Projection of root systems

Abstract

Let $a$ be a real euclidean vector space of finite dimension and $\hat{I}\pounds$ a root system in $a$ with a basis $\hat{I}”$. Let $\hat{I}˜\subset \hat{I}”$ and $M=M_{\hat{I}}˜$ be a standard Levi of a reductive group $G$ such that $a_{\hat{I}}˜=a_M/a_G$. Let us denote $d$ the dimension of $a_{\hat{I}}˜$, i.e the cardinal of $\hat{I}”-\hat{I}˜$ and $\hat{I}{\pounds }_{\hat{I}}˜$ the set of all non-trivial projections of roots in $\hat{I}\pounds$. We obtain conditions on $\hat{I}˜$ such that $\hat{I}{\pounds }_{\hat{I}}˜$ contains a root system of rank $d$.

The computations by hand concerning the exceptional groups were incomplete and contained a few errors leading to incorrect results in some cases