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

TL;DR

This paper provides a combinatorial criterion for the surjectivity of restriction maps of invariant polynomials in reflection groups, and applies it to characterize normality of closures of regular decomposition classes in semisimple Lie algebras.

Contribution

It offers a simple combinatorial characterization of restriction map surjectivity for Coxeter groups and completes a classification of normality of decomposition class closures in Lie algebras.

Findings

Characterization of when the restriction map is surjective based on exponents.

Complete classification of normality of decomposition class closures in Weyl groups.

Application to semisimple Lie algebra decomposition classes.

Abstract

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.