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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Elliptic centralizers in Weyl groups and their coinvariant representations
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by Mark Reeder PDF
Represent. Theory 15 (2011), 63-111 Request permission

Abstract:

The centralizer $C(w)$ of an elliptic element $w$ in a Weyl group has a natural symplectic representation on the group of $w$-coinvariants in the root lattice. We give the basic properties of this representation, along with applications to $p$-adic groups—classifying maximal tori and computing inducing data in $L$-packets—as well as to elucidating the structure of the centralizer $C(w)$ itself. We give the structure of each elliptic centralizer in $W(E_8)$ in terms of its coinvariant representation, and we refine Springer’s theory for elliptic regular elements to give explicit complex reflections generating $C(w)$. The case where $w$ has order three is examined in detail, with connections to mathematics of the nineteenth century. A variation of the methods recovers the subgroup $W(H_4)\subset W(E_8)$.
References
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Additional Information
  • Mark Reeder
  • Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
  • Email: reederma@bc.edu
  • Received by editor(s): June 9, 2009
  • Received by editor(s) in revised form: February 3, 2010
  • Published electronically: January 24, 2011
  • © Copyright 2011 American Mathematical Society
  • Journal: Represent. Theory 15 (2011), 63-111
  • MSC (2010): Primary 11E72, 20G05, 20G25
  • DOI: https://doi.org/10.1090/S1088-4165-2011-00377-0
  • MathSciNet review: 2765477