rt.representation theory - Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup

This is not a complete answer, but maybe it will be of use.

You're asking which $L$-high weights $mu$ occur in the $G$-irrep $V_lambda$. Let me say that $mu$ occurs classically if for some $N>0$, $Nmu$ occurs in $V_{Nlambda}$.

1. The set of such $mu$ form a rational polytope lying inside $L$'s positive Weyl chamber.




2. The vertices of this polytope strictly inside $L$'s chamber ("regular vertices") are exactly those of the form $wcdot lambda$ that are lucky enough to be in there.




3. The vertices of this polytope lying on $L$'s Weyl walls are very likely to be very complicated. In particular they may not be integral weights of $L$. As I recall this already happens for $GL(3) supset GL(2)times GL(1)$.

Parts 1 & 3 apply to any branching problem (and much further). Part 2 is special to your case that $L$ has the same rank as $G$ (I'm not actually using that it's a Levi).

If all you want is an upper bound, as your comment to Jim suggests, then that's easy: the $L$-high weights that can occur are a subset of the $T$-weights that occur, which you already described. Probably you want something better than that though. In principle it wouldn't be too hard to figure out the local structure of your polytope nearby the regular vertices, but I expect that not all facets contain regular vertices.

Littelmann describes (in the case of a Levi) the highest weights that occur and their multiplicities: one looks at all the Littelmann paths for the irrep $V_lambda$ that lie entirely inside the closed $L$-chamber.