We prove that for any finite dimensional vector space $V$ over an algebraically closed field $K$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the order $|G|$ is a unit in $K$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes |G|}$, where $\mathcal{O}(1)$ denotes the ample generator of the Picard group of $\mathbb P(V)$. We also prove that for the standard representation $V$ of 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 $\mathbb{C}$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V.$

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Last updated: 29 Mar 2023