Bases of the Contact-order Filtration of Derivations of Coxeter Arrangements

In a recent paper, we constructed a basis for the contact-order filtration of the module of derivations on the orbit space of a finite real reflection group acting on an -dimensional Euclidean space. Recently M. Yoshinaga constructed another basis for the contact-order filtration. In this note we give an explicit formula relating Yoshinaga’s basis to the basis we constructed earlier. The two bases turn out to be equal (up to a constant matrix). 1. The setup and the main result Let V be an -dimensional Euclidean vector space with inner product I. Then its dual space V ∗ is equipped with the inner product I∗, which is induced by I. Let S be the symmetric algebra of V ∗ over R. Identify S with the algebra of polynomial functions on V . Let DerS be the S-module of R-linear derivations of S. When X1, · · · , X denote a basis for V ∗, the partial derivations ∂i := ∂/∂Xi with respect to Xi (1 ≤ i ≤ ) naturally form a basis for DerS over S. Let K be the field of quotients of S and DerK be the K-vector space of R-linear derivations of K. Then the partial derivations ∂i (1 ≤ i ≤ ) naturally form a basis for DerK over K. Let W be a finite irreducible orthogonal reflection group (a Coxeter group) acting on V . The Coxeter group W naturally acts on V ∗, S and DerS . The W invariant subring of S is denoted by R. Then it is classically known [1, V.5.3, Theorem 3] that there exist algebraically independent homogeneous polynomials P1, · · · , P ∈ R with degP1 ≤ · · · ≤ degP , which are called basic invariants, such that R = R[P1, · · · , P ]. The primitive derivation D ∈ DerR is characterized by DPi = { 1 for i = , 0 otherwise. Received by the editors June 25, 2002 and, in revised form, March 1, 2004. 2000 Mathematics Subject Classification. Primary 32S22. The author was partially supported by the Grant-in-aid for scientific research (Nos. 14340018 and 13874005), the Ministry of Education, Sports, Science and Technology, Japan. c ©2005 American Mathematical Society Reverts to public domain 28 years from publication