On reductive automorphism groups of regular embeddings

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X 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 group AutA degrees(X, D), the connected automorphism group of X stabilizing all elements of D. We determine a Levi subgroup A(X, D) of AutA degrees(X, D), and also relevant invariants of X as a spherical A(X, D)-variety. As a byproduct, we obtain a complete description of the inclusion relation between closures of A(X, D)-orbits on X.