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Let G be a connected reductive complex algebraic group acting on a smoothcomplete complex algebraic variety X. We assume that X under the action of G isa regular embedding, a condition satisfied in particular by smooth toricvarieties and flag varieties. For any set D of G-stable prime divisors, westudy 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 computerelevant invariants of X as a spherical A-variety. As a byproduct, we obtain adescription of the open A-orbit on X and the inclusion relation between A-orbitclosures.

On reductive automorphism groups of regular embeddings