Paving Hessenberg varieties by affines
The paper under review studies the topology of regular nilpotent Hessenberg varieties, a family of subvarieties of the flag variety introduced by \textit{F. De Mari, C. Procesi} and \textit{M. A. Shayman} [Trans. Am. Math. Soc. 332, No. 2, 529--534 (1992; Zbl 0770.14022)]. Recall the definition of these varieties: let \(G\) be a complex linear algebraic group of classical type, \(B\) a fixed Borel subgroup, and \({\mathfrak g}\) and \({\mathfrak b}\) their Lie algebras. A Hessenberg space \(H\) is a linear subspace of \({\mathfrak g}\) that contains \({\mathfrak b}\) and that is closed under Lie braket with \({\mathfrak b}\). Then, fix and element \(X\) in \({\mathfrak g}\) and a Hessenberg space \(H\). The Hessenberg variety \({\mathcal H}(X,H)\) is the subvariety of the flag variety \(G/B\) consisting of \(gB/B\) satisfying \(g^{-1}Xg\in H\). Moreover, a regular nilpotent element \(N\) of \({\mathfrak g}\) is an element which is in the dense adjoint orbit within the nilpotents elements in \({\mathfrak g}\). Finally, a paving of an algebraic variety \(X\) is an ordered partition into disjoint \(X_0,X_1,X_2,\dotsc\) so that each finite union \(\bigcup_{i=0}^j X_i\) is Zariski closed in \(X\). The main result of this paper states the existence of paving by affines for regular nilpotent Hessenberg varieties for all classical types, generalizing results of \textit{C. De Concini, G. Lusztig} and \textit{C. Procesi} [J. Am. Math. Soc. 1, No. 1, 15--34 (1988; Zbl 0646.14034)] and \textit{B. Kostant} [Sel. Math., New Ser. 2, No. 1, 43--91 (1996; Zbl 0868.14024)]. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. The author uses the paving to prove that these Hessenberg varieties have no odd-dimensional homology.
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