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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?

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This needs a more careful formulation. For instance, why restrict attention just to classical groups? More important, all representations (here presumably meant to be finite dimensional) of $G$ automaticaly lift to an extension group; so it's unclear what the problem is. – Jim Humphreys Feb 8 at 11:49
I restrict for classical groups, because for exceptional types, there is not minuscule weights. For instance, if I take $PGL_{n}$ there is no minuscule irréducible representations, but for $GL_{n}$ yes – prochet Feb 8 at 18:32
the comment is in two part, the example is to answer your second question. I used classical types, because we now that in each type, the simply connected group associated to it has at least one minuscule representations. For exceptional type, a group of type $G_{2}$ doesn't have minuscule representations. – prochet Feb 8 at 18:36
and in fact, my question is rather for minuscule coweights, – prochet Feb 8 at 19:30
The original question still doesn't make sense to me. Also, your notion of "minuscule" doesn't agree with the standard one in Bourbaki, Chapter VIII, 7.3. Groups don't always have to be simply connected to have minuscule weights or coweights (sometimes in types $A_\ell, D_\ell$ for instance). And the exceptional types $E_6, E_7$ do have minuscule weights. What is your source? – Jim Humphreys Feb 9 at 14:39
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