Selecta Mathematica

, Volume 19, Issue 4, pp 903–922

Affine pavings of Hessenberg varieties for semisimple groups


DOI: 10.1007/s00029-012-0109-z

Cite this article as:
Precup, M. Sel. Math. New Ser. (2013) 19: 903. doi:10.1007/s00029-012-0109-z


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 \(\mathfrak{gl }_n(\mathbb C )\) 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.


Hessenberg varietiesAffine pavingBruhat decomposition

Mathematics Subject Classification (1991)

Primary 14L3514M15Secondary 14F25

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA