Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotan-gent bundle with certain other vector bundles over the flag vari-ety. We show that the analogue of the Springer sheaf has as di-rect summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques estab-lished by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual pic-ture, where we consider maps that generalize8Nff3/afNzqjAfFj1j+rOO5H5jlBthe Grothendieck– Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan. As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work of De Concini, Lusztig, and Procesi. We connect this relation in type A with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some conse-quences of that connection.