Solvable spherical subgroups: a review of different approaches to classification

Solvable spherical subgroups of semisimple complex algebraic groups were first classified by Luna in 1993. A completely different classification of these subgroups was obtained by the speaker in 2011. Moreover, there is a program of classifying arbitrary spherical subgrops (not necessarily solvable) that was initiated by Luna in 2001 and has been completed by now. All the three above-mentioned classifications are based on different approaches and each of them uses its own combinatorial invariants.
In the talk, we shall briefly present all the three classifications together with the corresponding combinatorial invariants and discuss how to pass between the invariants involved in different classifications.