Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement

TL;DR

The paper develops a comprehensive framework to study invariants and semi-invariants in the cohomology $H^*(M({\mathscr A}))$ of the complement of reflection arrangements, in the presence of a finite group $G$ acting by automorphisms. By reducing to irreducible reflection pairs and employing the Orlik–Solomon algebra, Brieskorn’s Lemma, and Euler-vector-field contractions, the authors construct explicit bases for $H^*(M({\mathscr A}))^G$ across irreducible cases, and derive Poincaré polynomials that encode invariant dimensions. A key advance is a Felder–Veselov–style construction that yields a basis compatible with Coxeter-group intuition and extends to all finite complex reflection groups, enabling a case-by-case yet principled determination of top-degree invariants. These results not only refine invariant descriptions but also simplify existing cohomology computations (Lehrer, Callegaro–Marin, Marin) and lead to applications in Lehrer's relative equivariant Poincaré polynomials and determinant-like character vanishing. Collectively, the work deepens understanding of how reflection symmetries constrain the cohomology of hyperplane complements and provides practical tools for explicit invariant calculus in the broader landscape of complex reflection groups.

Abstract

Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^*( . ) denotes rational singular cohomology, in the case when A is a reflection arrangement and the pair (A,G) arises from a reflection coset. Our main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to leading to a proof of the description of the space of invariants conjectured by Felder and Veselov for Coxeter groups that does not rely on computer calculations, this construction provides an extension of this description of the space of invariants to arbitrary finite, complex reflection groups. The main result also leads to simplifications of some cohomology computations of Lehrer, Callegaro-Marin, and Marin.

Theorems & Definitions (33)

• Proposition 2.5
• proof
• Proposition 2.9
• proof
• Lemma 3.3
• proof
• Lemma 3.6
• proof
• Proposition 3.8
• proof