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The Springer variety is the set of flags stabilized by a nilpotent operator.In 1976, T.A. Springer observed that this variety's cohomology ring carries asymmetric group action, and he offered a deep geometric construction of thisaction. Sixteen years later, Garsia and Procesi made Springer's work moretransparent and accessible by presenting the cohomology ring as a gradedquotient of a polynomial ring. They combinatorially describe an explicit basisfor this quotient. The goal of this paper is to generalize their work. Our mainresult deepens their analysis of Springer varieties and extends it to a familyof varieties called Hessenberg varieties, a two-parameter generalization ofSpringer varieties. Little is known about their cohomology. For the class ofregular nilpotent Hessenberg varieties, we conjecture a quotient presentationfor the cohomology ring and exhibit an explicit basis. Tantalizing new evidencesupports our conjecture for a subclass of regular nilpotent varieties calledPeterson varieties.

A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties