Projective normality of Weyl group quotients

In this note, we prove that for the standard representation Vof the Weyl group W of a semi-simple algebraic group of type A _n , B _n , C _n , D _n , F ₄ and G ₂ over mathbbCmathbb{C}, the projective variety mathbbP(V^m)/Wmathbb{P}(V^m)/W is projectively normal with respect to the descent of O(1)^?|W|mathcal{O}(1)^{otimes |W|}, where V ^m denote the direct sum of m copies of V.