Remark

Under the hypotheses of Prop. 5, the subspaces T_p for $0 \leq p \leq s$ are the isotypical components of the linear representation U of W on T (Algebra, Chap. VIII, §3, no. 4). It follows (loc. cit., Prop. 11) that every vector subspace V of T stable under W is the direct sum of the subspaces $V \cap T_p$ for $0 \leq p \leq s$; moreover (loc. cit., Prop. 10), every endomorphism commuting with the operators U(w) for $w \in W$ leaves stable each of the T_p for $0 \leq p \leq s$ and induces on them a homothety for $1 \leq p \leq s$. In particular, the bilinear forms $\Phi$ on T invariant under W are the bilinear forms given by

where $\Phi_0$ is a bilinear form on T_0 and $a_p$ (for $1 \leq p \leq s$) is a real number: indeed, such a form $\Phi$ can be written uniquely in the form $(t, t') \mapsto (A(t)|t')$, where A is an endomorphism of T commuting with the U(w) for $w \in W$.