  A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth
subvariety of the full flag variety $\mathrm{Fl}(\mathbb{C}^n)$ 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
$\mathrm{Hess}(S,h)$ is connected and not the entire space
$\mathrm{Fl}(\mathbb{C}^n)$, the reductive part of the identity component
$\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ of the automorphism group
$\mathrm{Aut}(\mathrm{Hess}(S,h))$ of $\mathrm{Hess}(S,h)$ is an algebraic
torus of dimension $n-1$ and
$\mathrm{Aut}(\mathrm{Hess}(S,h))/\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ is
isomorphic to a subgroup of $\mathfrak{S}_n$ or $\mathfrak{S}_n\rtimes \{\pm
1\}$, where $\mathfrak{S}_n$ is the symmetric group of degree $n$. As a
byproduct of our argument, we show that $\mathrm{Aut}(X)/\mathrm{Aut}^0(X)$ is
a finite group for any projective GKM manifold $X$.
