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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 2011: Volume 210, Number 988
ISBNs: 978-0-8218-4769-5 (print); 978-1-4704-0605-9 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00594-6
Published electronically: July 21, 2010
MathSciNet review: 2780340
MSC (2000): Primary 20G15, 20G41

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Table of Contents


Chapters

  • 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

Abstract


Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ whose characteristic is either $0$ or a good prime for $G$, and let $u\in G$ be unipotent. We study the centralizer $C_G(u)$, especially its centre $Z(C_G(u))$. We calculate the Lie algebra of $Z(C_G(u))$, in particular determining its dimension; we prove a succession of theorems of increasing generality, the last of which provides a formula for $\dim Z(C_G(u))$ in terms of the labelled diagram associated to the conjugacy class containing $u$. We proceed by using the existence of a Springer map to replace $u$ by a nilpotent element lying in the Lie algebra of $G$. The bulk of the work concerns the cases where $G$ is of exceptional type. For these we produce a set of nilpotent orbit representatives $e$ and perform explicit calculations. For each such $e$ we obtain not only the Lie algebra of $Z(C_G(e))$, but in fact the whole upper central series of the Lie algebra of $R_u(C_G(e))$; we write each term of this series explicitly as a direct sum of indecomposable tilting modules for a reductive complement to $R_u(C_G(e))$ in $C_G(e)^\circ $.

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