Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups

The extended weight semigroup of a homogeneous space $G/H$ of a connected semisimple algebraic group $G$ characterizes the spectra of the representations of $G$ on spaces of regular sections of homogeneous line bundles over $G/H$, including the space of regular functions on $G/H$. We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces $G/H$, where $G$ is a simply connected non-simple semisimple complex algebraic group and $H$ is a connected closed subgroup of $G$. In all cases we also find the highest-weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.