Symmetric pairs and branching laws

Let $G$ be a compact connected Lie group and let $H$ be a subgroup fixed by an involution. A classical result assures that the $H_{ℂ}$-action on the flag variety $F$ of $G$ admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair $(G,H)$ that is parametrized by $H_{ℂ}\setminus F$.