Chapter 1. Introduction ......................................... 1
Acknowledgements ............................................. 8
Chapter 2. Shifted Yangians ..................................... 9
2.1. Generators and relations ................................ 9
2.2. PBW theorem ............................................ 10
2.3. Some automorphisms ..................................... 12
2.4. Parabolic generators ................................... 13
2.5. Hopf algebra structure ................................. 18
2.6. The center of Yn(σ) .................................... 20
Chapter 3. Finite W-algebras ................................... 23
3.1. Pyramids ............................................... 23
3.2. Finite W-algebras ...................................... 24
3.3. Invariants ............................................. 25
3.4. Finite W-algebras are quotients of shifted Yangians .... 26
3.5. More automorphisms ..................................... 27
3.6. Miura transform ........................................ 28
3.7. Vanishing of higher Ti,j(r),S ............................ 30
3.8. Harish-Chandra homomorphisms ........................... 32
Chapter 4. Dual canonical bases ................................ 35
4.1. Tableaux ............................................... 35
4.2. Dual canonical bases ................................... 37
4.3. Crystals ............................................... 40
4.4. Consequences of the Kazhdan-Lusztig conjecture ......... 41
Chapter 5. Highest weight theory ............................... 47
5.1. Admissible modules ..................................... 47
5.2. Gelfand-Tsetlin characters ............................. 47
5.3. Highest weight modules ................................. 49
5.4. Classification of admissible irreducible
representations ........................................ 50
5.5. Composition multiplicities ............................. 51
Chapter 6. Verma modules ....................................... 53
6.1. Parametrization of highest weights ..................... 53
6.2. Characters of Verma modules ............................ 55
6.3. The linkage principle .................................. 57
6.4. The center of W(π) ..................................... 59
6.5. Proof of Theorem 6.2 ................................... 61
Chapter 7. Standard modules .................................... 67
7.1. Two rows ............................................... 67
7.2. Classification of finite dimensional irreducible
representations ........................................ 71
7.3. Tensor products ........................................ 72
7.4. Characters of standard modules ......................... 77
7.5. Grothendieck groups .................................... 78
Chapter 8. Character formulae .................................. 81
8.1. Skryabin's theorem ..................................... 81
8.2. Tensor identities ...................................... 82
8.3. Translation functors ................................... 85
8.4. Translation commutes with duality ...................... 89
8.5. Whittaker functor ...................................... 95
Notation ...................................................... 103
Bibliography .................................................. 105
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