Let $\lambda$ and $\mu$ be partitions with at most $n$ parts. Write $\mu = 0^{n_0}1^{n_1}2^{n_2}...$ with $\sum_i n_i = n$, to mean $\mu$ has $n_0$ parts of size $0$, $n_1$ parts of size $1$, and so on. Let $V(\lambda)$ be the irreducible finite dimensional polynomial representation of $GL_n(\mathbb{C})$ of highest weight $\lambda$, and let $V(\lambda)_\mu$ be its $\mu$-weight space.

$V(\lambda)_\mu$ naturally carries an action of $S_{n_0} \times S_{n_1} \times S_{n_2} \times \dots$ (a subgroup of $S_n$, conjugate to the obvious inclusion). I am wondering what is known about these spaces as $S_{n_0} \times S_{n_1} \times S_{n_2} \times \dots$ representations?

Here are a few initial things I've noticed:

The dimension of these spaces are Kostka numbers.

For polynomial representations the $S_{n_0}$ factor acts trivially, so we can just think of them as $S_{n_1} \times S_{n_2} \times S_{n_3} \times \dots$ representations.

If we fix partitions $\lambda$ and $\mu$ and let $n$ vary the $S_{n_1} \times S_{n_2} \times S_{n_3} \times \dots$ representations we get don't change (provided $n$ is large enough for this all to make sense).

Ideally I'd like some description of these representations in terms of some combinatorics on semistandard Young tableaux, but I'd be interested in hearing whatever is known.