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

Abstract Suppose V is a finite dimensional, complex vector space, $${\mathscr {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a finite set of codimension one subspaces of V , and G is a finite subgroup of the general linear group $$\operatorname {GL}(V)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>GL</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that permutes the hyperplanes in $${\mathscr {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . In this paper we study invariants and semi-invariants in the graded $${\mathbb {Q}}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> -module $$H^*(M({\mathscr {A}}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $$M({\mathscr {A}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> denotes the complement in V of the hyperplanes in $${\mathscr {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> and $$H^*(\,\cdot \,)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mspace/> <mml:mo>·</mml:mo> <mml:mspace/> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> denotes rational singular cohomology, in the case when $${\mathscr {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a reflection arrangement and the pair $$({\mathscr {A}},G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> arises from a reflection coset. The main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, $$H^*(M({\mathscr {A}}))^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:msup> </mml:mrow> </mml:math> . 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.