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

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

2010-07-08

arxiv.org/abs/1007.1350v6

Mathematics

Representation Theory

11 pages revised 2/2012; to appear in Compos. Math

Scientific paper

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.

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