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Let G be a finite, complex reflection group and f its discriminant polynomial. The fibers of f admit commuting actions of G and a cyclic group. The virtual $G\times C_m$ character given by the Euler characteristic of the fiber is a refinement of the zeta function of the geometric monodromy, calculated in a paper of Denef and Loeser. We compute the virtual character explicitly, in terms of the poset of normalizers of centralizers of regular elements of G, and of the subspace arrangement given by proper eigenspaces of elements of G. As a consequence, we compute orbifold Euler characteristics and find some new "case-free" information about the discriminant., Comment: 18 pages

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function l(W) associated to a given generator set S is completely described by the function n(W), introduced in [BM], associated to the root system of W. In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space HW in terms of Demazure operators, and show that HW is isomorphic to the coinvariant algebra SW, which enables us to define a homogeneous basis on SW parametrized by w ϵ W.

In the first part of this paper we study normal forms of elements of the imprimitive complex reflection group G(e,1,n). This allows to prove a conjecture of Broue on basis elements and the canonical symmetrizing form of the associated cyclotomic Hecke algebra. Secondly we introduce a root system for G(e,1,n) and study the associated length function. This has many properties in common with the usual length function for finite Weyl groups.