Affine pavings of Hessenberg varieties for semisimple groups
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Affine pavings of Hessenberg varieties for semisimple groups Martha Precup
Published online: 9 November 2012 © Springer Basel 2012
Abstract In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of gln (C) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology. Keywords
Hessenberg varieties · Affine paving · Bruhat decomposition
Mathematics Subject Classification (1991) Secondary 14F25
Primary 14L35 · 14M15;
1 Introduction and results This paper investigates the topological structure of Hessenberg varieties, a family of subvarieties of the flag variety introduced in [5]. We prove that under certain conditions Hessenberg varieties over a complex, linear, reductive algebraic group G have a paving by affines. This paving is given explicitly by intersecting these varieties with the
M. Precup (B) Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA e-mail: [email protected]
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M. Precup
Schubert cells corresponding to a particular Bruhat decomposition, which form a paving of the flag variety. This result generalizes results of J. Tymoczko in [11–13]. Let G be a linear, reductive algebraic group over C, B a Borel subgroup, and let g, b denote their respective Lie algebras. A Hessenberg space H is a linear subspace of g that contains b and is closed under the Lie bracket with b. Fix an element X ∈ g and a Hessenberg space H . The Hessenberg variety, B(X, H ), is the subvariety of the flag variety G/B = B consisting of all g · b such that g −1 · X ∈ H where g · X denotes the adjoint action Ad(g)(X ). We say that a nilpotent element N of a reductive Lie algebra m is a regular nilpotent element in m if N is in the dense adjoint orbit within the nilpotent elements of m. Suppose N is a regular nilpotent element in a Levi subalgebra m of g. In this case, we prove that there is a torus action on B(N , H ) with a fixed point set consisting of a finite collection of points. This action yields a vector bundle over each fixed point, giving an affine paving of B(N , H ) by its intersection with the Schubert cells paving B. Our argument is inspired by the proof by C. De Concini, G. Lusztig and C. Procesi that Springer fibers are paved by affines [4]. The main result is as follows. Theorem Fix a Hessenberg space H with respect to b. Let N ∈ g be a nilpotent element such that N is regular in some Levi subalgebra m of g. Then,
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