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A regular semisimple Hessenberg variety Hess (S, h) is a smooth subvariety of the full flag variety Fl (Cⁿ) associated with a regular semisimple matrix S of order n and a function h from \1, 2, , n\ to itself satisfying a certain condition. We show that when Hess (S, h) is connected and not the entire space Fl (Cⁿ), the reductive part of the identity component Aut⁰ (Hess (S, h) ) of the automorphism group Aut (Hess (S, h) ) of Hess (S, h) is an algebraic torus of dimension n-1 and Aut (Hess (S, h) ) /Aut⁰ (Hess (S, h) ) is isomorphic to a subgroup of Sₙ or Sₙ \ 1\, where Sₙ is the symmetric group of degree n. As a byproduct of our argument, we show that Aut (X) /Aut⁰ (X) is a finite group for any projective GKM manifold X.
Jang et al. (Sat,) studied this question.