Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 2

Hessenberg varieties for the minimal nilpotent orbit

Pages: 183 – 223

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n2.a1


Hiraku Abe (Advanced Mathematical Institute, Osaka City University, Osaka, Japan; and Department of Mathematics, University of Toronto, Ontario, Canada)

Peter Crooks (Institute of Differential Geometry, Leibniz Universität, Hannover, Germany)


For a connected, simply-connected complex simple algebraic group $G$, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincaré polynomials and irreducible components of these varieties in Lie type $A$. Furthermore, we show these Hessenberg varieties to be GKM with respect to the action of a maximal torus $T \subseteq G$. The corresponding GKM graphs are then explicitly determined. Finally, we present the ordinary and $T$-equivariant cohomology rings of our varieties as quotients of those of the flag variety.


minimal nilpotent orbit, Hessenberg variety, equivariant cohomology

2010 Mathematics Subject Classification

Primary 17B08. Secondary 55N91.

Received 17 November 2016

Published 9 February 2018