Degenerations of spherical subalgebras and spherical roots of spherical subgroups

Let $G$ be a connected semisimple algebraic group and let $H$ be a spherical subgroup of $G$ (that is, a Borel subgroup $B \subset G$ has an open orbit in $G/H$). The famous Luna–Vust theory describes all normal equivariant open embeddings of the homogeneous space $G/H$ in terms of three combinatorial invariants: the weight lattice, the set of spherical roots, and the so-called “colors”. In this situation, a natural problem is to compute the three above-mentioned invariants starting from an explicit form of the subgroup $H$; a standard way of specifying $H$ is by using a right inclusion in a parabolic subgroup of $G$. By now, a complete solution of this problem is known only for the weight lattice whereas approaches for computing the remaining two invariants have been found only for the following two “opposite” classes of spherical subgroups: reductive and solvable.
In the talk, we shall propose a general ideology that enables one to reduce the problem of computing the spherical roots for a given subgroup $H$ to the same problem for two (or more) other spherical subgroups for which the weight lattice has lower rank than that for $H$. Here, a key point consists in finding explicit constructions of degenerations of the Lie algebra of $H$ that satisfy certain strict conditions. Iterating this procedure in a finite number of steps leads to a finite number of “primitive” cases for which the sets of spherical roots are known.
Next we shall discuss a concrete realization of the above-mentioned ideology for a wide class of spherical subgroups that includes solvable spherical subgroups. For this class of subgroups, it is possible to prove analogues of some structure theorems for solvable spherical subgroups, which enables us to exhibit constructions of degenerations having all the desired properties. In the end this yields an algorithm (laborious and slow but effective) for computing the set of spherical roots for any spherical subgroup in the class under consideration.