Let $G$ a semisimple group over an algebraically closed field $k$. We assume that $G$ is classical.

We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by a torus $Z$ and such that his derived group is simply connected.

Can we find a $z$-extension $\tilde{G}$ of $G$ such that it admits minuscule weights?

`$G$`

automaticaly lift to an extension group; so it's unclear what the problem is. – Jim Humphreys Feb 8 '13 at 11:49`$A_\ell, D_\ell$`

for instance). And the exceptional types`$E_6, E_7$`

do have minuscule weights. What is your source? – Jim Humphreys Feb 9 '13 at 14:39