POINCARE POLYNOMIALS FOR UNITARY REFLECTION GROUPS

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)).