x Contents
§3. Example: The diamond Lie algebra g 126
3.1. The coadjoint orbits for g 126
3.2. Representations corresponding to generic orbits 128
3.3. Representations corresponding to cylindrical orbits 131
§4. Amendments to other rules 132
4.1. Rules 3–5 132
4.2. Rules 6, 7, and 10 134
Chapter 5. Compact Lie Groups 135
§1. Structure of semisimple compact Lie groups 136
1.1. Compact and complex semisimple groups 137
1.2. Classical and exceptional groups 144
§2. Coadjoint orbits for compact Lie groups 147
2.1. Geometry of coadjoint orbits 147
2.2. Topology of coadjoint orbits 155
§3. Orbits and representations 161
3.1. Overlook 161
3.2. Weights of a unirrep 164
3.3. Functors Ind and Res 168
3.4. Borel-Weil-Bott theorem 170
3.5. The integral formula for characters 173
3.6. Infinitesimal characters 174
§4. Intertwining operators 176
Chapter 6. Miscellaneous 179
§1. Semisimple groups 179
1.1. Complex semisimple groups 179
1.2. Real semisimple groups 180
§2. Lie groups of general type 180
2.1. Poincar´ e group 181
2.2. Odd symplectic groups 182
§3. Beyond Lie groups 184
3.1. Infinite-dimensional groups 184
3.2. p-adic and adelic groups 188
3.3. Finite groups 189
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