This is a follow-up to a recent question here concerning the natural representation of a Weyl group $W$ on the zero weight space of an irreducible representation $L(\lambda)$ of highest weight $\lambda$ for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$.
One small rank case of interest involves $\mathfrak{g}$ of type $\mathrm{G}_2$. Here $W$ is dihedral of order 12. There seem to be just 3 highest weights for which the zero weight space affords an irreducible representation of $W$: the trivial or sign representation or the natural $2$-dimensional one. (These happen to be the special representations of $W$ among the 5 Springer representations in the 6 element set of inequivalent irreducible representations of $W$.)
Another small example involves $\mathfrak{g}$ of type $\mathrm{F}_4$, where $|W| = 2^7 \, 3^2$. Here $W$ has 25 irreducible representations; the character table was computed by T. Kondo. (This and other Weyl groups are extensively discussed in the book Finite Groups of Lie Type by R.W. Carter.) Among these are 16 Springer representations, of which 11 are special. The degrees of the representations comprise the set $\{1,2,4,6,8,9,12,16\}$.
Which irreducible representations $L(\lambda)$ of the Lie algebra of type $\mathrm{F}_4$ afford irreducible representations of $W$ on their zero weight spaces?
Recall that in general the zero weight space of $L(\lambda)$ is nonzero iff $\lambda$ lies in the root lattice, which is automatic in $\mathrm{F}_4$ since its root lattice is the full weight lattice. Within some bounds one can check various tables for the dimensions involved. For example, the data posted by Frank Luebeck here (ignoring differences for some prime characteristics) suggest that only five irreducibles for $W$ might actually occur, having respective degrees $1,2,4,9,12$ for the highest weights (in Bourbaki numbering) $0, \varpi_4, \varpi_1, \varpi_3, 2\varpi_4$. Here the $\varpi_i$ are the fundamental weights.
It would be helpful to verify such calculations, but more interesting to explain why these particular representations occur and also whether there is any predictable combinatorics in the $W$-decomposition of larger zero weight spaces which fail to be irreducible. Are there patterns which generalize to all Lie types?
