2 added 16 characters in body

There are three types of subgroups of $PGL_2(\mathbb{C})$ that act on $\mathbb{P}^1$ non-transitively but with finitely many orbits:

(1) Type $T$: a one-dimensional torus

(2) Type $N$: the normalizer of a one-dimensional torus

(3) Type $U$: containing a non-trivial one-dimensional unipotent subgroup

This trichotomy plays a key role in the study of the geometry of spherical varieties, a class of algebraic varieties that includes grassmannians, flag varieties, toric varieties, algebraic monoids and symmetric spaces. It is particularly important in understanding the analogues of Schubert subvarieties (i.e. closures of orbits of a Borel subgroup) of a spherical variety.

In this example, there is no "middle" case as there is no intrinsic order to the three types.

There are three types of subgroups of $PGL_2(\mathbb{C})$ that act on $\mathbb{P}^1$ non-transitively but with finitely many orbits:

(1) Type $T$: a one-dimensional torus

(2) Type $N$: the normalizer of a one-dimensional torus

(3) Type $U$: containing a non-trivial unipotent subgroup

This trichotomy plays a key role in the study of the geometry of spherical varieties, a class of algebraic varieties that includes grassmannians, flag varieties, toric varieties, algebraic monoids and symmetric spaces. It is particularly important in understanding the analogues of Schubert subvarieties (i.e. closures of orbits of a Borel subgroup) of a spherical variety.

In this example, there is no "middle" case as there is no intrinsic order to the three types.