Proposition 4

Let $\Gamma$ be the Casimir element of G.

a) If $\tau$ is an irreducible representation of G$,\widetilde{\Gamma}(\tau )$ is real and positive. If $\tau$ is not the trivial representation, $\widetilde{\Gamma}(\tau )>0$.

b) There exists a unique quadratic form $Q_{\Gamma}$ on $X(T)\otimes \mathbf{R}$ such that, for every irreducible representation $\tau$ of G,

where $\lambda$ is the highest weight of $\tau$. The form $Q_{\Gamma}$ is positive, separating and invariant under W.

Let F be a separating negative symmetric bilinear form on $\mathfrak{g}$ defining $\Gamma$. Let $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ be an irreducible representation of G; let $\langle ,\rangle$ be a Hilbert scalar product on V invariant under G (§1, no. 1), and $(e_i)$ a basis of $\mathfrak{g}$ such that $F(e_i, e_j) =-\delta_{ij}$. Then, for every element $v$ of V not invariant under G we have

hence a).

Let B be the inverse form on $\mathfrak{t}^*_{\mathbf{C}}$ of the restriction to $\mathfrak{t}_{\mathbf{C}}$ of the bilinear form on $\mathfrak{g}_{\mathbf{C}}$ induced by F by extension of scalars. By the Cor. of Prop. 7 of Chap. VIII, §6, no. 4, we have$^5\widetilde{\Gamma}(\tau ) = B(\delta (\lambda ), \delta (\lambda +2\rho ))$. Extend $\delta : X(T)\rightarrow$ $\mathfrak{t}^*_{\mathbf{C}}$ to an $\mathbf{R}$-linear map from $X(T)\otimes \mathbf{R}$ to $\mathfrak{t}^*_{\mathbf{C}}$ and let $Q_{\Gamma}$ be the quadratic form $x \rightarrow B(\delta (x), \delta (x))$ on $X(T)\otimes \mathbf{R}$; it is separating and invariant under W, and

We show that the form $Q_{\Gamma}$ is positive. Indeed, if $x\in X(T)\otimes \mathbf{Q}$, the element $\delta (x)$ of $\mathfrak{t}^*_{\mathbf{C}}$ takes purely imaginary values on $\mathfrak{t}$, hence real values on $i\mathfrak{t}$; we conclude by remarking that, for $y\in i\mathfrak{t}$, we have $F(y, y)\geq 0$.

It remains to prove the uniqueness assertion in b). Let Q be a quadratic form on $X(T)\otimes \mathbf{R}$ satisfying the required condition, and let $\Phi$ (resp. $\Phi_{\Gamma}$) be the bilinear form associated to Q (resp. $Q_{\Gamma}$). For $\lambda , \mu\in X_{++}$, we have

Since $X(T)_{++}$ generates the $\mathbf{R}$-vector space $X(T)\otimes \mathbf{R}$, we have $\Phi =\Phi_{\Gamma}$, so $Q = Q_{\Gamma}$.