LECTURES ON CANONICAL AND CRYSTAL BASES OF HALL ALGEBRAS

LECTURES ON CANONICAL AND CRYSTAL BASES OF HALL ALGEBRAS OLIVIER SCHIFFMANN Contents Introduction 2 Lecture 1. 6 1.1. Recollections on quivers. 6 1.2. Moduli spaces of representations of quivers. 7 1.3. The induction and restriction functors. 9 1.4. The Lusztig sheaves and the Hall category. 16 1.5. The geometric pairing on the Hall category. 19 Lecture 2. 22 2.1. The simplest of all quivers. 22 2.2. The fundamental relations. 23 2.3. Finite type quivers. 27 2.4. The Jordan quiver and the cyclic quivers. 30 2.5. Affine quivers. 33 Lecture 3. 40 3.1. The graded Grothendieck group of the Hall category. 40 3.2. Relation to quantum groups. 42 3.3. Proof of Lusztig's theorem (finite type). 45 3.4. Fourier-Deligne transform. 46 3.5. Proof of Lusztig's theorem. 51 3.6. The Lusztig graph. 55 3.7. The trace map and purity. 57 Lecture 4. 62 4.1. Kashiwara crystals. 62 4.2. Lusztig's Lagrangian. 67 4.3. Hecke correspondences. 70 4.4. Geometric construction of the crystal. 72 4.5. Relationship to the Hall category. 75 4.6. Relationship to the Lusztig graph. 76 Lecture 5. 79 5.1. Moduli stacks of coherent sheaves on curves. 79 5.2. Convolution functors and the Hall category. 82 5.3.

• canonical bases has


• lusztig's nilpotent


• called canonical


• sheaves


• moduli spaces


• kashiwara's crystal graph


• perverse sheaves generating


• lusztig's theorem