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Centres of centralizers of unipotent elements in simple algebraic groups

About this Title

R. Lawther and D.M. Testerman

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 210, Number 988
ISBNs: 978-0-8218-4769-5 (print); 978-1-4704-0605-9 (online)
Published electronically: July 21, 2010
MathSciNet review: 2780340
MSC (2000): Primary 20G15, 20G41

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Table of Contents


  • 1. Introduction
  • 2. Notation and preliminary results
  • 3. Reduction of the problem
  • 4. Classical groups
  • 5. Exceptional groups: Nilpotent orbit representatives
  • 6. Associated cocharacters
  • 7. The connected centralizer
  • 8. A composition series for the Lie algebra centralizer
  • 9. The Lie algebra of the centre of the centralizer
  • 10. Proofs of the main theorems for exceptional groups
  • 11. Detailed results


Let be a simple algebraic group defined over an algebraically closed field whose characteristic is either or a good prime for , and let be unipotent. We study the centralizer , especially its centre . We calculate the Lie algebra of , in particular determining its dimension; we prove a succession of theorems of increasing generality, the last of which provides a formula for in terms of the labelled diagram associated to the conjugacy class containing . We proceed by using the existence of a Springer map to replace by a nilpotent element lying in the Lie algebra of . The bulk of the work concerns the cases where is of exceptional type. For these we produce a set of nilpotent orbit representatives and perform explicit calculations. For each such we obtain not only the Lie algebra of , but in fact the whole upper central series of the Lie algebra of ; we write each term of this series explicitly as a direct sum of indecomposable tilting modules for a reductive complement to in .

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