Automorphisms of GKM graphs and regular semisimple Hessenberg varieties

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.