Proposition 6

The assumptions and notation are those of Th. 4. Let H be the set of pseudo-reflections belonging to G and distinct from 1. Assume that H generates G. For all g $\in$ G, put $g(x) = x + f_g(x)e_g$ with $e_g \in V$, $f_g \in V^*$. Put

(i) The different of S over R is the principal ideal SD.

(ii) Identify S with the algebra $K[X_1, \ldots, X_l]$ by choosing a basis $(X_1, \ldots, X_l)$ of V, and let $P_1, \ldots, P_l$ be algebraically independent homogeneous elements generating the algebra R. Then the jacobian $J = \det \left( \frac{\partial P_i}{\partial X_j} \right)$ is of the form $\lambda D$ where $\lambda \in K^*$.

(iii) $\sum_{i=1}^l (\deg(P_i) - 1) = \mathrm{Card}(H)$.

(iv) The set of anti-invariant elements of S is RD.

Assertion (i) follows from Lemma 4. Assertion (ii) follows from the fact that SJ is the different of S over R. (Commutative Algebra). Assertion (iii) is obtained by writing down the fact that the homogeneous polynomials D and J are of the same degree. The proof of (iv) is then the same as that given in no. 4 (proof of Prop. 5, parts b) and d)).