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.

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