Automorphisms and deformations of regular semisimple Hessenberg varieties

TL;DR

The paper shows that regular semisimple Hessenberg varieties, viewed as codimension-one divisors in flag varieties, have nontrivial moduli and deformation spaces of dimension $r-1$ (except in type $A_2$), with a surjective Kodaira–Spencer map and trivial higher cohomology $H^i(X,TX)$ for $i\ge2$. In type A, the authors classify isomorphisms between $X(s)$ (and $Y(s)$) in terms of conjugation by $G$ and affine (or $GL_2$) actions, determine automorphism groups, and relate isomorphism classes to stabilizers in the Weyl group and to points in the moduli spaces $M_{0,n}$ and $M_{0,n+1}$; they also show Tymoczko’s dot action cannot be geometrically lifted to an action on $X(s)$. General types are treated via tangent/normal sequences and cohomology vanishing, yielding analogous deformation-vanishing results and a parabolic generalization; moduli stacks are described as quotient stacks by natural group actions, with good moduli spaces given by $M_{0,n+1}/S_n$ and $M_{0,n}/S_n$, and natural GIT compactifications discussed. Overall, the work connects Hessenberg geometry, deformation theory, automorphism groups, and moduli theory, illuminating how Hessenberg varieties sit inside the landscape of flag varieties and moduli of rational curves. The results provide explicit computations and structural descriptions that enable further exploration of Hessenberg moduli, automorphisms, and their geometric and representation-theoretic consequences.

Abstract

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 $G/B$, 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 $i \geq 2$. Along the way, we prove several theorems of independent interest about the cohomology of homogeneous vector bundles on~$G/B$. In type $A$, we can give an even more precise statement determining when two codimension $1$ regular semisimple Hessenberg varieties in $G/B$ 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 $G/P$, where $P$ is a parabolic subgroup of $G$. In type $A$, these results are used in the proofs of the results for $G/B$, but they are also of independent interest because the associated moduli stacks are related directly to the action of $S_n$ on $M_{0,n}$.

Theorems & Definitions (141)

• Theorem 1.2
• Lemma 1.3
• Lemma 1.4
• Theorem 1.5
• Theorem 1.7
• Corollary 1.8: No geometric lift of Tymoczko's dot action
• Theorem 1.9
• Conjecture 1.10
• Lemma 1.11
• Theorem 1.13