Geometry of regular semisimple Lusztig varieties
Patrick Brosnan, Jaehyun Hong, Donggun Lee
TL;DR
This work analyzes regular semisimple Lusztig varieties $Y_w(\mathbf{s})$ and their relations to Hessenberg varieties across Lie types. It proves fundamental geometric properties (normality, Cohen–Macaulayness, pure dimension $\ell(w)$, rational singularities) and derives cohomology vanishing for nef/ample line bundles, using Bott–Samelson resolutions and Frobenius splitting. The paper shows Lusztig cells are affine and establishes a flat degeneration to Hessenberg varieties, generalizing Abreu–Nigro’s type-$A$ results to arbitrary types and linking cohomology and $W$-representations via GKM graphs. It also identifies a natural degeneration framework and provides counterexamples illustrating limits of certain natural correspondences outside type $A$. These results unify geometric, cohomological, and combinatorial aspects of Lusztig and Hessenberg varieties, with implications for topological types and Weyl-group actions.
Abstract
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.