Automorphisms and deformations of regular semisimple Hessenberg varieties

We show that regular semisimple Hessenberg varieties can have moduli. To be precise, suppose $X$ is a regular semisimple Hessenberg variety of codimension $1$ in the flag variety $G\mathrm{/}B$, where $G$ is a simple algebraic group of rank $r$ over $\mathbb{C}$ and $B$ is a Borel subgroup. We show that the space~${\mathrm{H}}^1(X,TX)$ of first order deformations of $X$ has dimension $r-1$ except in type $A_2$. (In type $A_2$, the Hessenberg varieties in question are all isomorphic to the permutohedral toric surface, and $\dim {\mathrm{H}}^1(X,TX)=0$.) Moreover, we show that the Kodaira–Spencer map $\mathfrak{g}\to {\mathrm{H}}^1(X,TX)$ is onto, that the identity component of the automorphism group of $X$ is a maximal torus of $G$, and that ${\mathrm{H}}^i(X,TX)=0$ for $i\ge 2$. Along the way, we prove several theorems of independent interest about the cohomology of homogeneous vector bundles on~$G\mathrm{/}B$. In type $A$, we can give an even more precise statement determining when two codimension $1$ regular semisimple Hessenberg varieties in $G\mathrm{/}B$ are isomorphic. We also compute the automorphism groups explicitly in type~$A_{n-1}$ in the terms of stabilizer subgroups of the action of the symmetric group $S_n$ on the moduli space $M_{0,n+1}$ of smooth genus $0$ curves with $n+1$ marked points. Using this, we describe the moduli stack of the regular semisimple Hessenberg varieties $X$ explicitly as a quotient stack of $M_{0,n+1}$. We prove several analogous results for Hessenberg varieties in generalized flag varieties $G\mathrm{/}P$, where $P$ is a parabolic subgroup of $G$. In type $A$, these results are used in the proofs of the results for $G\mathrm{/}B$, but they are also of independent interest because the associated moduli stacks are related directly to the action of $S_n$ on $M_{0,n}$.