A regular semisimple Hessenberg variety \({{\,\textrm{Hess}\,}}(S,h)\) is a smooth subvariety of the full flag variety \({{\,\textrm{Fl}\,}}(\mathbb C^n)\) associated with a regular semisimple matrix S of order n and a function h from \(\{1,2,\dots ,n\}\) to itself satisfying a certain condition. We show that when \({{\,\textrm{Hess}\,}}(S,h)\) is connected and not the entire space \({{\,\textrm{Fl}\,}}(\mathbb C^n)\) , the reductive part of the identity component \({{\,\textrm{Aut}\,}}^0({{\,\textrm{Hess}\,}}(S,h))\) of the automorphism group \({{\,\textrm{Aut}\,}}({{\,\textrm{Hess}\,}}(S,h))\) of \({{\,\textrm{Hess}\,}}(S,h)\) is an algebraic torus of dimension \(n-1\) and \({{\,\textrm{Aut}\,}}({{\,\textrm{Hess}\,}}(S,h))/{{{\,\textrm{Aut}\,}}^0({{\,\textrm{Hess}\,}}(S,h))}\) is isomorphic to a subgroup of \(\mathfrak {S}_n\) or \(\mathfrak {S}_n \times \{\pm 1\}\) , where \(\mathfrak {S}_n\) is the symmetric group of degree n. As a byproduct of our argument, we show that \({{\,\textrm{Aut}\,}}(X)/{{{\,\textrm{Aut}\,}}^0(X)}\) is a finite group for any projective GKM manifold X.
