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Representation Theory
Representation Theory
ISSN 1088-4165

 

Elliptic centralizers in Weyl groups and their coinvariant representations


Author: Mark Reeder
Journal: Represent. Theory 15 (2011), 63-111
MSC (2010): Primary 11E72, 20G05, 20G25
Published electronically: January 24, 2011
MathSciNet review: 2765477
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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)$.


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Additional Information

Mark Reeder
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
Email: reederma@bc.edu

DOI: http://dx.doi.org/10.1090/S1088-4165-2011-00377-0
PII: S 1088-4165(2011)00377-0
Received by editor(s): June 9, 2009
Received by editor(s) in revised form: February 3, 2010
Published electronically: January 24, 2011
Article copyright: © Copyright 2011 American Mathematical Society