Spherical affine cones in exceptional cases and related branching rules

Given a complex simply connected simple algebraic group $G$ of exceptional type and a maximal parabolic subgroup $P \subset G$, we classify all triples $(G,P,H)$ such that $H \subset G$ is a maximal reductive subgroup acting spherically on $G/P$. In addition we derive branching rules for $\text{res}^G_H (V^*_{k\omega_i})$, $k \in \N$, where $\omega_i$ is the fundamental weight associated to $P$. This is the first of two parts of a project to classify all such triples and corresponding branching rules for all simply connected simple algebraic groups.