## Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie AlgebrasThis book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers. |

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### Contents

9 | |

Classical groups in good characteristic | 39 |

statement of results | 59 |

the symplectic and orthogonal cases p 2 | 65 |

Unipotent elements in symplectic and orthogonal groups p 2 | 91 |

Finite classical groups | 113 |

Tables of examples in low dimensions | 119 |

statement of results for nilpotent elements | 129 |

Nilpotent classes and centralizers in E8 | 219 |

Nilpotent elements in the other exceptional types | 263 |

statement of results for unipotent elements | 281 |

Corresponding unipotent and nilpotent elements | 287 |

Distinguished unipotent elements | 299 |

Nondistinguished unipotent classes | 317 |

Proofs of Theorems 1 2 and Corollaries 3 8 | 341 |

Tables of nilpotent and unipotent classes in the exceptional | 351 |

Reductive subgroups | 139 |

Annihilator spaces of nilpotent elements | 153 |

Standard distinguished nilpotent elements | 169 |

Exceptional distinguished nilpotent elements | 203 |

373 | |

379 | |

### Other editions - View all

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras Martin W. Liebeck,Gary M. Seitz No preview available - 2012 |

### Common terms and phrases

1-dimensional torus adjoint algebraic group annihilated points assume Borel subgroup CG(e Cg(T CG(u CL(e classical groups completes the proof component group conjugacy classes conjugate contains Corollary corresponding decomposition deﬁne denote dense orbit determined dim CG dim GL(Q)(e dim P/QZQ dimC dimension dimL distinguished nilpotent elements distinguished parabolic subgroup Dynkin diagram exceptional groups ﬁnd finite ﬁrst ﬁxed points ﬁxes follows from Lemma Frattini argument Frobenius morphism fundamental roots G I Sp(V Gg(e Gg(T graph automorphism Hence dim implies indecomposable involution irreducible isomorphic Jordan blocks labelling Lemma Let G I Levi subgroup Lie algebra maximal torus natural module nontrivial orthogonal groups P I QL P-linked product of root proof of Lemma result root elements root groups root subgroups Section semisimple simple algebraic group simple factor SO(V subgroup of G subsystem summand Suppose symplectic T-weight unipotent classes unipotent group unipotent radical vector weight space Write