Automorphisms of GKM graphs and regular semisimple Hessenberg varieties

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