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Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.

Let $\lambda$ a minuscule or quasiminuscule weight.

For which types and for which weights do we have that: $\forall w\in W, \lambda-w\lambda$ is a multiple of a root?

For classical groups, we know that for types $A$, $B$, $C$ it's true for $\omega_{1}$ and true for $\omega_{1}$ and $\omega_{2}$ for $G_{2}$.

So the question concerns type $E$ and $F_{4}$.

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up vote 2 down vote accepted

EDIT: Reading the question more carefully, I think the difference between the highest weight and an arbitrary Weyl group conjugate will almost never be a single root or muiltiple of a root. (What's true is that the difference between "adjacent" weights in that orbit across a single reflecting wall will be 0 or else a root. The adjoint representation in type $E_8$ illustrates this behavior. The saturation property in Bourbaki implies here that weight strings between such adjacent weights are of length 0 or 1.)

To go into more detail about your exceptional types, it's useful to have at hand both the tables for individual root systems in Bourbaki and the lists of positive roots at the end of Springer's paper here. In simply-laced cases (here type $E_n$), the adjoint representation is quasi-minuscule. This is easy to analyze from the tables, since subtracting arbitrary roots from the highest root usually doesn't give a multiple of a single root. In type $F_4$, the only quasi-minuscule highest weight is the highest short root. Here the weights other than 0 are the various short roots, so it's again easy to see that the difference need not be a multiple of a root. In your other minuscule cases (a pair of dual representations of dimension 27 for $E_6$, one representation of dimension 56 for $E_7$), more computation is needed than I've done. But for these small cases I think there are tables of weights available; I'll check.

By the way, since this is a roots-and-weights question, it's mainly about simple Lie algebras (not algebraic groups), whatever the application may be.

UPDATE: After tracking down lists for the minuscule weights in types $E_6, E_7$, I'm pretty sure I get a negative answer again: some differences of the highest weight and another weight fail to be multiples of a single root. The arithmetic is a bit complicated, since Springer's numbering of vertices in the Dynkin diagrams disagrees with the Bourbaki numbering which later became standard in most places. So caution is needed when making numertical comparisons.

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It doesn't always hold, because in case $D_{n}$ it never holds, but maybe there is sth that I don't understand in your answer? –  prochet May 13 '13 at 21:01
    
@prochet: I misread the question, so I've edited my answer. –  Jim Humphreys May 13 '13 at 21:09
    
thanks a lot for your detailed answer. –  prochet May 14 '13 at 17:33
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