Remarks concerning Linear Characters of Reflection Groups

Let G be a finite group generated by unitary reflections in a Hermitian space V , and let ζ be a root of unity. Let E be a subspace of V , maximal with respect to the property of being a ζ-eigenspace of an element of G, and let C be the parabolic subgroup of elements fixing E pointwise. If χ is any linear character of G, we give a condition for the restriction of χ to C to be trivial in terms of the invariant theory of G, and give a formula for the polynomial ∑ x∈G χ(x)T d(x,ζ), where d(x, ζ) is the dimension of the ζ-eigenspace of x. Applications include criteria for regularity, and new connections between the invariant theory and the structure of G. 1. Background and notation In this section we recall some basic and well-known facts concerning reflection groups, most of which may be found in [OT]. 1.1. Invariant theory. Let G be a finite group generated by (pseudo)reflections in a complex vector space V of dimension > 0. For any linear transformation A : V−→ V and element ζ ∈ C, denote by V (A, ζ) the ζ-eigenspace of A, and by d(A, ζ) its dimension. It is well known that if S denotes the coordinate ring of V (identified with the symmetric algebra on the dual V ∗) the ring S of polynomial invariants of G is free; if f1, f2, . . . , f is a set of homogeneous free generators of S, then the degrees di = deg fi (i = 1, . . . , ) are determined by G and are called the invariant degrees of G. Let F be the ideal of S generated by the elements of S which vanish at 0 ∈ V . The space S/F := SG is called the coinvariant algebra of (G, V ). It is clearly graded, and for each i, the graded component (SG)i of degree i is a G-module. By a classical result of Chevalley, SG realises the regular representation of G, and we have an isomorphism of CG-modules: S ∼ −→ S ⊗C SG. We shall identify the G-module SG with the G-invariant complement HG of F in S comprising the harmonic polynomials, i.e., those polynomials f ∈ S which are annihilated by S(V ), regarded as differential operators on S = S(V ∗). Then Chevalley’s isomorphism may be interpreted as the assertion that S = S ⊗HG. For any (finite-dimensional) CG-module M , the space (S ⊗M∗)G = S ⊗ (HG ⊗M∗)G Received by the editors December 12, 2003 and, in revised form, June 8, 2004 and June 14, 2004. 2000 Mathematics Subject Classification. Primary 20F55; Secondary 14G05, 20G40, 51F15. c ©2005 American Mathematical Society Reverts to public domain 28 years from publication 3163