Cohomology actions and centralisers in unitary reflection groups. (Q2766430)

Let \(V\) be a finite-dimensional complex vector space. A reflection in \(V\) is a semi-simple automorphism with fixed-point subspace of codimension 1. A reflection group \(G\) is a finite group generated by reflections. Let \(M_G\) be the manifold obtained by removing from \(V\) the reflecting hyperplanes of \(G\). If \(g\in\text{GL}(V)\) normalizes \(G\), it acts on \(M_G\) and on the de Rham cohomology. The paper concerns the equivariant Poincaré polynomial NEWLINE\[NEWLINEP_G(g,t)=\sum_{i=0}^l\text{trace}(g,H^i(M_G,\mathbb{C}))\cdot t^i.NEWLINE\]NEWLINE The origin of the authors' interest in this polynomial is the representation theory of reductive groups over finite fields.