Let
R =
Q[
x1,
x2, . . . ,
xn] be the ring of polynomials in the variables
x1,
x2, . . . ,
xn. Let
WB be the finite reflection group of type
Bn, let
IB be a basic set of invariants of
WB, and let
R*
B, denote the quotient of
R by the ideal generated by
IB. It is well known (see [Macdonald, 1991]) that the action of
WB on the quotient ring
R*
B, viewed as a vector space over
R, is isomorphic to the left regular representation of
WB. Using methods similar to those in [Allen, 1992, 1993] we construct a basis
of
R*
B which exhibits the decomposition of
R*
B into its irreducible components. Now let
WD be the finite reflection group of type
Dn, let
ID be a basic set of invariants for
WD, and let
R*
D be the quotient of
R with the ideal generated by
ID. We will show that the basis
has the remarkable property that when restricted to
R*
D, exactly one-half of the elements of
are non-zero and the non-zero polynomials
D form a basis for
R*
D. The action of
WD on the quotient ring
R*
D, viewed as a vector space of
R, is also isomorphic to the left regular representation of
WD (see [Macdonald, 1991]). This collection of polynomials
D gives the decomposition of
R*
D into its irreducible components when
n is odd. A slight modification of
D gives a basis for the decomposition of
R*
D when
n is even. We use these bases to construct the respective graded characters of
R*
B and
R*
D.
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doi:10.1006/jabr.1999.7875 
