Branching rules for compact Lie groups

Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know the multiplicity of the trivial representation $\mathbf{1}:H\to\mathbb{C}$ in the restricted representation $\pi|_H:H\to\mathrm{End}_\mathbb{C}(V)$. Let's call it $\mathrm{mult}(\mathbf{1},\pi|_H)$. By Frobenius reciprocity, it is the same as $\mathrm{mult}(\pi,\operatorname{Ind}_H^G\mathbf{1})$.

I know that branching rules for general pairs of representations are tricky. And I know that there is an extensive literature and tables for classical groups. But what I am interested in is special: general compact Lie group $G$ and closed subrgoup $H$, general irreducible representation of $G$, but only the trivial representation of $H$. A good starting point would be $G$ simple simply connected which reduces to the question for the corresponding complex Lie algebras. Even this I wasn't able to find for my scope: everything is either for special classical groups (general linear, unitary etc.) or for arbitrary representations of $H$ (in which case it is too abstract). I am pretty sure that for the trivial representation of $H$ things must be much easier than in general. There are a few articles by Patera and colleagues who deal with the general case in terms of weight systems, and there it is already clear that things get easier if one of the weight systems is just zero. But I hope there is an explicit description of $\mathrm{mult}(\mathbf{1},\pi|_H)$ somewhere in the literature. Thank you.

• What kind of answer do you expect? As you said, the question is equivalent to decomposing the induced representation --- roughly, harmonic analysis on $G/H$. For connected $H$, the multiplicity can be expressed via the Weyl character formula. This is not particularly illuminating or explicit in most cases, but then again, there are way too many pairs $(G,H)$ to hope for an explicit and uniform description. – Victor Protsak Dec 27 '17 at 8:51
• Thank you for your kind comment. If you honestly wonder what I expect as an answer, please, note the 'reference-request' tag below my question. Harmonic analysis on $G/H$ is exactly why I am asking this question. If you look into Chapter 8 of Goodman, Wallach, you see that for many classical groups multiplicity has to do with interlacing properties, and that becomes trivial if one of the weights is zero. More generally, Kostant's multiplicity formula in terms of partition functions becomes more symmetric when $\mu=0$, and it is not unreasonable to expect simplifications. – Bedovlat Dec 27 '17 at 10:48
• Someone must have systematically treated the $\mu=0$ case, because it is of special importance. It is the basic block of harmonic analysis on $G/H$. Disentangling this as a special case from general branching laws is a remarkable effort, and should not be performed by everyone who does analysis on $G/H$. It should be done somewhere once and forever. And presumably not by an analyst. – Bedovlat Dec 27 '17 at 13:34