Restricting invariants of unitary reflection groups

TL;DR

This paper characterizes when the restriction map of invariant polynomials is surjective for finite unitary reflection groups, linking algebraic properties to geometric smoothness and normality of orbit varieties.

Contribution

It extends previous work by providing a complete characterization of the surjectivity of restriction maps for all finite unitary reflection groups based on exponents and arrangements.

Findings

Restriction map surjectivity characterized by exponents and arrangements.

Orbit variety smoothness equivalent to normality.

Generalization from Coxeter to all unitary reflection groups.

Abstract

Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is the fixed point subspace of an element of G. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. Extending earlier work by Douglass and Roehrle for Coxeter groups, we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups G in terms of the exponents of G and C, and their reflection arrangements. A consequence of our main result is that the variety of G-orbits in the G-saturation of X is smooth if and only if it is normal.