Fundamental representations and weight space dimension

Let $\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $\varpi_1, \ldots, \varpi_l$, where $l$ is the rank of $\mathfrak{g}$ and the ordering of the $\varpi_i$ is the usual one (Bourbaki).

The cases where an irreducible representation of highest weight $\lambda$ has all weight spaces $1$-dimensional are the following:

• Type $A_l$: $\lambda = \varpi_i, c \varpi_1$, or $c\varpi_l$.
• Type $B_l$ ($l \geq 2$): $\lambda = \varpi_1$ or $\varpi_l$.
• Type $C_l$ ($l \geq 3$): $\lambda = \varpi_1$. For type $C_3$, also $\lambda = \varpi_3$.
• Type $D_l$ ($l \geq 4$): $\lambda = \varpi_1$, $\varpi_{l-1}$ or $\varpi_{l}$.
• Type $G_2$: $\lambda = \varpi_1$.
• Type $E_6$: $\lambda = \varpi_1$ or $\varpi_6$.
• Type $E_7$: $\lambda = \varpi_7$.

So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.

For a reference, see Chapter 6 in Seitz, The maximal subgroups of classical algebraic groups (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well.

Seitz actually proves a result over an algebraically closed field of characteristic $p \geq 0$. In characteristic $p > 0$ we get some additional examples, such as $\lambda = d \varpi_i + (p-d-1) \varpi_{i+1}$ in type $A_l$ (for $1 \leq i < l$ and $0 \leq d < p$).