Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formula for the equivariant Poincaré polynomial of a linear transformation g which normalises a unitary reflection group G, acting on the cohomology of the corresponding hyperplane complement. This formula involves a certain function (called a Z‐function below) on the centraliser CG(g), which was proved to exist only in certain cases, for example, when g is a reflection, or is G‐regular, or when the centraliser is cyclic. In this work we prove the existence of Z‐functions in full generality. Applications include reduction and product formulae for the equivariant Poincaré polynomials. The method is to study the poset L(CG(g)) of subspaces which are fixed points of elements of CG(g). We show that this poset has Euler characteristic 1, which is the key property required for the definition of a Z‐function. The fact about the Euler characteristic in turn follows from the ‘join‐atom’ property of L(CG(g)), which asserts that if [X1,…, Xk} is any set of elements of L(CG(g)) which are maximal (set theoretically) then their setwise intersection ⋂i=1kXi lies in L(CG(g)). 2000 Mathematical Subject Classification: primary 14R20, 55R80; secondary 20C33, 20G40.