Projective normality of Weyl group quotients

Abstract

In this note, we 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 $V$. We also prove that for any finite group $G$ and for any finite dimentional representation $V$ over $\mathbb{C}$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal{O}(1{)}^{\otimes n!}$ as a consequence.

10 pages