Affine pavings of Hessenberg varieties for semisimple groups (Q387652)

Let \(G\) be a complex linear reductive algebraic group and let \(B\subset G\) be a Borel subgroup. A subspace \(H\) of the Lie algebra \({\mathbf g}\) of \(G\) is a Hassemberg space with respect to \( {\mathbf b}\) (the Lie algebra of \(B\)) if it contains \({\mathbf b}\) and it is closed under Lie bracket with \({\mathbf b}\). Given a Hessenberg space \(H\) and an element \(X\in {\mathbf g}\), the Hessenberg variety \({\mathcal B}(X,H)\) is the subvariety of the flag variety \({\mathcal B}=G/B\) consisting of all \(gB/B\) such that \(\mathrm{Ad}_{g^{-1}}X\in H\). The paper is concerned with affine pavings of Hessenberg varieties (a paving of an algebraic variety \(Y\) is a filtration \(Y_0\subset Y_1\subset\ldots\subset Y_d=Y \) by closed subvarieties \(Y_i\), and it is affine if \(Y_i\setminus Y_{i-1}\) is a finite disjoint union of affine spaces. Pavings by affine cells compute the Betti numbers of algebraic varieties). When \(H={\mathbf g}\), the Hessenberg variety coincides with the flag \({\mathcal B}\), and an affine paving of \({\mathcal B}\) is given by the Bruhat decomposition and its Schubert cells. In general, intersecting an Hessenberg variety \({\mathcal B}(X,H)\) with each Schubert cell in \({\mathcal B}\) yields a paving of \({\mathcal B}(X,H)\). For some elements \(X\in{\mathbf g}\) such a paving can be proved to be affine. The main results of the paper are the following theorems: { Theorem.} Fix a Hessenberg space \(H\) with respect to \({\mathbf b}\). Let \(N\in{\mathbf g}\) be a nilpotent element such that \(N\) is regular in some Levi subalgebra \({\mathbf m}\) of \({\mathbf g}\). Then there is an affine paving of \({\mathcal B}(X,H)\) given by intersection of each Schubert cell in \({\mathcal B}\) with \({\mathcal B}(X,H)\). { Theorem.} Suppose \(X\in{\mathbf g}\) has Jordan decomposition \(X=S+N\) and \(N\) is regular in some Levi subalgebra of \({\mathbf m}\), where \({\mathbf m}\) is the Lie algebra of the Levi subgroup \(M=Z_G(S)\). Then \({\mathcal B}(X,H)\) is paved by affines. The above theorems generalize results of J. Tymoczko proving the existence of affine pavings for Hessenberg varieties of regular nilpotent elements of classical Lie algebras ([\textit{J. S. Tymoczko}, Sel. Math., New Ser. 13, No. 2, 353--367 (2007; Zbl 1144.14043)] and for Hessenberg varieties of arbitrary elements of \(\mathfrak{gl}(n,{\mathbb{C}})\) [Am. J. Math. 128, No. 6, 1587--1604 (2006; Zbl 1106.14038)]).