  Let G be a connected reductive complex algebraic group acting on a smooth
complete complex algebraic variety X. We assume that X under the action of G is
a regular embedding, a condition satisfied in particular by smooth toric
varieties and flag varieties. For any set D of G-stable prime divisors, we
study the action on X of the connected automorphism group of X stabilizing D.
We determine a Levi subgroup A of this automorphism group, and we compute
relevant invariants of X as a spherical A-variety. As a byproduct, we obtain a
description of the open A-orbit on X and the inclusion relation between A-orbit
closures.
