Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras. (Q2894209)

Let \(W\) be a finite complex reflection group acting on the vector space \(V=\mathbb C^l\), \(X\) a subset of \(V\), \(N_X\) denote the setwise stabilizer of \(X\), \(Z_X\) its pointwise stabilizer, and let \(C_X=N_X/Z_X\). Then restriction defines a homomorphism \(\rho\) from the algebra of \(W\)-invariant polynomial functions on \(V\) to the algebra of \(C_X\)-invariant functions on \(X\).NEWLINENEWLINE The authors 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 \(\mathcal A\) of the reflection hyperplanes of \(W\). The main result of the paper is a combinatorial criterion for surjectivity of the map \(\rho\) in terms of the exponents of \(W\) and \(C_X\). In the course of the proof the authors obtain a complete list of finite irreducible Coxeter groups \(W\) for which the conditions of the criterion are satisfied.NEWLINENEWLINE As an application, the authors consider the case when \(W\) is the Weyl group of a complex semisimple Lie algebra \(\mathfrak g\). Using a theorem of \textit{R. W. Richardson} [Lect. Notes Math. 1271, 243-264 (1987; Zbl 0632.14011)] and the main result of the present paper they give a complete classification of the regular decomposition classes in \(\mathfrak g\) whose closure is a normal variety.