Projection of root systems and the generalized injectivity conjecture for exceptional groups

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 $d$. When considering the case of $\hat{I}\pounds$ of type exceptional, we give a list of all exceptional root systems that can occur in $\hat{I}{\pounds }_{\hat{I}}˜$ and use it to prove the generalized injectivity conjecture in most exceptional groups cases.

The first part of this article has a substantial overlap with arXiv:1904.01884. Comments welcome!