Geometry of regular semisimple Lusztig varieties

Lusztig varieties are subvarieties in flag manifolds $G/B$ associated to an element $w$ in the Weyl group $W$ and an element $x$ in $G$, introduced in Lusztig’s papers on character sheaves. We study the geometry of these varieties when $x$ is regular semisimple. In the first part, we establish that they are normal, Cohen-Macaulay, of pure expected dimension and have rational singularities. We then show that the cohomology of ample line bundles vanishes in positive degrees, in arbitrary characteristic. This extends to nef line bundles when the base field has characteristic zero or sufficiently large characteristic. Along the way, we prove that Lusztig varieties are Frobenius split in positive characteristic and that their open cells are affine. We also prove that the open cells in Deligne-Lusztig varieties are affine, settling a question that has been open since the foundational paper of Deligne and Lusztig. In the second part, we explore their relationship with regular semisimple Hessenberg varieties. Both varieties admit Tymoczko’s dot action of $W$ on their (intersection) cohomology. We associate to each element $w$ in $W$ a Hessenberg space using the tangent cone of the Schubert variety associated with $w$, and show that the cohomology of the associated regular semisimple Lusztig varieties and Hessenberg varieties is isomorphic as graded $W$-representations when they are smooth. This relationship extends to the level of varieties: we construct a flat degeneration of regular semisimple Lusztig varieties to regular semisimple Hessenberg varieties. In particular, this proves a conjecture of Abreu and Nigro on the homeomorphism types of regular semisimple Lusztig varieties in type $A$, and generalizes it to arbitrary Lie types.

💡 Research Summary

The paper studies Lusztig varieties—subvarieties of the flag manifold G/B defined from a Weyl group element w and a group element x—under the hypothesis that x is regular semisimple. The authors split the work into two parts.

Part 1: Singularities, line bundles and affine cells
For a regular semisimple element s∈G the fibre Y₍w₎(s) (the “regular semisimple Lusztig variety”) is shown to be normal, Cohen‑Macaulay, of pure dimension ℓ(w) (the length of w), and to have at worst rational singularities. Moreover Y₍w₎(s) is smooth (indeed Gorenstein) precisely when the associated Schubert variety X₍w₎ is smooth.

The authors then investigate cohomology of line bundles L_λ on Y₍w₎(s). If λ∈X(T) is regular dominant, then for any characteristic the higher cohomology groups Hⁱ(Y₍w₎(s),L_λ) vanish for i>0, and the restriction map to any closed Lusztig subvariety is surjective. When λ is merely dominant, the same vanishing holds in characteristic 0 and in sufficiently large positive characteristic. Two different proofs are given: in characteristic 0 a Kawamata–Viehweg vanishing theorem is applied to a Bott–Samelson resolution; in positive characteristic the authors prove that Y₍w₎(s) is Frobenius split (and even D‑split for a suitable ample divisor) and deduce the vanishing from standard splitting arguments.

A key geometric consequence is that the open Lusztig cell Y₍w₎⁰(s)=Y₍w₎(s)∖∂Y₍w₎(s) is affine. The same argument shows that the open Deligne–Lusztig cell (the fibre of the Frobenius map) is also affine, settling a long‑standing open problem.

Part 2: Relation to regular semisimple Hessenberg varieties
The paper turns to Hessenberg varieties X_H(s) defined by a linear Hessenberg subspace H⊂𝔤 (a B‑submodule containing 𝔟) and a regular semisimple element s∈𝔤. These varieties are smooth, GKM manifolds, and carry Tymoczko’s dot action of the Weyl group W on their cohomology.

For each w∈W that is smooth (i.e. X₍w₎ smooth) the authors construct a Hessenberg space H_w by taking the tangent space of X₍w₎ at the identity and embedding it into 𝔤/𝔟. They prove that the Lusztig variety Y₍w₎(s) and the Hessenberg variety X_{H_w}(s) have identical GKM graphs; consequently their (intersection) cohomology rings are isomorphic as graded W‑representations. Moreover, if w and w′ are smooth and give the same Hessenberg space H_w=H_{w′}, then the cohomology of Y₍w₎(s) and Y₍w′₎(s′) coincide for any regular semisimple s,s′.

To explain this relationship geometrically, the authors construct a flat degeneration. They blow up G at the identity, obtaining a space bG whose open part is G^{rs} and whose closed part is the projectivized regular semisimple cone P(𝔤)^{rs}. Pulling back the family of Lusztig varieties to bG and applying a deformation‑to‑the‑normal‑cone construction yields a smooth projective morphism bY₍w₎→bG. Over G^{rs} this restricts to the original family Y₍w₎^{rs}→G^{rs}, while over P(𝔤)^{rs} it restricts to the family of Hessenberg varieties X_{H_w}^{rs}→P(𝔤)^{rs}. Thus Y₍w₎(s) degenerates flatly to X_{H_w}(s). Using Ehresmann’s theorem, the authors deduce that if two smooth elements w,w′ have the same tangent space at the identity, then Y₍w₎(s) and Y₍w′₎(s) are diffeomorphic. This proves the conjecture of Abreu–Nigro (originally formulated for type A) in full generality for any reductive group.

Techniques
The work combines several modern tools: Bott–Samelson resolutions to control singularities and boundary divisors, Frobenius splitting and D‑splitting in positive characteristic, Kawamata–Viehweg vanishing in characteristic 0, GKM theory to translate torus actions into combinatorial graphs, and deformation‑to‑the‑normal‑cone to build flat families. The authors also exploit the relationship between Schubert cells (which are B‑orbits and hence affine) and Lusztig cells (which are not B‑stable) to prove affineness of the latter.

Overall contribution
The paper establishes that regular semisimple Lusztig varieties enjoy the same excellent geometric properties as Schubert varieties, extends vanishing theorems for line bundles to this new setting, proves affineness of Lusztig and Deligne–Lusztig cells, and uncovers a deep link with regular semisimple Hessenberg varieties. The flat degeneration construction not only resolves the Abreu–Nigro conjecture in all types but also provides a conceptual bridge between two previously distinct families of varieties, enriching the interplay between representation theory, algebraic geometry, and combinatorics.