Tilting Modules and the p-canonical Basis

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