Suppose $G$ is a semisimple group, and $V\_{\lambda}$ is an irreducible finite-dimensional representation of highest weight $\lambda$. Suppose $H \subset G$ is a semisimple subgroup. What is the multiplicity of the trivial representation in $V\_{\lambda}|\_{H}$? Is there a simple way to read this off from $\lambda$ and the Dynkin diagrams of $G$ and $H$?



Occurrence of the trivial representation in restrictions of Lie group representations
The short answer is: no. In theory, there are explicit formulae for branchings based on the Weyl character formula, but no reasonable person would call these simple.


There are interesting results which give you some information about branching multiplicities, but all involve real work. For instance, if G and H are compact, you can obtain asymptotic information about H-invariants in the representations $V\_{n\lambda}$ as a function of $n$: it's a polynomial, whose leading order is the dimension of $\mathcal{O}\_{\lambda}//H $, the symplectic reduction of the coadjoint orbit through $\lambda$ by $H$, and whose leading coefficient is the symplectic volume of this manifold.


If you prefer algebraic geometry, this polynomial is the Hilbert polynomial of the corresponding GIT quotient.


If $H$ is a root subalgebra, then life is a bit easier, and you can use combinatorial methods like crystals, but this is still not easy.


