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~A₍-₁ in the terms of stabilizer subgroups of the action of the symmetric group S₍ on the moduli space M₀, ₍+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, ₍+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, ₍.