Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

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American Mathematical Soc., Jan 25, 2012 - Mathematics - 380 pages
This 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

Preliminaries
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

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