TITLE: Occurrence of the trivial representation in restrictions of Lie group representations
QUESTION [12 upvotes]: 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 $H$?

REPLY [3 votes]: I want to refer you to the following article by B. Cahen, treating the case of compact Lie groups, where an approach based on the realization of the irreducible representations of G according to the Borel-Weil theorem as reproducing kernel Hilbert spaces within the framework of Berezin quantization is adopted. The article describes a multiplicity formula (equation - 3.3) of an irreducible representation of a compact group G in the decomposition of a reproducing kernel Hilbert space on which G acts unitarily whose elements are functions on a manifold M.
When applied to the case in the question, the reproducing kernel Hilbert space can be taken as the space of holomorphic sectionsof the line bundle associated with lambda (according to the Borel-Weil theorem) on the flag manifold G/T. The multiplicity formula contains a double integral, the first is over the maximal torus of H which acts as a projector onto the required representation space and the second is over the flag manifold G/T which acts as a multiplicity counter. 
In this method no explicit sum on the Weyl group is needed, but it is replaced by the task of the double integration. The integration on the maximal torus is quite easy. The integration on the flag manifold looks like a formidable task but at least for the case of weight multiplicity computation it can be replaced by a computation of the rank of
a Hermitian matrix (proposition 3.6). In the article some low dimensional examples are included.
This method does not give nice closed formulas like the combinatorial methods, but may be it has a reduced complexity in some special cases. I think that at least the construction of the reproducing kernel in the affine coordinates of the big cell of the flag manifold can be implemented relatively efficiently by polynomial multiplication using 
a multidimensional fast Fourier transform.