Containment Relations among Spherical Subgroups

A closed subgroup $H$ of a connected reductive group $G$ is called $\mathit{\text{spherical}}$ if a Borel subgroup in $G$ has an open orbit on $G\mathrm{/}H$. We give a combinatorial characterization for a spherical subgroup to be contained in another one which generalizes previous work by Knop. As an application, we compute the Luna datum of the identity component of a spherical subgroup which yields a characterization of connectedness for spherical subgroups.