Branching multiplicities for spherical pairs of minimal rank

Given a complex reductive algebraic group $G$ and a reductive subgroup $H$, the branching problem asks to decompose irreducible representations of $G$ under the natural $H$ action. When $H$ is diagonally embedded into $H \times H$ this is equivalent to decomposing the tensor product of irreducible representations of $H$. Solving the branching problem is equivalent to understanding the so called “multiplicity spaces”. We will present a geometric interpretation of these spaces for $(H,G)$ spherical of minimal rank. As a special case, for the tensor product decomposition we find a classical result of Partasarathy, Ranga Rao and Varadarajan which is the starting point for the construction of good basis and thus for the realization of the $i$-trails models for multiplicities of Berenstein-Zelevinsky.