Let $G$ be a simple Lie group over ${\mathbb C}$, $P \subset G$ a parabolic subgroup, and $L \subset P$ its Levi subgroup. Let $\lambda$ be a $G$-dominant weight and $V_G^\lambda$ an irreducible representation of $G$ with highest weight $\lambda$. I am interested in restrictions on highest weights of irreducible components of the restriction
$$
(V_G^\lambda)_{|L}.
$$
In the simplest case, when $P = B$ is the Borel subgroup, $L = T$ is the maximal torus, and there is a well-known restriction — the weights of $V_G^\lambda$ all lie in
$$
Conv(\lbrace w\lambda \rbrace_{w \in W}),
$$
where $W$ is the Weyl group. I would be happy to know something of the same sort for arbitrary parabolic subgroup.
Best Answer
The standard classical question concerns multiplicities of the irreducible representations of $L$ (or its derived group) in the restriction: these are given by branching rules. This is complicated to work out in detail but is treated in many textbooks and other sources. I'm not sure exactly how much information you are asking for. The original $W$-invariant set of weights relative to $T \subset L$ is unchanged, but the original representation decomposes into a direct sum of irreducibles for $L$, with the subgroup $W_L \subset W$ acting on the separate weight diagrams within the convex region you describe. (It's easy to picture this in restricting from rank 2 to rank 1, for example.)
By the way, your $G$-dominant weight means a weight of $T$ which is dominant relative to a fixed Borel subgroup $B$ for which $T \subset B \subset P$.