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Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras

Part of: Projective and enumerative geometry General commutative ring theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  19 March 2012

J. Matthew Douglass  and
Gerhard Röhrle
J. Matthew Douglass
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203, USA (email: [email protected])
Gerhard Röhrle
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (email: [email protected])
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Abstract

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Suppose that W is a finite, unitary, reflection group acting on the complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of W-invariant polynomial functions on V to the algebra of C-invariant functions on X. In this note we consider the special case when W is a Coxeter group, V is the complexified reflection representation of W, and X is in the lattice of the arrangement of W, and give a simple, combinatorial characterization of when the restriction mapping is surjective in terms of the exponents of W and C. As an application of our result, in the case when W is the Weyl group of a semisimple, complex Lie algebra, we complete a calculation begun by Richardson in 1987 and obtain a simple combinatorial characterization of regular decomposition classes whose closure is a normal variety.

Keywords

arrangementsCoxeter groupsdecomposition classesinvariants

MSC classification

Secondary: 20F55: Reflection and Coxeter groups 14N20: Configurations and arrangements of linear subspaces 13A50: Actions of groups on commutative rings; invariant theory
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 921 - 930
Copyright
Copyright © Foundation Compositio Mathematica 2012

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