Unitary reflection groups and their associated hyperplane complements
have been shown recently (cf. [LS], [BM], [Lu]) to have empirical conn
ections with the representation theory of Hecke algebras and reductive
groups over finite fields. Against this backdrop we give in this work
a general formula ((2.2) below) for the trace of a linear transformat
ion g on the cohomology of the hyperplane complement M of a reflection
group G which it normalises. This formula involves a sum over the cen
traliser C-G(g). The main application (Theorem (5.9)) is an explicit f
ormula for the Poincare series of g (the transformation above) in the
case where g has an eigenspace which contains a G-regular vector. Othe
r significant applications are to the case when C-G(g) is cyclic (Theo
rem (3.3)) and when g is a unitary reflection (Theorem (4.1)). Another
application of our formula is to the proof of two reduction Theorems
((6.3) and (6.5)) which under certain conditions on C-G(g) reduce the
computation of P-M(g, t) to cases of lower rank. A by-product of our r
esults is an explicit formula (see (2.6) below) for the Mobius functio
n of L(g), the fixed points under g of the lattice L associated with t
he reflecting hyperplanes of G. In the course of proving these results
, some general facts concerning unitary reflection groups and their no
rmalisers emerge which may have some independent interest (cf. (1.5),(
5.1),(5.2), (5.6), (5.7), (5.8)).