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In type $A$, the simple modules all of whose weight spaces are $1$-dimensional are the $L(n\varpi_1)$ and $L(\varpi_k)$. This can be seen from the fact that dimensions of weight spaces are given by number of semistandard young tableaux with shape $\lambda$ and content = weight.

Suppose $\mathfrak{g}$ is a complex semisimple Lie algebra. Is there a classification of simple modules all whose weight spaces are $1$-dimensional?

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The following result of Howe [Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond] answers this completely:

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. Then a non-trivial irreducible $\mathfrak{g}$-module $V(\lambda)$ has one-dimensional weight spaces if and only if

  1. $\lambda$ is minuscule,
  2. $\lambda$ is quasi-minuscule and $\mathfrak{g}$ has only one short simple root,
  3. $\mathfrak{g}=C_{3}=\mathfrak{sp}_{6}$ and $\lambda=\omega_{3}$, or
  4. $\mathfrak{g}=A_{l}=\mathfrak{sl}_{l+1}$ and $\lambda=m\omega_{1}$ or $\lambda=m\omega_{l}$ for some $m\in \mathbb{N}$.

I was able to find this via a paper of Stembridge [Multiplicity-free products and restrictions of Weyl characters].

As I needed exactly this property in a paper of my own, you will find definitions of minuscule and quasi-minuscule as well as references from which I drew a table of these on pages 20-21 of it (arXiv:0409359, On Lie induction and the exceptional series). They are as follows, taken from [Visual basic representations: An atlas] by Plotkin–Semenov–Vavilov:

Non-zero minuscule weights:

$\begin{array}{ll} A_{l} & \omega_{i},\ 1\leq i \leq l \\ B_{l} & \omega_{l} \\ C_{l} & \omega_{1} \\ D_{l} & \omega_{1},\ \omega_{l-1},\ \omega_{l} \\ E_{6} & \omega_{1},\ \omega_{6} \\ E_{7} & \omega_{7} \end{array} $

Quasi-minuscule weights:

$ \begin{array}{ll} \begin{array}[t]{ll} A_{l} & [1,0,0,\ldots,0,1]\ (\mbox{adjoint}) \\ B_{l} & \omega_{1} \\ C_{l} & \omega_{2} \\ D_{l} & \omega_{2} \\ E_{6} & \omega_{2} \\ E_{7} & \omega_{1} \\ E_{8} & \omega_{8} \\ F_{4} & \omega_{4} \\ G_{2} & \omega_{1} \end{array} \end{array} $

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  • $\begingroup$ Just to check, under point 2., with 'only one short simple root', you mean 'exactly one short simple root'? That is, in the list of 'quasi-minuscle weights', the modules corresponding to the weights of the simply laced cases do not have the property that all weight space are one dimensional (just as the cases for $F_4$ and $C_{l > 2}$? $\endgroup$
    – eddy ardonne
    Commented Oct 28 at 18:53
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    $\begingroup$ Indeed. Looking back at 4.3 in my paper, a result whose proof is due to Bazlov, it says: "Of the remaining quasi-minuscule weights, we exclude [because they don't have the multiplicity-free property I wanted] the algebra-weight pairs corresponding to adjoint repre- sentations, namely (Al (l ≥ 2), [1, 0, 0, . . . , 0, 1]), (Dl, ω2), (E6, ω2), (E7, ω1) and (E8, ω8), since in these cases the zero weight occurs with multiplicity l, the rank of g, which is greater than one." $\endgroup$
    – Jan Grabowski
    Commented Oct 28 at 21:51

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