Let $V$ be the irreducible representation of $GL_n$ with highest $\lambda$, and $|\lambda|=n$. It is well known that the representation of $S_n$ on the $(1,1,\ldots,1)$ weight space is the Specht module $S_{\lambda}$. What is known about the representation of $S_n$ on the rest of $V$?
I am also interested in the case where $|\lambda| < n$.
Best Answer
Each direct sum of an $S_n$-orbit of weight spaces is an induction from the representation of the stabilizer of one of the weight spaces, which is always a Young subgroup (just coming from how many of your entries are the same). That representation of the stabilizer should be a sum of Specht modules depending on how your representation branches to the semi-simple part of the Levi (block diagonal matrices where each block as determinant 1) corresponding to the Young subgroup.
When the weight space is (1,...,1), then everything is in the stabilizer, so you're looking at the branching from the group to itself, and nothing happens. On the other hand, if you take a generic weight, its stabilizer will be trivial, and you'll just get a regular representation tensored with the weight space (the semi-simple part of the Levi is trivial in this case).
UPDATE: This is David Speyer abusing the edit power to warn people that I say a lot of false things in the comments below. I don't want to delete them because that will make Ben's responses look like nonsense, but you shouldn't rely on anything I say here.