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, which is a divisor in the flag variety $G/B$, where G is a simple algebraic group of rank r over C. Then we show that the space H¹(X,TX) of first order deformations of X has dimension $r-1$ except in type A₂. (In type A₂, the Hessenberg varieties in question are all isomorphic to the permutahedral toric surface. So ¹(X,TX) = 0.) We also show that the connected component of the automorphism group of X is the maximal torus of G, and we show that Hⁱ(X,TX) = 0 for i ≥ 2. In type A, we can give an even more precise statement determining when two regular semisimple Hessenberg varieties, which are divisors in $G/B$, are isomorphic. We also compute the automorphism groups explicitly in type~Aₙ₋₁ in the terms of stabilizer subgroups of the action of the symmetric group Sₙ on the moduli space M0,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 M0,n+1. We prove several analogous results for Hessenberg varieties in generalized flag varieties. In type A, these are used in the proofs of the results for $G/B$, but they are also independently interesting because the associated moduli stacks are related directly to the action of Sₙ on M0,n.