Contents
IX
Notes 71
Chapter 4. Highest Weight Modules I 73
§4.1. Simple Submodules of Verma Modules 74
§4.2. Homomorphisms between Verma Modules 75
§4.3. Special Case: Dominant Integral Weights 76
§4.4. Simplicity Criterion: Integral Case 77
§4.5. Existence of Embeddings: Preliminaries 78
§4.6. Existence of Embeddings: Integral Case 79
§4.7. Existence of Embeddings: General Case 81
§4.8. Simplicity Criterion: General Case 82
§4.9. Blocks of O Revisited 83
§4.10. Example: Antidominant Projectives 84
§4.11. Application to sl(3, C) 85
§4.12. Shapovalov Elements 86
§4.13. Proof of Shapovalov's Theorem 88
§4.14. A Look Back at Verma's Thesis 90
Notes 91
Chapter 5. Highest Weight Modules II 93
§5.1. BGG Theorem 93
§5.2. Bruhat Ordering 94
§5.3. Jantzen Filtration 95
§5.4. Example: sl(3,C) 97
§5.5. Application to BGG Theorem 98
§5.6. Key Lemma 98
§5.7. Proof of Jantzen's Theorem 100
§5.8. Determinant Formula 102
§5.9. Details of Shapovalov's Proof 103
Notes 106
Chapter 6. Extensions and Resolutions 107
§6.1. BGG Resolution of a Finite Dimensional Module 108
§6.2. Weak BGG Resolution 109
§6.3. Exactness of the Sequence 110
§6.4. Weights of the Exterior Powers 111
56.5. Extensions of Verma Modules 113
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