under consideration, assuming that ` > n and ` is odd; the existence of such an action is suggested in [24, Remark 7.27].) Combining these approaches with the main result of [30], in this way one can obtain direct algebraic proofs of the KazhdanLusztig conjecture and of Lusztig’s and Soergel’s quantum conjectures in type A, which bypass geometry completely (and even use few results from Representation Theory). In case (3) the similar approach does not apply (because we cannot embed the category in a larger category as in the other cases). However, since we have checked the appropriate relations on Rep(G), and since the translation functors in Soergel’s modular category O are obtained from the translation functors on Rep(G) by restricting to a subcategory and then taking the induced functors on the quotient, from the relations on Rep(G) one can deduce the relations on the modular category O . In this way one gets an alternative proof of the main result of [72], which does not rely on any result from [8] (contrary to Soergel’s proof). Note that these results are the main ingredient in the second author’s construction of counterexamples to the expected bound in Lusztig’s conjecture in [79]. Remark 1.7.1. — Case (2) above has recently been considered by Andersen-Tubbenhauer [11]. In particular, they have obtained by more explicit methods a diagrammatic description of the principal block in type A1 which is similar to the one that can be deduced from our methods.

1.8. Simple characters We conclude this introduction with some comments on another motivation for the current work, which is to establish a formula for the simple characters in Rep0 (G) in terms of p-Kazhdan-Lusztig polynomials. (See also [78] for more details and references.) The question of computing the characters of simple modules has a long history. For a fixed characteristic p, Steinberg’s tensor product theorem reduces this question to the calculation of the characters of the (finitely many) simple modules with restricted highest weight. As long as p ≥ h, classical results of Jantzen and Andersen reduce this question further to the calculation of simple characters corresponding to restricted highest weights in a regular block (a finite set which is independent of p ≥ h). As for tilting modules (see § 1.4), it is then natural to seek an expression (in the Grothendieck group of such a regular block) for the classes of simple modules in terms of costandard (dual Weyl) modules, since the characters of the latter modules are known.

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CHAPTER 1. INTRODUCTION

In 1979, Lusztig [50] gave a conjectural expression for this decomposition in terms of affine Kazhdan-Lusztig polynomials. This conjecture was then proved in 1994/95, combining works of Andersen-Jantzen-Soergel [8], Kashiwara-Tanisaki [44], KazhdanLusztig [45] and Lusztig [52], but only under the assumption that p is bigger than a non-explicit bound depending on the root system of G. More recently Fiebig [34] obtained an explicit bound above which the conjecture holds. However this bound is difficult to compute and in any case several orders of magnitude bigger than h. On the other hand, the second author [79] has recently exhibited examples showing that Lusztig’s conjecture cannot hold for all p ≥ h. In fact these examples show that there does not exist any polynomial P ∈ Z[X] such that Lusztig’s conjecture holds for the group G = GLn (k) in all characteristics p > P (n). (3) On the other hand, in this case the Coxeter number h is equal to n. Following a strategy due to Andersen, we can use a small part of the information on tilting characters above to deduce character formulas for simple modules. Of course, it is enough to do this when G is quasi-simple, which we will assume for the rest of this subsection. Let us assume that p ≥ 2h − 2. As above, we denote by ρ the half sum of positive roots, by α0∨ the highest coroot of G, and we set f W0 := {w ∈ f W | hw r λ0 + ρ, α0∨ i < p(h − 1)}. (More concretely, this means that w r λ0 belongs to an alcove which lies below the hyperplane orthogonal to α0∨ containing the “Steinberg weight” (p − 1)ρ. Clearly, this subset of f W does not depend on the choice of λ0 , nor on p.) Following Soergel [70] we define a map f

(1.8.1)

W → f W : y 7→ yb

as follows. Let Σ be the set of simple roots, and let + ∨ X+ 1 := {λ ∈ X | ∀α ∈ Σ, hλ, α i ≤ p − 1}

be the set of restricted dominant weights. Then, for λ ∈ (p − 1)ρ + X+ , we set ˇ = (p − 1)ρ + pγ + w0 η where γ ∈ X+ and η ∈ X+ are characterized by the fact that λ 1 ∼ λ = (p − 1)ρ + pγ + η. This map induces a bijection (p − 1)ρ + X+ − → X+ , and we denote its inverse by µ 7→ µ b. Then the map (1.8.1) is characterized by the fact that r λ0 = yb r λ0 . y[ The following result is closely related to [6, Proposition 2.6]. Proposition 1.8.1. — For any x, y ∈ f W0 we have [∆(x r λ0 ) : L(y r λ0 )] = (T(b y r λ0 ) : ∇(x r λ0 )). Proof. —

(4)

Let us consider + ∨ X+ e(π)s. Then G we set d0 (π) := d(π) − 1, and we define ϕπ as the composition s

Ee0 (π)

adj

−−→ (q s )∗ (q s )∗ Ees0 (π) → (q s )∗ Ee(π) [−1] (q s )∗ (ϕF )[−1]

−−−−−−π−−−→ (q s )∗ F [d(π) − 1] = G [d0 (π)], where the second morphism is induced by the second map in (9.4.6). In other words, G ϕπ is the image of the composition of ϕFπ [−1] with the projection (q s )∗ Ees0 (π) → Ee(π) [−1] under the isomorphism Hom b ((q s )∗ E s0 , F [d(π) − 1]) ∼ (E s0 , G [d0 (π)]) s = Hom b D(B) (X ,F)

D(B) (X ,F)

e (π)

e (π)

induced by adjunction. Now, assume that e(π) < e(π)s, or in other words that e(π) ∈ W s . Then we set G 0 d (π) := d(π), and we define ϕπ as the composition s

Ee0 (π)

(q s )∗ (ϕF )

π → (q s )∗ Ee(π) −−−−−− → (q s )∗ F [d(π)] = G [d0 (π)],

where the first map is the first morphism in (9.4.5).

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G

Remark 9.6.1. — An important point for us is that in both cases the morphism ϕπ factors through a shift of (q s )∗ (ϕFπ ) : (q s )∗ Ee(π) → G [d(π)]. G

Proposition 9.6.2. — The quadruple (Π0 , e0 , d0 , (ϕπ )π∈Π0 ) constructed above is a section of the !-flag of G = (q s )∗ F . Before proving Proposition 9.6.2 in general we consider a special case. Lemma 9.6.3. — Let w ∈ W , and assume that F is isomorphic to a direct sum of G shifts of ∇w . Then the quadruple (Π0 , e0 , d0 , (ϕπ )π∈Π0 ) constructed above is a section of the !-flag of G . Proof. — Let us fix a non-zero morphism fw : Ew → ∇w (which is unique up to scalar); then the morphism M M ϕπ [−d(π)] : Ew [−d(π)] → F π∈Π

factors through

L

π∈Π

π∈Π fw [−d(π)], and induces an isomorphism M ∼ ∇w [−d(π)] − →F; π∈Π

therefore we can assume that F = ∇w , Π = {w}, e(w) = w, d(w) = 0 and ϕFw = fw . In this setting G is isomorphic to ∇sw or ∇sw [1], so that to conclude it suffices to prove G that ϕw 6= 0. G If w > ws, then the fact that ϕw 6= 0 follows from the facts that the composition G of ϕFw [−1] with the projection (q s )∗ Ees0 (π) → Ee(π) [−1] is nonzero, and that ϕw is obtained from the latter morphism by adjunction. If w < ws, it is easy to see that G G (isw )∗ (ϕw ) is an isomorphism, hence is non-zero; it follows that ϕw 6= 0 also in this case. Proof of Proposition 9.6.2. — We have to prove that for any w ∈ W s the images of G the morphisms ϕπ with e0 (π) = w form a basis of Hom•Db (X s ,F) (Ews , G ). For this we (B)

fix w and choose some closed subvariety Y ⊂ X such that Y ⊃ Xw ,

≥w

Y 6⊃ Xws ,

and such that both Y 0 := Y \ Xw

and Y 00 := Y ∪ Xws

are closed subvarieties of X . We denote by i: Y → X ,

i0 : Y 0 → X ,

i00 : Y 00 → X

the embeddings, so that we have natural morphisms (i0 )! (i0 )! F → i! i! F → (i00 )! (i00 )! F → F induced by adjunction. Note that all of these objects are !-parity.

ASTÉRISQUE 397

9.6. SECTIONS OF THE !-FLAG AND PUSHFORWARD TO X s

139

One can easily check, using the standard triangle [1]

i! i! F → F → j∗ j ∗ F −→ where j : X \ Y → X is the (open) embedding, resp. the similar triangle for i00 , that every morphism ϕFπ where e(π) = w, resp. e(π) = ws, factors (uniquely) through a morphism !

ϕiπ! i F : Ew → i! i! F [d(π)],

resp. ϕ(i π

00

)! (i00 )! F

: Ews → (i00 )! (i00 )! F [d(π)]. G

Then by construction the corresponding morphism ϕπ factors through a morphism ϕπ(q

s

)∗ i! i! F

: Ews → (q s )∗ i! i! F [d0 (π)], resp. ϕ(q π

s

)∗ (i00 )! (i00 )! F

: Ews → (q s )∗ (i00 )! (i00 )! F [d0 (π)].

It is clear also that the morphisms obtained in this way coincide with the morphisms obtained by the above procedure from the section of the !-flag of i! i! F , resp. (i00 )! (i00 )! F , obtained (in the obvious way) from (Π, e, d, (ϕFπ )π∈Π ) by restriction to Y , resp. to Y 00 . Now if e(π) = w, resp. e(π) = ws, we can consider the composition i i! F

ϕπ!

!

ϕπ(iw )∗ (iw ) F : Ew −−−−→ i! i! F [d(π)] → (iw )∗ (iw )! F [d(π)]
where the second morphism is induced by the adjunction (iw )∗ , (iw )∗ , resp. the composition (i00 )! (i00 )! F

ϕπ

!

ϕπ(iws )∗ (iws ) F : Ews −−−−−−−→ (i00 )! (i00 )! F [d(π)] → (iws )∗ (iws )! F [d(π)]
where the second morphism is induced by the adjunction (iws )∗ , (iws )∗ , and the (q s ) (i ) (i )! F

(q ) (i

) (i

)! F

corresponding morphisms ϕπ ∗ w ∗ w , resp. ϕπ s ∗ ws ∗ ws , obtained by the procedure above, which can also be described as the compositions (q s )∗ i! i! F

s ϕπ

Ew

−−−−−−−→ (q s )∗ i! i! F [d0 (π)] → (q s )∗ (iw )∗ (iw )! F [d0 (π)],

resp. (q s )∗ (i00 )! (i00 )! F

s ϕπ

Ew

−−−−−−−−−−→ (q s )∗ (i00 )! (i00 )! F [d0 (π)] → (q s )∗ (iws )∗ (iws )! F [d0 (π)].

Finally, let us denote by i0w : Xw t Xws → X the inclusion. Then if e(π) = w, resp. e(π) = ws, we can consider the composition (i0 )∗ (i0w )! F

ϕπ w

!F

ϕ(iw )∗ (iw )

: Ew −−π−−−−−−→ (iw )∗ (iw )! F [d(π)] → (i0w )∗ (i0w )! F [d(π)],

resp. the composition (i0 )∗ (i0w )! F

ϕπ w

(i00 )! (i00 )! F

ϕπ

: Ews −−−−−−−→ (i00 )! (i00 )! F [d(π)] → (i0w )∗ (i0w )! F [d(π)],

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where in both cases the second morphism is again induced by adjunction, and the corresponding morphisms (q s )∗ (i0w )∗ (i0w )! F

ϕπ

: Ews → (q s )∗ (i0w )∗ (i0w )! F [d0 (π)].

Now, consider the natural distinguished triangle [1]

(iw )∗ (iw )! F → (i0w )∗ (i0w )! F → (iws )∗ (iws )! F −→ and its image [1]

(q s )∗ (iw )∗ (iw )! F → (q s )∗ (i0w )∗ (i0w )! F → (q s )∗ (iws )∗ (iws )! F −→ under the functor (q s )∗ . Since the image under (q s )∗ of any morphism ∇ws → ∇w [k] is zero, this triangle is split. Hence, taking the image under the functor s HomDb (X≥w s ,F) (Ew , −) we obtain an exact sequence of F-vector spaces (B)

0 → Hom•Db

(B)

s s ! s ,F) (Ew , (q )∗ (iw )∗ (iw ) F (X≥w

→ Hom•Db

(B)

)

s s 0 0 ! s ,F) (Ew , (q )∗ (iw )∗ (iw ) F (X≥w

→ Hom•Db

(B)

)

s s ! s ,F) (Ew , (q )∗ (iws )∗ (iws ) F (X≥w

) → 0.

(q s ) (i ) (i )! F

It follows from Lemma 9.6.3 that the images of the morphisms ϕπ ∗ w ∗ w , (q s ) (i ) (i )! F resp. ϕπ ∗ ws ∗ ws , with e(π) = w, resp. e(π) = ws, form a basis of the first, (q s ) (i0 ) (i0 )! F

respectively third, term in this exact sequence. Hence the morphisms ϕπ ∗ w ∗ w with e(π) ∈ {w, ws} form a basis of the middle term. Now we remark that the natural morphisms (i00 )! (i00 )! F → F and (i00 )! (i00 )! F → (i0w )∗ (i0w )! F induce isomorphisms Hom•Db

(B)

s s s ,F) (Ew , (q )∗ F (X≥w

∼

)− → Hom•Db

(B)

∼

− → Hom•Db

(B)

s s 00 00 ! s ,F) (Ew , (q )∗ (i )! (i ) F (X≥w

)

s s 0 0 ! s ,F) (Ew , (q )∗ (iw )∗ (iw ) F (X≥w

),

and we deduce the desired claim. 9.7. Sections of the !-flag and pullback from X s b As in § 9.6 we fix a simple reflection s ∈ S. Let F ∈ D(B) (X s , F) be an object which is !-parity, and let (Π, e, d, (ϕFπ )π∈Π ) be a section of the !-flag of F . It is clear that the object H := (q s )! F is !-parity, and the goal of this subsection is to explain how one can define a section of the !-flag for this object out of (Π, e, d, (ϕFπ )π∈Π ). We set Π0 := Π × {0, 1}. We define a map e0 : Π0 → W as follows. Given π ∈ Π, the elements e0 (π, 0) and e0 (π, 1) are characterized by the following properties: — the images in W/Ws of both e0 (π, 0) and e0 (π, 1) are equal to e(π); — e0 (π, 1) = e0 (π, 0)s;

ASTÉRISQUE 397

9.7. SECTIONS OF THE !-FLAG AND PULLBACK FROM X s

141

— e0 (π, 0) < e0 (π, 1) in the Bruhat order. H Then we define a map d0 : Π0 → Z and morphisms ϕH (π,0) and ϕ(π,1) for π ∈ Π as

follows. First, we set d0 (π, 1) = d(π) − 1, and ϕH (π,1) is defined as the composition Ee0 (π,1)

(q s )! (ϕF )[−1]

s → (q s )! Ee(π) [−1] −−−−−π−−−→ (q s )! F [d(π) − 1] = H [d0 (π, 1)],

where the first morphism is the first map in (9.4.6). On the other hand, we set d0 (π, 0) = d(π) and we define ϕH (π,0) as the composition Ee0 (π,0)

adj

−−→ (q s )! (q s )! Ee0 (π,0) = (q s )! (q s )∗ Ee0 (π,0) → (q s )! (ϕF )

π s (q s )! Ee(π) −−−−−− → (q s )! F [d(π)] = H [d0 (π, 0)].

Here the the third morphism is induced by the second map in (9.4.5). Remark 9.7.1. — As in Remark 9.6.1, an important point for us is that both ϕH (π,0) s ! s s ! F and ϕH (π,1) factor through a shift of the morphism (q ) (ϕπ ) : (q ) Ee(π) → H [d(π)].

Proposition 9.7.2. — The quadruple (Π0 , e0 , d0 , (ϕH π )π∈Π0 ) constructed above is a section of the !-flag of H = (q s )! F . Before proving Proposition 9.7.2 in general we consider a special case. Lemma 9.7.3. — Let w ∈ W s , and assume that F is isomorphic to a direct sum of shifts of ∇sw . Then the quadruple (Π0 , e0 , d0 , (ϕH π )π∈Π0 ) constructed above is a section of the !-flag of H . Proof. — As in the proof of Lemma 9.6.3, we can assume that F = ∇sw , Π = {w}, e(w) = w, d(w) = 0. In this case, since (q s )−1 (Xws ) is the disjoint union of the closed subset Xw and the open subset Xws , adjunction provides a canonical distinguished triangle [1]

∇w [−2] → H → ∇ws [−1] −→ . Moreover, the morphisms in this triangle induce isomorphisms Hom•Db

(X≥w ,F) (Ew , H )

∼ = Hom•−2 Db

(X≥w ,F)

(X≥ws ,F) (Ews , H )

∼ = Hom•−1 Db

(X≥ws ,F)

(B)

Hom•Db

(B)

(B)

(B)

(Ew , ∇w ), (Ews , ∇ws ),

and all of these spaces are 1-dimensional. Hence to conclude it suffices to prove that H the restriction of ϕH (π,0) to Xw and the restriction of ϕ(π,1) to Xws are non-zero; in both cases this is clear from construction. Proof of Proposition 9.7.2. — The proof is very similar to, and in fact simpler than, that of Proposition 9.6.2; details are therefore left to the reader.

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9.8. Morphisms between “Bott-Samelson type” parity complexes We set b b Υs := (q s )! (q s )∗ [−1] : D(B) (X , F) → D(B) (X , F).

By Lemma 9.4.2(1) and (9.4.2), we have a canonical isomorphism Υs ∼ = (−) ?B Es .

(9.8.1)

Recall the parity complexes Ew (where w is an expression) defined in § 9.4. Proposition 9.8.1. — Let x and v be expressions, and assume that x is a reduced expression for some element x ∈ W . 1. Assume that x ∈ W s , so that xs is a reduced expression for xs ∈ W . Let (fi )i∈I be a family of homogeneous elements in Hom•Db (X ,F) (Ex , Ev ) whose im(B)

ages span the vector space Hom•Db of homogeneous elements in tor space fi0 : Ex

(X≥x ,F) (Ex , Ev ), (B) • HomDb (X ,F) (Exs , Ev )

Hom•Db (X≥xs ,F) (Exs , Ev ). (B)

and let (gj )j∈J be a family whose images span the vec-

(B)

Then there exist integers ni and morphisms

→ Exs [ni ] (for i ∈ I) and integers mj and morphisms gj0 : Ex → Exss [mj ] (for j ∈ J) such that the images of the compositions Ex

fi ?B Es [ni ]

f0

i −−−−−−→ Evs [ni + deg(fi )] −→ Exs [ni ] −

together with the images of the compositions Ex

gj ?B Es [mj ]

gj0

−→ Exss [mj ] −−−−−−−→ Evs [mj + deg(gj )]

span the vector space Hom•Db

(B)

(X≥x ,F) (Ex , Evs ).

2. Assume that x = ys for some expression y (which is automatically a reduced expression for xs). Let (fi )i∈I be a family of homogeneous elements in Hom•Db (X ,F) (Ex , Ev ) whose images span Hom•Db (X≥x ,F) (Ex , Ev ), and let (B)

(B)

(gj )j∈J be a family of homogeneous elements in Hom•Db ages span fi0 : Ex

Hom•Db (X≥xs ,F) (Ey , Ev ). (B)

(B)

(X ,F) (Ey , Ev )

whose im-

Then there exist integers ni and morphisms

→ Exs [ni ] (for i ∈ I) and integers mj and morphisms gj0 : Ex → Ex [mj ] (for j ∈ J) such that the images of the compositions Ex

fi ?B Es [ni ]

f0

i −→ Exs [ni ] − −−−−−−→ Evs [ni + deg(fi )]

together with the images of the compositions Ex

gj ?B Es [mj ]

gj0

−→ Ex [mj ] −−−−−−−→ Evs [mj + deg(gj )]

span the vector space Hom•Db

(B)

ASTÉRISQUE 397

(X≥x ,F) (Ex , Evs ).

9.8. MORPHISMS BETWEEN “BOTT-SAMELSON TYPE” PARITY COMPLEXES

143

b ∼ Ex in Db (X≥x , F), and Exs ∼ Proof. — (1) We have Ex = (X≥xs , F). = Exs in D(B) (B) Hence we can fix split embeddings Ex → Ex and Exs → Exs , and assume that the compositions

Ex

fi

gj

→ Ex −→ Ev [deg(fi )] and Exs → Exs −→ Ev [deg(gj )]

are part of a section of the !-flag of Ev . Then Proposition 9.6.2 provides a section of the !-flag of (q s )∗ Ev whose morphisms in Hom• (Exs , (q s )∗ Ev ) are parametrized by I t J, in such a way that the morphism associated with i ∈ I factors through a shift of (q s )∗ (fi ) : (q s )∗ Ex → (q s )∗ Ev [deg(fi )], and the morphism associated with j ∈ J factors through a shift of (q s )∗ (gj ) : (q s )∗ Exs → (q s )∗ Ev [deg(gj )] (see in particular Remark 9.6.1). Applying Proposition 9.7.2, we then obtain a section of the !-flag of Υs Ev whose morphisms in Hom• (Ex , Υs Ev ) are parametrized by I t J, in such a way that the morphism associated with i ∈ I factors through a shift of Υs (fi ) : Υs Ex → Υs Ev [deg(fi )], and the morphism associated with j ∈ J factors through a shift of Υs (gj ) : Υs Exs → Υs Ev [deg(gj )] (see in particular Remark 9.7.1). Composing with an arbitrarily chosen split projection Ex → Ex and using (9.8.1), we deduce the desired claim. (2) The proof is identical to the proof of (1), and is therefore omitted.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2018

CHAPTER 10 PARITY COMPLEXES AND THE HECKE CATEGORY

10.1. Diagrammatic category associated with G We now define a realization of (W, S) over Z (in the sense of [31, Definition 3.1]) as follows: — the underlying Z-module is hZG := HomZ (Λ, Z); — for any s ∈ S, the elements “αs ” and “αs∨ ” are the simple root and coroot attached to s respectively. This realization is balanced in the sense of [31, Definition 3.6]. Z For any integral domain K one can consider the realization hK G := hG ⊗Z K over K. If the “Demazure surjectivity” condition [31, Assumption 3.7] holds for hZG , we set 0 Z0 := Z; otherwise we set Z0 := Z[ 21 ]. Then Demazure surjectivity holds for hZG , hence 0 for hK G for any integral domain K such that there exists a ring morphism Z → K. For K such a ring, we denote by DBS (G ) the diagrammatic category of [31, Definition 5.2] associated with this realization. (The definition of this category is similar to the definition of the category DBS in § 4.2. In particular, the morphisms are generated by the same diagrams as in § 4.2, with now S and W defined as in § 9.1, and h replaced by hK G .) By construction, for K as above and any M, N in DK BS (G ), the morphism space M HomDKBS (G ) (M, N hii) Hom•DK (G ) (M, N ) := BS

i∈Z

is a graded bimodule over K

O (hG )

(10.1.1)


= Sym K ⊗Z Λ .

Note also that if K → K0 is a ring morphism (where again K0 is an integral domain), K0 then there exists a natural functor DK BS (G ) → DBS (G ) which is the identity on objects. Moreover, this functor induces an isomorphism (10.1.2)

∼

K0 ⊗K HomDKBS (G ) (M, N ) − → HomDK0 (G ) (M, N ) BS

for any M, N in Theorem 6.11].)

DK BS (G ).

(In fact, this follows from the double leaves theorem [31,

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If K is in addition a complete local ring, we denote by DK (G ) the Karoubi envelope • of the additive hull of DK BS (G ). We define HomDK (G ) (M, N ) in the obvious way. Remark 10.1.1. — In this case, contrary to the situation in § 4.2, in general the roots do not generate K ⊗Z Λ. Hence we need to consider the “polynomial” morphisms in DK BS (G ); they cannot be expressed in terms of the other generators in general. 10.2. More on Bott-Samelson parity complexes Let us come back to the setting of § 9.3. Below we will also use an analog of the b construction of ?B in the case of 3 variables: given F1 , F2 , F3 in DB (X , K), we set e F2  e F3 ), Conv3 (F1 , F2 , F3 ) := m3∗ (F1  where m3 : G ×B G ×B X → X is the morphism induced by multiplication in G , e F2  e F3 is the unique object in Db (G ×B G ×B X , K) whose pullback and F1  B to G × G × X is µ∗ F1  µ∗ F2  F3 . Of course, there exist canonical isomorphisms ∼

∼

(F1 ?B F2 ) ?B F3 − → Conv3 (F1 , F2 , F3 ) − → F1 ?B (F2 ?B F3 ) whose composition is the associativity constraint for the product ?B . The “Bott-Samelson parity complex” Ew defined in § 9.4 is defined only up to (canonical) isomorphism, since one needs to choose the order in which the convolution products are taken. To remedy this we introduce a canonical object EK (w) as follows. For this, recall the Demazure (or Bott-Samelson) resolution νw : BS(w) → X defined in § 9.1. Then we set EK (w) := (νw )∗ KBS(w) [`(w)]. Lemma 10.2.1. — For any expressions w and v, there exists a canonical isomorphism EK (w) ?

B

EK (v)

∼ = EK (wv),

where wv is the concatenation of w and v. e EK (v) Proof. — Write w = s1 · · · sr . It can be easily checked that the complex EK (w)  is canonically isomorphic to (the extension by 0 of) the direct image of the constant sheaf on
Ps1 ×B · · · ×B Psr ×B BS(v) = BS(wv) under the natural projection to G ×B X . The composition of this projection with the morphism induced by m identifies with νwv , and the claim follows. We will consider the category ParityBS B (X , K) whose objects are the pairs (w, n) where w is an expression and n ∈ Z, and whose morphisms are defined as follows:

HomParityBS (w, n), (v, m) := HomDB b (X ,K) EK (w)[n], EK (v)[m] . B (X ,K)

ASTÉRISQUE 397

10.3. STATEMENT OF THE EQUIVALENCES

147

We endow this category with a monoidal product ? by declaring that (w, n) ? (v, m) := (wv, n + m) and using the canonical isomorphism in Lemma 10.2.1 to define the product of morphisms. If K and K0 are Noetherian commutative rings of finite global dimension, and if we are given a ring morphism K → K0 , then we have an “extension of scalars” functor L

b b K0 := K0 ⊗K (−) : DB (X , K) → DB (X , K0 ).

Standard compatibility properties of extension of scalars with !-pushforward functors (see e.g., [42, Proposition 2.6.6] in the classical context) show that for any expression w there exists a canonical isomorphism K0 (EK (w)) ∼ = EK0 (w). In particular, it follows that the functor K0 defines in a natural way a monoidal functor from ParityBS B (X , K) 0 (X , K to ParityBS ), which we will denote by the same symbol. The following result is B standard, see e.g., [61, Lemma 2.2(2)]. Lemma 10.2.2. — For any expressions w and v and any n, m ∈ Z, the K-module
HomParityBS (w, n), (v, m) B (X ,K) is free, and the functor K0 induces an isomorphism
∼
K0 ⊗K HomParityBS (w, n), (v, m) − → HomParityBS (w, n), (v, m) . 0 B (X ,K) B (X ,K ) Assume now in addition that K is a complete local ring, and recall the category ParityB (X , K) defined in § 9.4. We can define a natural fully-faithful monoidal functor (10.2.1)

ParityBS B (X , K) → ParityB (X , K)

sending (w, n) to EK (w)[n]. In this case any indecomposable object in the category ParityB (X , K) is isomorphic to a direct summand of an object EK (w)[n], see [41]. We deduce the following lemma. Lemma 10.2.3. — Assume that K is a Noetherian commutative complete local ring. The functor (10.2.1) realizes ParityB (X , K), as a monoidal category, as the Karoubi envelope of the additive hull of the monoidal category ParityBS B (X , K). 10.3. Statement of the equivalences We continue with the setting of § 9.3 and § 10.2, and assume in addition that K is an integral domain. The main result of this subsection is the following. Theorem 10.3.1. — Assume that K is a complete local ring and that there exists a ring morphism Z0 → K. Then there exists an equivalence of additive monoidal categories ∼

∆ : DK (G ) − → ParityB (X , K).

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In view of Lemma 10.2.3 and the construction of the category DK (G ), Theorem 10.3.1 will follow from the following result, which applies to more general coefficients. Theorem 10.3.2. — Assume that there exists a ring morphism Z0 → K. Then there exists an equivalence of monoidal categories ∼

∆BS : DK → ParityBS BS (G ) − B (X , K). The remainder of this section is devoted to the proof of Theorem 10.3.2: in § 10.4– BS 10.5 we construct a monoidal functor ∆BS : DK BS (G ) → ParityB (X , K) which is obviously essentially surjective. Then in § 10.6 we show that this functor is fully-faithful, which completes the proof of Theorem 10.3.2, hence also of Theorem 10.3.1. To fix notation we assume that we work in the classical setting. The étale setting can be treated in a similar way, replacing Z0 by a ring of p-adic integers. 10.4. Construction of the functor ∆BS 10.4.1. Principle of the construction. — In this subsection we assume that K is a Noetherian integral domain of finite global dimension, and that there exists a ring morphism Z0 → K. Our goal is to construct a monoidal functor BS ∆BS : DK BS (G ) → ParityB (X , K).

The definition on objects is obvious: we simply set ∆BS (Bw hni) = (w, n). To define ∆BS on morphisms, we will explain how to define the image of a morphism φ : Bw hni → Bw0 hni where w0 is either equal to w or obtained from w by one of the substitutions s 99K ∅, ∅ 99K s, ss 99K s, s 99K ss, st · · · 99K ts · · ·

(10.4.1)

(where s, t ∈ S, s 6= t, and st has finite order mst , the number of terms on each side of the last substitution being mst ), and φ is induced by the corresponding “elementary” morphism (polynomial, upper dot, lower dot, trivalent morphism or 2mst -valent morphism). Then we will check that these images satisfy the relations from [31]. In fact we will only consider the case when K = Z0 . Then using Lemma 10.2.2 one can deduce the definition of the morphisms in the case of any K, and the fact that the relations hold over Z0 implies that they also hold over K. We only need to define the images of the morphisms associated with the polynomials and the substitutions in (10.4.1). For instance, if one knows the definition of the image ψ : E (s) → E (∅)[1] of the “upper dot morphism” Bs → B∅ h1i, for any expressions u and v one defines the image of the induced morphism Busv → Buv h1i as the composition E (usv)

∼

− → Conv3 (E (u), E (s), E (v)) Conv3 (E (u),ψ,E (v))

∼

−−−−−−−−−−−−→ Conv3 (E (u), E (∅), E (v))[1] − → E (uv)[1]

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149

where the first and third morphisms are the canonical isomorphisms (given by the obvious analog of Lemma 10.2.1). The definition of these images occupies the rest of this subsection. Then in § 10.5 we prove that these morphisms satisfy the required relations. 10.4.2. Polynomials. — As noted in (9.4.3), for m ∈ Z≥0 the Borel isomorphism gives us a canonical identification
m 0 0 ∼ 2m HomDB b (X ,Z0 ) (EZ0 (∅), EZ0 (∅)[2m]) = HB (pt; Z ) = SymZ0 Z ⊗Z Λ .
0 We send the morphism B∅ → B∅ h2mi given a region labeled by f ∈ Symm Z0 Z ⊗Z Λ to the corresponding map EZ0 (∅) → EZ0 (∅)[2m] under this identification:   f ∆BS := EZ0 (∅) − → EZ0 (∅)[2m]. f 10.4.3. Dot morphisms. — Recall the inclusion i1 : X≤1 = B/B ,→ G /B. We define the image of the upper dot morphism attached to a simple reflection s ∈ S to be the adjunction morphism: ! • := a : E 0 (s) → (i ) (i )∗ E 0 (s) = E 0 (∅)[1]. ∆ ∗

BS

Z

1 ∗

1

Z

Z

s

(Because (f ◦ g)∗ ∼ = g ∗ f ∗ for two maps f and g, we have canonically (i1 )∗ EZ0 (s) = (i1 )∗ (ps )∗ Z0 pt [1] = Z0 B/B [1] where ps : Ps /B → pt denotes the projection.) We define the image of the lower dot morphism attached to s ∈ S to be the adjunction morphism:   s := a! : EZ0 (∅) = (i1 )! (i1 )! EZ0 (s)[1] → EZ0 (s)[1]. ∆BS • (Because (f ◦ g)! ∼ = g ! f ! for two maps f and g, we have canonically (i1 )! EZ0 (s)[1] = 0 ! Z (i1 ) DPs /B = (i1 )! (ps )! Z0 pt = Z0 B/B , where as above ps : Ps /B → pt denotes the 0

projection. Here DZX denotes the dualizing complex for coefficients Z0 on a variety X, 0 and the identification EZ0 (s) = Z0 Ps /B [1] = DZPs /B [−1] is canonical because Ps /B is √ smooth of complex dimension 1 and we have chosen once and for all −1 ∈ C.) 10.4.4. Trivalent vectices. — We fix a simple reflection s ∈ S. Lemma 10.4.1. — There exists an isomorphism EZ0 (ss) ∼ = EZ0 (s)[1] ⊕ EZ0 (s)[−1]. Proof. — There exist isomorphisms BS(ss) ∼ = Ps /B × Ps /B ∼ = P1 × P1 such that the morphism νss identifies with the second projection P1 × P1 → P1 . Then the decomposition follows e.g., from the projection formula together with the known description of the cohomology of P1 .

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In particular, from Lemma 10.4.1 we deduce the following isomorphisms: Hom(EZ0 (s), EZ0 (ss)[−1]) = Hom(EZ0 (s), EZ0 (s)) = H0B (Ps /B; Z0 ) = Z0 , Hom(EZ0 (ss), EZ0 (s)[−1]) = Hom(EZ0 (s)[−1], EZ0 (s)[−1]) = H0B (Ps /B; Z0 ) = Z0 . Lemma 10.4.2. — EZ0 (ss)[−1]

1. Composition with the morphism E 0 (s)?

∼

B

a∗ [−1]

∼

− → EZ0 (s) ?B EZ0 (s)[−1] −Z−−−−−−−−→ EZ0 (s) ?B EZ0 (∅) − → EZ0 (s)

induces an isomorphism ∼

HomDB → HomDB b (X ,Z0 ) (EZ0 (s), EZ0 (ss)[−1]) − b (X ,Z0 ) (EZ0 (s), EZ0 (s)). 2. Composition with the morphism EZ0 (s)[−1]

E 0 (s)?

∼

B

a! [−1]

∼

Z − → EZ0 (s) ?B EZ0 (∅)[−1] −− −−−−−−−→ EZ0 (s) ?B EZ0 (s) − → EZ0 (ss)

induces an isomorphism ∼

HomDB → HomDB b (X ,Z0 ) (EZ0 (ss), EZ0 (s)[−1]) − b (X ,Z0 ) (EZ0 (s)[−1], EZ0 (s)[−1]). Proof. — Statement (1) follows from the observation that one can choose the decomposition EZ0 (ss) ∼ = EZ0 (s)[1] ⊕ EZ0 (s)[−1] of Lemma 10.4.1 so that our morphism identifies with the shift by [−1] of the projection on the first factor. The proof of (2) is similar. We now define b1 ∈ HomDB b (X ,Z0 ) (EZ0 (s), EZ0 (ss)[−1]) and b2 ∈ HomD b (X ,Z0 ) (EZ0 (ss), EZ0 (s)[−1]) B to be the unique elements which map to the identity in HomDB b (X ,Z0 ) (EZ0 (s), EZ0 (s)) 0 0 and HomDB b (X ,Z0 ) (EZ (s)[−1], EZ (s)[−1]) respectively under the isomorphisms of Lemma 10.4.2. We set:     s

s

∆BS 

s

 := b1 s

and ∆BS 

 := b2 . s

s

10.4.5. 2mst -valent vectices. — Fix s, t ∈ S and define  ∨ ∨  2 if hαis , αit i = hαit , αis i = 0;    3 if hαi∨s , αit ihαi∨t , αis i = 1;  mst := 4 if hαi∨s , αit ihαi∨t , αis i = 2;     6 if hαi∨s , αit ihαi∨t , αis i = 3;    ∞ if hαi∨s , αit ihαi∨t , αis i > 3. Then mst is the order of st ∈ W . From now on we fix a pair s, t ∈ S with mst < ∞ and abbreviate m := mst . Set wI := sts · · ·

ASTÉRISQUE 397

(with m terms)

10.4. CONSTRUCTION OF THE FUNCTOR ∆BS

151

and let WI := hs, ti. To simplify notation we also set Fs

:= EZ0 (st · · · )

(m terms),

Ft

:= EZ0 (ts · · · )

(m terms).

Lemma 10.4.3. — The Z0 -module HomDB b (X ,Z0 ) (Fs , Ft ) is free of rank 1. Proof. — By Lemma 10.2.2, the Z0 -module under consideration is free, and we have ∼ (Q(Fs ), Q(Ft )). Q ⊗Z0 Hom b 0 (Fs , Ft ) = Hom b DB (X ,Q)

DB (X ,Z )

By Kazhdan-Lusztig theory and a calculation in the Hecke algebra we have isomorphisms: M M ⊕m ⊕m IC x x IC x x , Q(Ft ) ∼ Q(Fs ) ∼ = IC wI ⊕ = IC wI ⊕ x∈WI , x6=wI ,tx 0. Write v = us where s ∈ S, and assume the result is known for u. We distinguish two cases: Case 1: ws < w. In this case, w has a reduced expression w0 ending with s. It is clear from definitions that the restriction to Xw of the image under ∆BS of the morphism EF (w) → EF (w 0 ) induced by a rex move w w0 (see § 4.3) is invertible. Hence it is enough to prove the surjectivity of δw0 ,v ; in other words we can assume (replacing w by w0 if necessary) that w = xs for some reduced expression x (expressing ws). By

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157

induction, there exists a family (fi )i∈I of homogeneous elements in the image of γw,u whose image spans the vector space Hom•Db

(B)

(X≥w ,F) (EF (w), EF (u)),

and a family (gj )j∈J of homogeneous elements in the image of γx,u whose image spans the vector space Hom•Db

(B)

(X≥ws ,F) (EF (x), EF (u)).

By Proposition 9.8.1(2), there exist morphisms fi0 : EF (w) → EF (ws)[ni ],

gj0 : EF (w) → EF (w)[mj ]

such that the images of the compositions (fi ?B EF (s)[ni ]) ◦ fi0

and (gj ?B EF (s)[mj ]) ◦ gj0

span the vector space Hom•Db

(10.6.1)

(B)

(X≥w ,F) (EF (w), EF (v)).

By Lemma 10.6.4 we can assume that the morphisms fi0 are in the image of γw,ws , and that the morphisms gj0 are in the image of γw,w ; then we obtain that (10.6.1) is spanned by images of vectors in the image of γw,v , and the claim follows. Case 2: ws > w. By induction, there exists a family (fi )i∈I of homogeneous elements in the image of γw,u whose image spans the vector space Hom•Db

(B)

(X≥w ,F) (EF (w), EF (u)),

and a family (gj )j∈J of homogeneous elements in the image of γws,u whose image spans the vector space Hom•Db

(B)

(X≥ws ,F) (EF (ws), EF (u)[n]).

By Proposition 9.8.1(1), there exist morphisms fi0 : EF (w) → EF (ws)[ni ],

gj0 : EF (w) → EF (wss)[mj ]

such that the compositions (fi ?B EF (s)[ni ]) ◦ fi0

and (gj ?B EF (s)[mj ]) ◦ gj0

span the vector space (10.6.2)

Hom•Db

(B)

(X≥w ,F) (EF (w), EF (v)).

By Lemma 10.6.4 we can assume that the morphisms fi0 are in the image of γw,ws , and that the morphisms gj0 are in the image of γw,wss ; then we obtain that (10.6.2) is spanned by images of vectors in the image of γw,v , and the claim follows.

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10.7. The case of the affine flag variety We now explain the relation between the results stated in § 10.3 and the setting considered in Sections 3–4. We consider a connected reductive algebraic group G over an algebraically closed field F of characteristic p, and use the same notation as in § 3.1 (but do not assume that p ≥ h). We denote by G∨ the connected reductive group over L which is Langlands dual to G. That is to say, G∨ has a fixed maximal torus T ∨ ⊂ G∨ endowed with a fixed isomorphism X ∗ (T ∨ ) ∼ = X∗ (T ), such that the root datum of (G∨ , T ∨ ) identifies with the dual of the root datum of (G, T ). We also denote by B ∨ ⊂ G∨ the Borel subgroup whose roots are the negative coroots of G. Next, we denote by G∧ the simply connected cover of the derived subgroup of G∨ . In particular, we have a natural group morphism G∧ → G∨ , and we denote by T ∧ , resp. B ∧ , the inverse image of T ∨ , resp. B ∨ , under this morphism. Let K := L((z)) and O := L[[z]]. Then we can consider the ind-group scheme G∧ (K ) over L and its group subscheme G∧ (O). We define the Iwahori subgroup I ∧ ⊂ G∧ (O) as the inverse image of B ∧ under the evaluation morphism G∧ (O) → G∧ (at z = 0). We can similarly define G∨ (K ), G∨ (O) and I ∨ , and consider the affine flag varieties Fl∨ := G∨ (K )/I ∨ , Fl∧ := G∧ (K )/I ∧ and their natural ind-variety structure. The morphism G∧ → G∨ induces a closed embedding Fl∧ → Fl∨ which identifies Fl∧ with the connected component of the base point I ∨ /I ∨ in Fl∨ . For a detailed account of these constructions, see e.g., [37]. Both in the classical or étale setting of § 9.3, we can consider the equivariant derived category DIb∧ (Fl∧ , K). It is well-known that we have a “Bruhat decomposition” G Fl∧ = Fl∧ w w∈W

where is the I ∧ -orbit of the point of Fl∧ associated naturally with w, and that moreover Fl∧ w is isomorphic to an affine space of dimension `(w); see [37, Theorem 2.18]. We also have “Bott-Samelson varieties” BS(w), hence we can consider ∧ the corresponding objects EK (w) in DIb∧ (Fl∧ , K), and the category ParityBS I ∧ (Fl , K) ∧ ∧ of I -equivariant Bott-Samelson parity complexes on Fl . This category has a natural ∧ convolution product ?I , which makes it a monoidal category. Finally for any s ∈ S we have a corresponding partial affine flag variety Fl∧,s and a morphism Fl∧ → Fl∧,s ; see e.g., [65]. In case K is complete local, one can also consider the monoidal category of all ∧ I -equivariant parity complexes on Fl∧ , denoted ParityI ∧ (Fl∧ , K), which identifies ∧ with the Karoubi envelope of the additive hull of ParityBS I ∧ (Fl , K). The proof of the following theorem, which provides a geometric description of the category D used in Sections 4–5, is identical to the proof of Theorem 10.3.1 and Theorem 10.3.2. Here we set Z0 = Z if the morphisms α : ZΦ∨ → Z and α∨ : ZΦ → Z Fl∧ w

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159

are surjective for any simple root α, and Z0 = Z[ 21 ] otherwise. (In this statement, since we only consider the categories D and DBS , we do not need to assume that (4.2.1) is satisfied.) Theorem 10.7.1. — Assume that there exists a ring morphism Z0 → K. Then there exists a canonical equivalence of monoidal categories DBS

∼

∧ − → ParityBS I ∧ (Fl , K).

If K is furthermore a complete local ring, this equivalence induces an equivalence of additive monoidal categories D

∼

− → ParityI ∧ (Fl∧ , K).

Remark 10.7.2. — 1. Assume for simplicity that G is quasi-simple. Let Σ∨ be the set of simple coroots of G, which constitute a basis of the root system of (G∧ , T ∧ ). To avoid confusion, for α ∈ Φ∨ we will denote by α e ∈ Φ the cor∧ ∧ responding coroot of G . Let θ be the highest root of G , and let A be the generalized Cartan matrix with rows and columns parametrized by Σ∨ ∪ {0} and coefficients  hβ, α ei if α, β ∈ Σ∨ ;    2 if α = β = 0; aα,β = e  −hβ, θi if β ∈ Σ∨ and α = 0;    −hθ, α ei if α ∈ Σ∨ and β = 0. There exists a natural Kac-Moody root datum for A with underlying Z-module HomZ (ZΦ, Z), and with the simple roots and coroots defined in a way similar to the construction in § 4.2. It is likely that the Kac-Moody group over L associated with this root datum is the group ind-scheme G∧ (K ) considered above. This fact would make Theorem 10.7.1 an actual special case of Theorem 10.3.1 and Theorem 10.3.2. However, such a statement does not seem to be known. (See [65, § 9.h] for a comparison of the corresponding flag varieties.) 2. Assume again that G is quasi-simple. Instead of the “degenerate” Kac-Moody root datum of (1), one can consider the more traditional Kac-Moody root datum consisting of Λ = HomZ (ZΦ ⊕ Zc ⊕ Zd, Z), the simple roots Σ∨ ∪ {δ − θ} (where δ(d) = 1 and δ | = 0, and where Σ∨ is considered as a subset of Λ in the ZΦ⊕Zc e In § 1.4 we recalled the definition obvious way) and the simple coroots Σ∪{c− θ}. of the p-canonical basis of H using the category D. It might be more natural to define this p-canonical basis using the category Dk (G ), where G is the KacMoody group associated with Λ and the above roots and coroots. But in fact the two definitions coincide. Indeed, we have natural W -invariant morphisms ZΦ ⊕ Zc ⊕ Zd  ZΦ ⊕ Zd and ZΦ ,→ ZΦ ⊕ Zd, which allow to construct monoidal functors η1 : D0 → Dk (G ) and η2 : D0 → D,

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where D0 is the category from [31] associated with the natural realization of W with underlying k-module k⊗Z (ZΦ⊕Zd). Hence to prove our claim it suffices to prove that η1 and η2 send indecomposable objects to indecomposable objects. However, if we define the category D with the same objects as D and morphisms from M to N given by the degree-0 part of k ⊗O(k⊗Z ZΦ) Hom•D (M, N ), then D is a Krull-Schmidt category, and an object in D is indecomposable if and only if its image in D is indecomposable. Similar remarks apply to D0 and Dk (G ). This suffices to conclude since, by the double leaves theorem [31, Theorem 6.11], for any M, N in D0 the morphisms k ⊗O(k⊗Z (ZΦ⊕Zd)) Hom•D0 (M, N ) → k ⊗O(k⊗Z (ZΦ⊕Zc⊕Zd)) Hom•Dk (G ) (η1 (M ), η1 (N )) induced by η1 and k ⊗O(k⊗Z (ZΦ⊕Zd)) Hom•D0 (M, N ) → k ⊗O(k⊗Z ZΦ) Hom•D (η2 (M ), η2 (N )) induced by η2 are isomorphisms.

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CHAPTER 11 WHITTAKER SHEAVES AND ANTISPHERICAL DIAGRAMMATIC CATEGORIES

11.1. Definition of Whittaker sheaves Let us come back to the general setting of Sections 9–10. In § 10.3 we have obtained a description of the category DK (G ) in terms of parity complexes on a flag variety. The goal of this section is to obtain a similar description for the antispherical quotient defined in Remark 4.5.4. This relies on the consideration of some “Whittaker sheaves,” which exist only in the étale context. So, from now on we restrict to the étale context of § 9.3, assuming in addition that char(L) > 0, and that there exists a nontrivial (additive) character ψ from the prime subfield of L to K× (which we fix once and for all). Then we can consider the corresponding Artin-Schreier local system Lψ on Ga,L , see [12, 20, 2], i.e., the rank-1 local system defined as the ψ-isotypic component in the direct image of the constant sheaf under the Artin-Schreier map Ga,L → Ga,L defined by x 7→ xchar(L) − x. Let J ⊂ I be a subset of finite type. We will denote by JW ⊂ W the subset of elements v which are minimal in WJ v (i.e., the image of W J under w 7→ w−1 ). To J we have associated the parabolic subgroup PJ of G . We denote by U J the prounipotent radical of PJ (i.e., the base change to L of the group scheme denoted Uma+ J in [67]), and by LJ its Levi factor (i.e., the base change to L of the group scheme denoted G(J) in [67]), which is a connected reductive L-group. Finally, let UJ− be the unipotent radical of the Borel subgroup of LJ which is opposite to B ∩ LJ . Then the UJ− U J -orbits on X are parametrized by W (in the obvious way). For w ∈ W , we will denote by Xw,J the orbit corresponding to w, and by dJw its dimension. For any s ∈ {sj : j ∈ J} we have a root subgroup Us− ⊂ UJ− . Moreover, the natural embedding induces an isomorphism of algebraic groups Y ∼ Us− − → UJ− /[UJ− , UJ− ]. s∈J ∼

If we choose once and for all, for any s ∈ J, an isomorphism Ga,L − → Us− , we deduce a morphism of algebraic groups Y ∼ ∼ + UJ− U J → UJ− → UJ− /[UJ− , UJ− ] − → Us− − → (Ga,L )J − → Ga,L s∈J

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which we will denote χJ . The local system Lψ is multiplicative in the sense of [2, Appendix A], hence so is (χJ )∗ Lψ . We will denote by b DWhit,J (X , K)

the triangulated category of (UJ− U J , (χJ )∗ Lψ )-equivariant complexes on the variety X (see [2, Definition A.1]). Note that if w ∈ W , then the UJ− U J -orbit parametrized by w supports a (UJ− U J , (χJ )∗ Lψ )-equivariant local system iff w ∈ JW . In this case there exists a unique such local system of rank one (up to isomorphism), which we will denote LwJ , and we have dJw = `(w) + `(w0J ). We denote by iJw,◦ : Xw,J ,→ X the embedding, and set ∆Jw,◦ := (iJw,◦ )! LwJ [`(w) + `(w0J )],

∇Jw,◦ := (iJw,◦ )∗ LwJ [`(w) + `(w0J )].

By minimality, when w = 1 is the neutral element we have ∆J1,◦ ∼ = ∇J1,◦ , see [20, Corollary 4.2.2]. These objects satisfy ( K if w = v and n = 0; n J J HomDb (∆w,◦ , ∇v,◦ ) = Whit,J (X ,K) 0 otherwise. There exists a natural convolution functor b b b (−) ?B (−) : DWhit,J (X , K) × DB (X , K) → DWhit,J (X , K), b defined in a way similar to the convolution on DB (X , K). Then we can define the “averaging functor” b b AvJ : DB (X , K) → DWhit,J (X , K),

as the functor F 7→ ∆J1,◦ ?B F . (Alternatively, this functor can be described as the composition of the forgetful functor to the constructible derived category, followed by any choice of the functors defined as in [2, § A.2]; see [20, Lemma 4.4.3].) By construcb tion, for F , G in DB (X , K) there exists a canonical and functorial isomorphism AvJ (F ?B G ) ∼ = AvJ (F ) ?B G .

(11.1.1)

For s ∈ S, we can consider in a similar way the category b DWhit,J (X s , K).

The UJ− U J -orbits on X s are parametrized in a natural way by W s ; those which support a (UJ− U J , (χJ )∗ Lψ )-equivariant local system correspond to the elements in J

W◦s := {w ∈ W s | w ∈ JW and ws ∈ JW };

s,J we denote the corresponding standard and costandard sheaves ∆s,J w,◦ and ∇w,◦ respectively. We also have functors (q s )! = (q s )∗ , (q s )∗ , (q s )! between the (UJ− U J , (χJ )∗ Lψ )-equivariant categories.

ASTÉRISQUE 397

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11.2. WHITTAKER PARITY COMPLEXES

Lemma 11.1.1. — Let w ∈ JW . 1. If w ∈ W s and ws ∈ / JW , then (q s )! ∆Jw,◦ = (q s )! ∇Jw,◦ = 0. 2. If w ∈ W s and ws ∈ JW , then (q s )! ∆Jw,◦ ∼ = ∆s,J w,◦

and

(q s )! ∇Jw,◦ ∼ = ∇s,J w,◦ .

3. If ws < w (which implies that ws ∈ JW ), then (q s )! ∆Jw,◦ ∼ = ∆s,J ws,◦ [−1]

and

(q s )! ∇Jw,◦ ∼ = ∇s,J ws,◦ [1].

Proof. — Consider the restriction of q s to Xw,J . If w ∈ W s and ws ∈ / JW , then this 1 J restriction is an A -fibration, and the restriction of Lw to the fibers are isomorphic to Lψ , so that (q s )! ∆Jw,◦ = (q s )! ∇Jw,◦ = 0. If w ∈ W s and ws ∈ JW , then the restriction of q s is an isomorphism, so that s,J s,J s (q )! ∆Jw,◦ ∼ = ∆w,◦ and (q s )! ∇Jw,◦ ∼ = ∇w,◦ . Finally, if w ∈ / W s , then the restriction of q s to Xw,J is an A1 -fibration, and the s,J J restriction of Lw to the fibers are constant; it follows that (q s )! ∆Jw,◦ ∼ = ∆ws,◦ [−1] and s,J (q s )! ∇Jw,◦ ∼ = ∇ws,◦ [1]. Lemma 11.1.2. — Let w ∈ JW◦s . There exists distinguished triangles [1]

J ∆Jws,◦ [−1] → (q s )∗ ∆s,J w,◦ → ∆w,◦ −→

and

[1]

J ∇Jw,◦ → (q s )! ∇s,J w → ∇ws,◦ [1] −→ .

Proof. — These distinguished triangles are obtained by using the base change theorem and applying the standard distinguished triangles of functors attached to the decomposition of the inverse image under q s of the orbit corresponding to w as the disjoint union of the closed piece Xw,J and the open piece Xws,J . 11.2. Whittaker parity complexes The general theory of parity complexes developed in [41] applies verbatim in the b setting of Whittaker sheaves. In particular, a complex F in DWhit,J (X , K) is said to be ∗-even, resp. !-even, if for any w ∈ W we have

odd H (iJw,◦ )∗ F = 0, resp. H odd (iJw,◦ )! F = 0, and if moreover for w ∈ W and k even the local system

k J ∗ k J ! H (iw,◦ ) F , resp. H (iw,◦ ) F , is projective (equivalently, free) over K. (These conditions are automatic if w ∈ / JW J ∗ J ! b since in this case (iw,◦ ) F = (iw,◦ ) F = 0 for any F in DWhit,J (X , K).) Then an object F is called a parity complex if F ∼ = F0 ⊕ F1 where the objects F0 and F1 [1] are both ∗-even and !-even. We will denote by ParityWhit,J (X , K)

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b the additive category of parity complexes in DWhit,J (X , K). For any s ∈ S, we can similarly consider the category

ParityWhit,J (X s , K) b of parity complexes in DWhit,J (X s , K). b Lemma 11.2.1. — Let F ∈ DWhit,J (X , K) and s ∈ S.

1. If F is ∗-even, then (q s )∗ F is ∗-even. 2. If F is !-even, then (q s )∗ F is !-even. 3. If F is a parity complex, then (q s )∗ F is a parity complex. Proof. — Clearly, (3) follows from (1) and (2). We explain the proof of (1); the proof of (2) is similar. So, let us assume that F is ∗-even. We proceed by induction on the support of F ; so we assume that F is supported on a closed union Y ⊂ X of UJ− U J -orbits, and that w ∈ W is such that Xw,J is open in Y and (iJw,◦ )∗ F 6= 0. Necessarily we have w ∈ JW . Let k : Y \ Xw,J ,→ Y be the closed embedding, and consider the canonical distinguished triangle [1]

(iJw,◦ )! (iJw,◦ )! F → F → k∗ k ∗ F −→ . The object k∗ k ∗ F is ∗-even and supported on Y \ Xw,J , hence by induction we know that (q s )∗ k∗ k ∗ F is ∗-even. On the other hand, we have an isomorphism M
⊕ni (iJw,◦ )! (iJw,◦ )! F ∼ ∆Jw,◦ [i] , = i∈Z

where ni ∈ Z≥0 and ni = 0 unless dim(Xw,J ) + i is even. Now it follows from Lemma 11.1.1 that under this condition the object (q s )∗ (∆Jw,◦ [i]) is ∗-even, and the claim follows. Using arguments similar to those for Lemma 9.4.2, one can check that the functors (q ) (q s )∗ and (q s )∗ (q s )∗ are isomorphic up to cohomological shift by 2. (In our present setting Verdier duality exchanges the Whittaker categories defined in terms of ψ and ψ −1 ; but this does not cause any trouble.) We deduce the following. s !

b (X , K) is parity, then so are (q s )∗ (q s )∗ F and Lemma 11.2.2. — If F ∈ DWhit,J s ! s (q ) (q )∗ F .

Corollary 11.2.3. — For any E in ParityB (X , K) the object AvJ (E ) is a parity complex. Proof. — Since the functor (q s )∗ (q s )∗ commutes with AvJ (see (11.1.1)) and since any object in ParityB (X , K) is isomorphic to a direct sum of shifts of direct summands of objects obtained from ∆1 by repeated application of functors (q s )∗ (q s )∗ for various s, the corollary follows from the observation that AvJ (∆1 ) = ∆J1,◦ is parity (since ∆J1,◦ ∼ = ∇J1,◦ ) and Lemma 11.2.2.

ASTÉRISQUE 397

11.3. SECTIONS OF THE !-FLAG FOR WHITTAKER PARITY COMPLEXES

165

Remark 11.2.4. — For any w ∈ JW , the general theory of parity complexes of [41] guarantees the uniqueness (up to isomorphism) of an indecomposable parity complex supported on Xw,J and whose restriction to Xw,J is LwJ [dim(Xw,J )], but not its existence. This existence follows from Corollary 11.2.3: in fact AvJ (Ew ) is a parity complex supported on Xw,J and whose restriction to Xw,J is LwJ [dim(Xw,J )]; hence it admits an indecomposable direct summand whose restriction to Xw,J is LwJ [dim(Xw,J )]. Similar comments apply to the variety X s . Using this, as in Lemma 9.4.3 one can check that the functors (q s )∗ and (q s )! send parity complexes to parity complexes. Lemma 11.2.5. — If w ∈ W \ JW , then AvJ (Ew ) = 0. Proof. — Let w be a reduced expression for w starting with a simple reflection sj for some j ∈ J. Then Ew is isomorphic to a direct summand of Ew . On the other hand b (X , K) of an object in the equivariant derived category of the Ew is the image in DB parabolic subgroup Psj . Therefore we have AvJ (Ew ) = 0 (see e.g., [20, Lemma 4.4.6 and its proof]), which implies that AvJ (Ew ) = 0. 11.3. Sections of the !-flag for Whittaker parity complexes In this subsection we assume that K is a field. Our goal is to explain how to develop a theory of “sections of the !-flag” for Whittaker parity complexes. Since the constructions are close to those performed in Chapter 9, we leave most of the details to the reader. J : w ∈ JW ) the (normalized) indecomposable objects We will denote by (Ew,◦ s,J in the category ParityWhit,J (X , K) and by (Ew,◦ : w ∈ JW◦s ) the similar objects s in ParityWhit,J (X , K). Then one can define the sections of the !-flag for !-parity b b objects in DWhit,J (X , K) and DWhit,J (X s , K) in the obvious way. (In this case, one replaces the Bruhat order which appears in the condition for the morphisms ϕFπ by the order on W induced by inclusions of orbit closures in X or in X s ; the corresponding open ind-subvarieties of X will be denoted X≥w,J .) Next, we fix some split embeddings and projections s,J

Ew,◦

J s,J → (q s )∗ Ew,◦ → Ew,◦

for w ∈ JW◦s , and a split embedding and split projection J

Ews,◦

s,J → (q s )! Ew,◦ [−1],

s,J J (q s )∗ Ew,◦ [1] → Ews,◦

for w ∈ JW◦s respectively. This allows to obtain analogs of Proposition 9.6.2 and Proposition 9.7.2. Namely, if (Π, e, d, (ϕFπ )π∈Π ) is a section of the !-flag of b F ∈ DWhit,J (X , K), then one can define a section of the !-flag of (q s )∗ F as follows. 0 We set Π := {π ∈ Π | e(π)s ∈ JW }, and define e0 : Π0 → JW◦s as sending π to the (q s ) F shortest element among e(π) and e(π)s. Then we can define d0 and (ϕπ ∗ )π∈Π0 by the same procedure as in § 9.6. (The proof that this indeed defines a section of the !-flag of (q s )∗ F is similar to the proof of Proposition 9.6.2; details are left

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to the reader.) On the other hand, if (Π, e, d, (ϕFπ )π∈Π ) is a section of the !-flag b of F ∈ DWhit,J (X s , K), then one can define a section of the !-flag of (q s )! F by exactly the same procedure as in § 9.7. Using these constructions one can prove the following analog of Proposition 9.8.1. Proposition 11.3.1. — Let x and v be expressions, and assume that x is a reduced expression for some element x ∈ JW . Let also s ∈ S, and assume that xs ∈ JW . 1. Assume that x ∈ W s , so that xs is a reduced expression for xs ∈ JW . Let (fi )i∈I be a family of homogeneous elements in Hom•Db

Whit,J (X

,K) (AvJ (Ex ), AvJ (Ev ))

whose images span Hom•Db

Whit,J (X≥x,J ,K)

(AvJ (Ex ), AvJ (Ev )),

and let (gj )j∈J be a family of homogeneous elements in Hom•Db

Whit,J (X

,K) (AvJ (Exs ), AvJ (Ev ))

whose images span the vector space Hom•Db

Whit,J (X≥xs,J ,K)

(AvJ (Exs ), AvJ (Ev )).

Then there exist integers ni and morphisms fi0 : AvJ (Ex ) → AvJ (Exs )[ni ] (for i ∈ I) and integers mj and morphisms gj0 : AvJ (Ex ) → AvJ (Exss )[mj ] (for j ∈ J) such that the images of the compositions fi ?B Es [ni ]

f0

i AvJ (Ex ) −→ AvJ (Exs )[ni ] −−−−−−−→ AvJ (Evs )[ni + deg(fi )]

together with the images of the compositions gj0

gj ?B Es [mj ]

AvJ (Ex ) −→ AvJ (Exss )[mj ] −−−−−−−→ AvJ (Evs )[mj + deg(gj )] span the vector space Hom•Db

Whit,J (X≥x,J ,K)

(AvJ (Ex ), AvJ (Evs )).

2. Assume that x = ys for some expression y (which is automatically a reduced expression for xs). Let (fi )i∈I be a family of homogeneous elements in Hom•Db

Whit,J (X

,K) (AvJ (Ex ), AvJ (Ev ))

whose images span Hom•Db

Whit,J (X≥x,J ,K)

(AvJ (Ex ), AvJ (Ev )),

and let (gj )j∈J be a family of homogeneous elements in Hom•Db

Whit,J (X

,K) (AvJ (Ey ), AvJ (Ev ))

whose images span Hom•Db

Whit,J (X≥xs,J ,K)

ASTÉRISQUE 397

(AvJ (Ey ), AvJ (Ev )).

11.4. SURJECTIVITY

167

Then there exist integers ni and morphisms fi0 : AvJ (Ex ) → AvJ (Exs )[ni ] (for i ∈ I) and integers mj and morphisms gj0 : AvJ (Ex ) → AvJ (Ex )[mj ] (for j ∈ J) such that the images of the compositions fi ?B Es [ni ]

f0

i AvJ (Ex ) −→ AvJ (Exs )[ni ] −−−−−−−→ AvJ (Evs )[ni + deg(fi )]

together with the images of the compositions gj0

gj ?B Es [mj ]

AvJ (Ex ) −→ AvJ (Ex )[mj ] −−−−−−−→ AvJ (Evs )[mj + deg(gj )] span the vector space Hom•Db

Whit,J (X≥x,J ,K)

(AvJ (Ex ), AvJ (Evs )).

11.4. Surjectivity We drop the assumption that K is a field. Corollary 11.2.3 shows that the functor AvJ restricts to a functor ParityB (X , K) → ParityWhit,J (X , K); we will denote this restriction also by AvJ . Proposition 11.4.1. — For any E , F in ParityB (X , K), the morphism Hom•ParityB (X ,K) (E , F ) → Hom•ParityWhit,J (X ,K) (AvJ (E ), AvJ (F )) induced by the functor AvJ is surjective. Sketch of proof. — Using the Nakayama lemma and the appropriate analog of Lemma 10.2.2, it suffices to consider the case when K is a field, to which we restrict from now on. In this setting, the surjectivity can be proved by using once again the technique of the proof of § 5.5 and § 10.6. Namely, one can assume that E and F are of BottSamelson type, and then using adjunction that E = EK (∅) and F = EK (v) for some expression v. To treat this case we prove more generally that if w is a reduced expression for w ∈ JW and v is any expression, the composition Av

J J δw,v : Hom•ParityB (X ,K) (EK (w), EK (v)) −−→

Hom•ParityWhit,J (X ,K) (AvJ (EK (w)), AvJ (EK (v))) → Hom•ParityWhit,J (X≥w,J ,K) (AvJ (EK (w)), AvJ (EK (v))) (where the second map is induced by restriction) is surjective. For this we first consider J J J J as in Lemma 10.6.4 the case of morphisms δys,ys , δys,yss , δy,ys , δy,yss where y is a reduced expression for an element y ∈ JW◦s . Then we treat the general case by induction on `(v) as follows. If `(v) = 0 the claim is obvious. If `(v) > 0, we write v = us. If ws < w, then ws ∈ JW , and it suffices to treat the case when w = xs for some reduced expression x for ws. In this case we invoke Proposition 11.3.1(2) as in the proof of Proposition 10.6.3. If ws > w and ws ∈ JW , then we can invoke Proposition 11.3.1(1).

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Finally, it remains to consider the case when ws > w and ws ∈ / JW . In this case, s ! b since AvJ (EK (v)) is the image under (q ) of an object in DWhit,J (X s , K), its corestriction to a stratum Xx,J can be nonzero only if x and xs are both in JW . In J particular, its co-restriction to Xw,J vanishes, so that the codomain of δw,v vanishes. Hence there is nothing to prove in this case. Corollary 11.4.2. — If w ∈ JW , then AvJ (Ew ) is a nonzero indecomposable parity complex. Proof. — If w ∈ JW , then AvJ (Ew ) is supported on Xw,J , and its restriction to the orbit Xw,J is LwJ [dim(Xw,J )], see Remark 11.2.4. It is indecomposable by Proposition 11.4.1 and the fact that a quotient of a local ring is local.

11.5. Description of the antispherical diagrammatic category in terms of Whittaker sheaves By Corollary 11.2.3, we can define ParityBS Whit,J (X , K) as the full (but not strictly full) subcategory of ParityWhit,J (X , K) whose objects are the parity complexes AvJ (EK (w))[n] for w an expression and n ∈ Z. Using (11.1.1), we see that this category admits a natural action of the monoidal category ParityBS B (X , K) on the right. Moreover, using Remark 11.2.4, one can easily check that the Karoubi envelope of the additive hull of ParityBS Whit,J (X , K) is ParityWhit,J (X , K). On the other hand, consider the antispherical category Dasph,K BS,J (G ) defined as in K Remark 4.5.4, for the Coxeter group (W, S), its realization hG , and the subset {sj : j ∈ J} ⊂ S. We also denote by Dasph,K (G ) the Karoubi envelope of the additive hull J asph,K of DBS,J (G ). Theorem 11.5.1. — Assume that there exists a ring morphism Z0 → K. Then there exists a canonical equivalence of categories ∼

∆JBS : Dasph,K → ParityBS Whit,J (X , K) BS,J (G ) − BS which is compatible with the right actions of DK BS (G ) and ParityB (X , K) through the equivalence of Theorem 10.3.2. As a consequence, we obtain an equivalence of categories ∼

∆J : DJasph,K (G ) − → ParityWhit,J (X , K) compatible with the right actions of DK (G ) and ParityB (X , K) through the equivalence of Theorem 10.3.1.

ASTÉRISQUE 397

11.6. APPLICATION TO THE LIGHT LEAVES BASIS IN THE ANTISPH. CATEGORY

169

Proof. — We consider the diagram DK BS (G )

/ ParityBS B (X , K)

∆BS ∼





asph,K DBS,J (G )

AvJ

ParityBS Whit,J (X

, K),

where the left vertical arrow is the natural quotient functor, and the horizontal arrow is the equivalence of Theorem 10.3.2. Since Avχ (EK (w)) = 0 for any expression w starting with a simple reflection in {sj : j ∈ J} (see the proof of Lemma 11.2.5), and ∗ (∆J1,◦ , ∆J1,◦ [2]) = 0), AvJ (∆BS (f )) = 0 for any f ∈ (hK G ) (because HomParityBS Whit,J (X ,K) the composition AvJ ◦ ∆BS factors through a functor ∼

∆JBS : Dasph,K → ParityBS Whit,J (X , K) BS,J (G ) −

(11.5.1)

which is the natural bijection on objects. It follows from Theorem 10.3.2 and Proposition 11.4.1 that this functor induces surjections on morphisms, and what remains is to prove that these surjections are in fact isomorphisms. To prove this property, using adjunction (as e.g., in § 5.5), it suffices to consider morphisms (of any degree) from B w to B ∅ where w is any expression. In this case, the version of Proposition 4.5.1 in our present setting (see Remark 4.5.4) provides a finite generating family for the K-module Hom•Dasph,K (G ) (B w , B ∅ ). Now, as in Lemma 10.6.1, BS,J

one can prove that the K-module M Hom(AvJ (EK (w)), AvJ (EK (∅))[n]) n∈Z

is free, and compute its rank in terms of the antispherical module for the Hecke algebra of (W, S) associated with J (based on Lemma 11.1.1 and Lemma 11.1.2). The rank one obtains in this way is precisely the cardinality of the generating family of Hom•Dasph,K (G ) (B w , B ∅ ) considered above (by the appropriate generalization of BS,J

Lemma 4.1.1). Since any surjective morphism from a K-module generated by m elements to Km must be an isomorphism, we deduce the desired claim, and the first equivalence of the theorem. The second equivalence follows by taking the Karoubi envelope of the additive hull of each of these categories.

11.6. Application to the light leaves basis in the antispherical category As noticed in the course of the proof of Theorem 11.5.1, it follows from the constructions of the present section that the generating family for the K-module Hom•Dasph,K (G ) (B w , B ∅ ) considered in Proposition 4.5.1 actually forms a basis of this BS,J

K-module, when the étale derived category with coefficients in K makes sense. In particular, this implies that for any expressions w and v, the graded K-module Hom•Dasph,K (G ) (B w , B v ) is free. We conclude the book with the following lemma, which BS,J

allows to deduce that the same property holds for more general rings of coefficients.

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Lemma 11.6.1. — Let K and K0 be integral domains such that there exists a ring morphism Z0 → K, and consider a ring morphism ϕ : K → K0 . 1. Assume that ϕ is injective, and that the family of Proposition 4.5.1 forms a basis of Hom•Dasph,K0 (G ) (B w , B ∅ ) for coefficients K0 . Then the same property BS,J

holds for K. 2. Assume that K0 is free over K, and that the family of Proposition 4.5.1 forms a basis of Hom•Dasph,K (G ) (B w , B ∅ ) for coefficients K. Then the same property BS,J

holds for K0 .

Proof. — Consider the following diagram: K0 ⊗K Hom•DK

BS (G )

0

(Bw , B∅ )

∼



•

K ⊗K HomDasph,K (G ) (B w , B ∅ ) BS,J

/ Hom• K0 (Bw , B∅ ) (G ) D BS



•

HomDasph,K0 (G ) (B w , B ∅ ). BS,J

Here the upper horizontal morphism is as in (10.1.2), and the vertical arrows are induced by the respective quotient functors. From this diagram we deduce a natural surjective morphism K0 ⊗K Hom•Dasph,K (G ) (B w , B ∅ )  Hom•Dasph,K0 (G ) (B w , B ∅ ).

(11.6.1)

BS,J

BS,J

Now we can prove (1). In fact if the property is known for K0 then the surjection in (11.6.1) must be an isomorphism. Hence the image of our family is free (over K0 ) in K0 ⊗K Hom•Dasph,K (G ) (B w , B ∅ ), which implies that the family is free (over K) BS,J

in Hom•Dasph,K (G ) (B w , B ∅ ) since ϕ is injective. BS,J

Finally, we prove (2). Consider the K-basis of Hom•DK (G ) (Bw , B∅ ) constructed BS as in the proof of Lemma 5.7.2. The image of this basis in Hom•DK0 (G ) (Bw , B∅ ) is BS a K0 -basis of this K0 -module. As in the proof of Lemma 5.7.2, what we have to prove is that any homogeneous element in Hom•DK0 (G ) (Bw , B∅ ) which factors through BS an object Bx hki where x starts with a simple reflection sj with j ∈ J is a linear combination of the elements ϕe where e does not avoid W \ JW and the elements fi · ϕe . Choosing a basis for K0 over K, from the knowledge of this property over K one easily deduces the property over K0 , and the proof is complete. Using Lemma 11.6.1(2) one obtains that the property holds over any field admitting a ring morphism from Z0 , and then using (1) one deduces that it holds over any integral domain satisfying this condition. 11.7. Iwahori-Whittaker sheaves on the affine flag variety Let us consider the setting of § 10.7 (in the étale context), and let us assume in addition that char(L) > 0, that K satisfies (4.2.1), that there exists a ring morphism

ASTÉRISQUE 397

11.7. IWAHORI-WHITTAKER SHEAVES ON THE AFFINE FLAG VARIETY

171

Z0 → K (so that (4.2.2) is satisfied), and that there exists a nontrivial (additive) character ψ from the prime subfield of L to K× (which we fix once and for all). Then asph we can consider the categories DBS and D asph of § 4.4, with our present choice of ring of coefficients K. Let I◦∧ ⊂ G∧ (O) be the inverse image under the evaluation map G∧ (O) → G∧ of the unipotent radical of the Borel subgroup which is opposite to B ∧ (with respect to T ∧ ). Then after choosing isomorphisms between Ga,L and the appropriate root subgroups, we obtain a morphism χ : I◦∧ → Ga,L , and we can consider the corresponding category Parity(I◦∧ ,χ) (Fl∧ , K) of Whittaker parity complexes. (Of course, in this setting we can also consider more general parahoric subgroups, but we restrict to this special case for simplicity and concreteness.) The proof of the following result is very similar to the proof of Theorem 11.5.1; see the remarks in § 10.7. Theorem 11.7.1. — Assume that there exists a ring morphism Z0 → K. Then there exists a canonical equivalence of categories asph ∼

DBS

∧ − → ParityBS (I◦∧ ,χ) (Fl , K)

∧ which is compatible with the right actions of DBS and ParityBS I ∧ (Fl , K) through the first equivalence in Theorem 10.7.1. As a consequence, we obtain an equivalence of categories asph ∼ D − → Parity(I◦∧ ,χ) (Fl∧ , K)

compatible with the right actions of D and ParityI ∧ (Fl∧ , K) through the second equivalence in Theorem 10.7.1.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2018

LIST OF NOTATION

Chapter 2 Notation § AΩ 2.1 AΩ 2.1 Tilt(A ) 2.3

Description Subcategory of A generated by simples with label in Ω Quotient A /AΛ\Ω Category of tilting objects in A

Chapter 3 Notation k G T ⊂B h X, X+ Φ, Φ∨ Σ ⊂ Φ+ ρ Sf ⊂ Wf S⊂W CZ λ0 µs X+ 0 X+ s Rep(G) ∆(λ), ∇(λ) L(λ), T(λ) Rep0 (G) Reps (G) Ts , Ts Θs T(w)

§ 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.2 3.5 3.5

Description Algebraically closed field of characteristic p Reductive group with simply-connected derived subgroup Maximal torus and Borel subgroup Coxeter number Weight lattice and dominant weights Roots and coroots Simple and positive roots half sum of positive roots (Finite) Weyl group and simple reflections Affine Weyl group and simple reflections Intersection of X with fundamental alcove Fixed weight in CZ Fixed weight on the s-wall Intersection of X+ and W r λ0 Intersection of X+ and W r µs Category of finite-dimensional algebraic G-modules Standard and costandard modules in Rep(G) Simple and tilting modules in Rep(G) Serre subcategory generated by the L(λ)’s with λ ∈ X+ 0 Serre subcategory generated by the L(λ)’s with λ ∈ X+ s Fixed functors isomorphic to translation functors Wall-crossing functor Ts Ts Bott-Samelson-type tilting module

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174

LIST OF NOTATION

Chapter 4 Notation H , Hf Hw , H w Hw M asph

Nw , N w h R DBS , D Bw , Bw ı, τ Lw,e asph DBS , D asph Bw , Bw

§ 4.1 4.1 4.1 4.1 4.1 4.2 4.2 4.2 4.2 4.2 4.2 4.4 4.4

Description Hecke algebras of (W, S) and (Wf , Sf ) Standard and Kazhdan-Lusztig basis elements in H Bott-Samelson-type element in H Antispherical H -module Standard and Kazhdan-Lusztig basis elements in M asph Realization of (W, S) over k Symmetric algebra of h∗ Diagrammatic Hecke categories Bott-Samelson and indecomposable objects in D Autoequivalences of DBS Elements of the light leaves basis Antispherical diagrammatic Hecke categories Bott-Samelson and indecomposable objects in D asph

Chapter 5 Notation Ψ αx,y , βx,y DBS,R asph

DBS,R

§ 5.2 5.3 5.7 5.7

Description Functor in Theorem 5.2.1 Morphisms induced by the functor Ψ Version of DBS over R Version of D asph over R

Chapter 6 Notation bN gl K, d ei , hi , fi εi P αi natN mλ G B T χi ς

ASTÉRISQUE 397

§ 6.1 6.1 6.1 6.1 6.1 6.1 6.2 6.2 6.3 6.3 6.3 6.3 6.3

Description Affine Lie algebra associated with glN (C) bN Elements in gl bN Chevalley elements in gl Basis of the dual of diagonal matrices in glN (C) bN Weight lattice of gl bN Simple roots for gl bN Natural module for gl Basis of nat Group GLn (k) Borel subgroup of lower triangular matrices Maximal torus of diagonal matrices Standard basis of X = X ∗ (T ) Element in X

LIST OF NOTATION

V E, F η,  X Ea , Fa Rc (G) ın ∂i Hm Γ CΓ

tij Hm (Γ) Hm (Γ)−Mod0 b p) U (gl T ω µsj , µs∞

6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4

175

Natural representation of V Functors of tensoring with V and V ∗ Adjunction morphisms for (E, F ) Endomorphism of E and F Generalized eigenspaces of X on E and F Direct factor of Rep(G) associated with c ∈ X/(W, r ) Bijection between weights of ∧n nat and X/(W, r ) Demazure operators Degenerate affine Hecke algebra Prime ring of the base field, considered as a quiver Category of H m -modules supported on Γm Scalars in the definition of the KLR algebra KLR algebra Category of locally nilpotent modules for Hm (Γ) bp KLR 2-category attached to gl Endomorphism of EE Weight in P Choice of weights for G

Chapter 7 Notation Rep[n] (G) Ym b p) U+ (gl

§ 7.2 7.3 7.3

Description Subcategory of Rep(G) Subset of P b p) Quotient of U (gl

Chapter 8 Notation § Description [n] b b n) U (gln ) 8.1 Quotient of U (gl σ 8.1 Functor from DBS to endomorphisms of ω Chapter 9 Notation A, I (Λ, {αi }, {αi∨ }) W, S G , B, T X

§ 9.1 9.1 9.1 9.1 9.1

Description Generalized Cartan matrix and set of parameters Kac-Moody root datum Weyl group and simple reflections Kac-Moody group, Borel subgroup, maximal torus Flag variety of G

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LIST OF NOTATION

Xw , iw BS(w) νw PJ XJ XwJ , iJw qJ K b DB (X , K) b DB (X J , K) b D(B) (X , K) b D(B) (X J , K) ∇w , ∇Jw ∆w , ∆Jw µ ?B ParityB (X , K) Parity(B) (X , K) ParityB (X J , K) Parity(B) (X J , K) Ew Ew Ews

Υs

9.1 9.1 9.1 9.2 9.2 9.1 9.1 9.3 9.3 9.3 9.3 9.3 9.3 9.3 9.3 9.3 9.4 9.4 9.4 9.4 9.4 9.4 9.4 9.8

Bruhat cell and embedding Bott-Samelson resolution Morphism from BS(w) to X Parabolic subgroup attached to J ⊂ I Partial flag variety attached to J Bruhat cell in X J and embedding Projection X → X J ring of coefficients B-equivariant derived category of X B-equivariant derived category of X J B-constructible derived category of X B-constructible derived category of X J Costandard perverse sheaves on X and X J Standard perverse sheaves on X and X J Projection from G to X b Convolution action of DB (X , K) Category of B-equiv. parity complexes on X Category of B-const. parity complexes on X Category of B-equiv. parity complexes on X J Category of B-const. parity complexes on X J Bott-Samelson-type parity complex on X Indecomposable parity complex on X Indecomposable parity complex on X s b Functor on D(B) (X , F)

Chapter 10 Notation K DK BS (G ), D (G ) EK (w) ParityBS B (X , K) ∆, ∆BS H BsB γw,v , δw,v G∨ T ∨, B∨ G∧ T ∧, B∧ K,O I ∨, I ∧ Fl∨ , Fl∧

ASTÉRISQUE 397

§ 10.1 10.2 10.2 10.3 10.5 10.5 10.6 10.7 10.7 10.7 10.7 10.7 10.7 10.7

Description Diagrammatic categories attached to G Canonical Bott-Samelson parity complex on X Category of Bott-Samelson parity complexes on X Functors from diagrams to parity complexes Functor from parity complexes to R-bimodules Soergel bimodule attached to s (over Q) Morphisms induced by the functor ∆BS Reductive group Langlands-dual to G Maximal torus and Borel subgroup in G∨ S.-c. cover of the derived subgroup of G∨ Maximal torus and Borel subgroup in G∧ Laurent series and power series over L Iwahori subgroups for G∨ and G∧ Affine flag variety for G∨ and G∧

LIST OF NOTATION

Fl∧ w DIb∧ (Fl∧ , K) ∧ ?I ∧ ParityBS I ∧ (Fl , K) ParityI ∧ (Fl∧ , K)

10.7 10.7 10.7 10.7 10.7

177

Bruhat cell in Fl∧ I ∧ -equivariant derived category of Fl∧ Convolution bifunctor I ∧ -equ. Bott-Samelson parity compl. on Fl∧ I ∧ -equ. parity complexes on Fl∧

Chapter 11 Notation ψ

§ 11.1 Lψ 11.1 UJ 11.1 LJ 11.1 − UJ 11.1 Xw,J 11.1 χJ 11.1 b DWhit,J (X , K) 11.1 J Lw 11.1 ∆Jw,◦ 11.1 ∇Jw,◦ 11.1 AvJ 11.1 b DWhit,J (X s , K) 11.1 ParityWhit,J (X , K) 11.2 ParityWhit,J (X s , K) 11.2 J Ew,◦ 11.3 s,J Ew,◦ 11.3 ∆J , ∆JBS 11.5 I◦∧ 11.7 Parity(I◦∧ ,χ) (Fl∧ , K) 11.7

Description Nontrivial character of the prime subfield of F Artin-Schreier local system attached to ψ Pro-unipotent radical of PJ Levi factor of PJ Unip. rad. of the Borel subgr. opp. to LJ ∩ B UJ− U J -orbit on X Character of UJ− U J Whittaker derived category of X Local system on Xw,J b Standard perverse sheaf in DWhit,J (X , K) b Costandard perverse sheaf in DWhit,J (X , K) Averaging functor Whittaker derived category of X b (X , K) Parity complexes in DWhit,J b Parity complexes in DWhit,J (X s , K) Indec. Whittaker parity complexes on X Indec. Whittaker parity complexes on X Functors from diag. to Whittaker par. compl. Opposite Iwahori subgroup for G∧ Iwahori-Whittaker parity complexes on Fl∧

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2018

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ASTÉRISQUE 397

ASTÉRISQUE 2018 397. S. RICHE & G. WILLIAMSON – Tilting modules and the p-canonical basis

2017 396. 395. 394. 393. 392. 391. 390. 389. 388. 387. 386.

Y. SAKELLARIDIS & A. VENKATESH – Periods and harmonic analysis on spherical varieties V. GUIRARDEL & G. LEVITT – JSJ decompositions of groups J. XIE – The Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane G. BIEDERMANN, G. RAPTIS & M. STELZER – The realization space of an unstable coalgebra G. DAVID, M. FILOCHE, D. JERISON & S. MAYBORODA – A Free Boundary Problem for the Localization of Eigenfunctions S. KELLY – Voevodsky motives and l dh-descent SÉMINAIRE BOURBAKI, volume 2015/2016, exposés 1104-1119 S. GRELLIER & P. GÉRARD – The cubic Szegő equation and Hankel operators T. LÉVY – The master field on the plane R. M. KAUFMANN, B. C. WARD – Feynman Categories B. LEMAIRE, G. HENNIART – Représentations des espaces tordus sur un groupe réductif connexe p-adique

2016 385. A. BRAVERMAN, M. FINKELBERG & H. NAKAJIMA – Instanton moduli spaces and W -algebras 384. T. BRADEN, A. LICATA, N. PROUDFOOT & B. WEBSTER – Quantizations of conical symplectic resolutions 383. S. GUILLERMOU, G. LEBEAU, A. PARUSIŃSKI, P. SCHAPIRA & J.-P. SCHNEIDERS – Subanalytic sheaves and Sobolev spaces 382. F. ANDREATTA, S. BIJAKOWSKI, A. IOVITA, P. L. KASSAEI, V. PILLONI, B. STROH, Y. TIAN & L. XIAO – Arithmétique p-adique des formes de Hilbert 381. L. BARBIERI-VIALE & B. KAHN – On the derived category of 1-motives 380. SÉMINAIRE BOURBAKI, volume 2014/2015, exposés 1089-1103 379. O. BAUES & V. CORTÉS – Symplectic Lie groups 378. F. CASTEL – Geometric representations of the braid groups 377. S. HURDER & A. RECHTMAN – The dynamics of generic Kuperberg flows 376. K. FUKAYA, Y.-G. OH, H. OHTA & K. ONO – Lagrangian Floer theory and mirror symmetry on compact toric manifolds

2015 375. 374. 373. 372. 371. 370.

F. FAURE & M. TSUJII – Prequantum transfer operator for symplectic Anosov diffeomorphism T. ALAZARD & J.-M. DELORT – Sobolev estimates for two dimensional gravity water waves F. PAULIN, M. POLLICOTT & B. SCHAPIRA – Equilibrium states in negative curvature R. FRIGERIO, J.-F. LAFONT & A. SISTO – Rigidity of High Dimensional Graph Manifolds K. KEDLAYA & R. LIU – Relative p-adic Hodge theory: Foundations De la géométrie algébrique aux formes automorphes (II), J.-B. BOST, P. BOYER, A. GENESTIER, L. LAFFORGUE, S. LYSENKO, S. MOREL & B. C. NGO, éditeurs 369. De la géométrie algébrique aux formes automorphes (I), J.-B. BOST, P. BOYER, A. GENESTIER, L. LAFFORGUE, S. LYSENKO, S. MOREL & B. C. NGO, éditeurs 367-368. SÉMINAIRE BOURBAKI, volume 2013/2014, exposés 1074-1088

2014 366. J. MARTÍN, M. MILMAN – Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation 365. B. KLEINER, J. LOTT – Local Collapsing, Orbifolds, and Geometrization 363-364. L. ILLUSIE, Y. LASZLO & F. ORGOGOZO avec la collaboration de F. DÉGLISE, A. MOREAU, V. PILLONI, M. RAYNAUD, J. RIOU, B. STROH, M. TEMKIN et W. ZHENG – Travaux de Gabber

362. 361. 360. 359.

sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. (Séminaire à l’École polytechnique 2006–2008) M. JUNGE, M. PERRIN – Theory of Hp -spaces for continuous filtrations in von Neumann algebras SÉMINAIRE BOURBAKI, volume 2012/2013, exposés 1059-1073 J. I. BURGOS GIL, P. PHILIPPON, M. SOMBRA – Arithmetic Geometry of Toric Varieties. Metrics, Measures and Heights M. BROUÉ, G. MALLE, J. MICHEL – Split Spetses for Primitive Reflection Groups

2013 358. 357. 356. 355. 354. 353. 352. 351.

A. AVILA, J. SANTAMARIA, M. VIANA, A. WILKINSON – Cocycles over partially hyperbolic maps D. SCHÄPPI – The formal theory of Tannaka duality A. GETMANENKO, D. TAMARKIN – Microlocal properties of sheaves and complex WKB J.-P. RAMIS, J. SAULOY, C. ZHANG – Local Analytic Classification of q-Difference Equations S. CROVISIER – Perturbation de la dynamique de difféomorphismes en topologie C 1 N.-G. KANG, N. G. MAKAROV – Gaussian free field and conformal field theory SÉMINAIRE BOURBAKI, volume 2011/2012, exposés 1043-1058 R. MELROSE, A. VASY, J. WUNSCH – Diffraction of singularities for the wave equation on manifolds with corners 350. F. LE ROUX – L’ensemble de rotation autour d’un point fixe 349. J. T. COX, R. DURRETT, E. A. PERKINS – Voter model perturbations and reaction diffusion equations

2012 348. SÉMINAIRE BOURBAKI, volume 2010/2011, exposés 1027-1042 347. C. MŒGLIN, J.-L. WALDSPURGER – Sur les conjectures de Gross et Prasad, II 346. W. T. GAN, B. H. GROSS, D. PRASAD, J.-L. WALDSPURGER – Sur les conjectures de Gross et Prasad 345. M. KASHIWARA, P. SCHAPIRA – Deformation quantization modules 344. M. MITREA, M. WRIGHT – Boundary value problems for the Stokes system in arbitrary Lipschitz domains 343. K. BEHREND, G. GINOT, B. NOOHI, P. XU – String topology for stacks 342. H. BAHOURI, C. FERMANIAN-KAMMERER, I. GALLAGHER – Phase-space analysis and pseudodifferential calculus on the Heisenberg group 341. J.-M. DELORT – A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on S1

2011 340. 339. 338. 337. 336. 335.

T. MOCHIZUKI – Wild harmonic bundles and wild pure twistor D-modules SÉMINAIRE BOURBAKI, volume 2009/2010, exposés 1012-1026 G. ARONE, M. CHING – Operads and chain rules for the calculus of functors U. BUNKE, T. SCHICK, M. SPITZWECK – Periodic twisted cohomology and T-duality P. GYRYA, L. SALOFF-COSTE – Neumann and Dirichlet Heat Kernels in Inner Uniform Domains P. PELAEZ – Multiplicative Properties of the Slice Filtration

2010 334. 333. 332. 331.

J. POINEAU – La droite de Berkovich sur Z K. PONTO – Fixed point theory and trace for bicategories SÉMINAIRE BOURBAKI, volume 2008/2009, exposés 997-1011 Représentations p-adiques de groupes p-adiques III : méthodes globales et géométriques, L. BERGER, C. BREUIL, P. COLMEZ, éditeurs 330. Représentations p-adiques de groupes p-adiques II : représentations de GL2 (Qp ) et (ϕ, Γ)-modules, L. BERGER, C. BREUIL, P. COLMEZ, éditeurs 329. T. LÉVY – Two-dimensional Markovian holonomy fields

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In this book we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the antispherical quotient of the Hecke category. We prove our conjecture for GLn (k) using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding Kac-Moody group.